Searching for Localized Black-Hole solutions in Brane-World models
Theodoros Nakas
gr-qc





UNIVERSITY OF IOANNINA




Searching for Localized Black-Hole
 solutions in Brane-World models




by

Theodoros Nakas




A thesis submitted in partial fulfilment for the 

Master's degree




in the





    < g r a p h i c s >





PHYSICS DEPARTMENT
 SCHOOL OF NATURAL SCIENCES
 UNIVERSITY OF IOANNINA
 GREECE
 JULY 2017







roman









“Every atom in your body came from a star that exploded. And, the atoms in your left hand probably came from a different star than your right hand. It really is the most poetic thing I know about physics: You are all stardust. You couldn't be here if stars hadn't exploded, because the elements - the carbon, nitrogen, oxygen, iron, all the things that matter for evolution and for life - weren't created at the beginning of time. They were created in the nuclear furnaces of stars, and the only way for them to get into your body is if those stars were kind enough to explode. So, forget Jesus. The stars died so that you could be here today."
 


Lawrence M. Krauss












Abstract


tocchapterAbstract



In the context of this thesis, the question that is going to occupy us, is the existence of a 5-dimensional braneworld black hole solution that is localized close to the 3-brane and has the properties of a regular 4-dimensional one. For this purpose, the 4-dimensional part of the complete 5-dimensional spacetime is considered to be a generalized Vaidya metric, in the context of which, the mass parameter m is allowed to vary with respect to time, while it is also allowed to have both y and r dependence. The dependence on the r-coordinate essentially means that our black hole solution can deviate from the conventional Schwarzschild solution. Additionally, the dependence on the y-coordinate leads to a non-trivial profile of the black hole along the extra dimension. In order to justify physically the existence of such general mass parameter, we consider the case of two scalar fields ϕ(v,r,y), χ(v,r,y) which interact with each other and they are also non-minimally coupled to gravity via a general coupling function f(ϕ,χ). In all the cases that were investigated in the context of this particular scenario, the result for the existence of a viable 5-dimensional localized black hole solution was negative, a result that causes concern about the compatibility of brane-world models with basic predictions of General Theory of Relativity.

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Acknowledgements


tocchapterAcknowledgements


First and foremost, I would like to thank my parents; without their love and support this thesis would never have been started.
I would also like to thank my supervisor, Panagiota Kanti, not only for the guidance, help and support, but also for providing me the opportunity to work with her and learn more about the enchanting field of General Relativity.
I am very thankful to Charalambos Kolasis as well. His passion for mathematics and his clear proofs in every theorem or every problem that he mentioned, helped me understand the way of reasoning as an undergraduate student.
Last but not least, I would like to say a big thank you to Ilias for letting me evaluate my mathematica codes on his “monster" computer and Georgia, Konstantinos and my friends for helping me maintain my sanity. This journey would not have been so enjoyable without you (cliché but also true).

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tocchapterContents



arabic



CHAPTER: BASIC NOTATION

tocchapterBasic Notation



  * The signature of the metric tensor (g_μν) that is going to be used in the context of this thesis is chosen to be: (-,+,+,…,+). Therefore, a flat and 4-D (four-dimensional) space-time has the following line-element.

    ds^2=-(cdt)^2+(dx)^2+(dy)^2+(dz)^2


  * Upper-case Latin indices M,N,… will denote bulk coordinates, thus for a 6-D space-time they will take the values 0,1,2,…,5. Greek indices μ,ν,… will be used for brane coordinates, hence they will take the values 0,1,2,3. Finally, lower-case Latin indices a,b,… will denote the three spatial coordinates taking the values 1,2,3.

  * Most of the times, we are going to use natural units or Planck units in order mathematical expressions to be less complicated, i.e. c=ħ=1 or c=ħ=G=1 respectively.

  * The partial derivative of a function f=f(x^0,x^1,x^2,x^3) with respect to x^μ can be expressed as

    f/ x^μ=_μ f=f_,μ


  * The covariant derivative of a tensor quantity with respect to x^ is denoted as

    T^μν_;    or     ∇_ T^μν

where we arbitrarily chose the contravariant tensor T^μν to present the notation. Of course, the same notation holds for any kind of tensor.






CHAPTER: INTRODUCTION

The fundamental similarity of people throughout the history of humankind is curiosity. Since ancient times, this special characteristic has made humans wonder about nature, the origin of the universe and consequently the origin of life. The questions that immediately come to mind are: How did the universe begin? Which are the fundamental building blocks of the universe? Are there extra dimensions?  These questions are still open and they may remain open for many more years.
Our understanding about the universe is limited. It is worth mentioning that everything that is possible to be observed in the night sky with a naked eye or by using telescopes adds up to about 4.9% of the entire universe, the other 95.1% of the universe is made of dark matter <cit.> (26.8%) and dark energy <cit.> (68.3%)[The name dark is related to the fact that they do not interact electromagnetically. Therefore, they are invisible to the entire electromagnetic spectrum.]. Almost nothing is known about dark matter and dark energy, but it is certainly known that both of them interact gravitationally. Dark matter behaves like ordinary matter (in terms of gravitation) but it is not luminous, its existence and properties are deduced by astrophysical and cosmological measurements i.e. galaxy rotation curves, gravitational lensing, cosmic microwave background (CMB), etc. On the other hand, dark energy is associated with the accelerated expansion of the universe, hence it is needed to act repulsively. The most popular method and historically the first one that was formulated in order to describe dark energy, is via the cosmological constant, which can be identified to a perfect fluid of constant energy density and negative pressure that fills the entire universe homogeneously[An alternative method to describe dark energy is via quintessence <cit.>, a scalar field that it is allowed to be both space and time varying.]. The evidence of the existence of dark energy is indicated also by astrophysical and cosmological observations, i.e. high redshifted observations from supernovae, CMB, Large-scale structures, observational Hubble constant data (OHD), etc.
So far we have discussed about what we do not know, thus, it is now reasonable to discuss about what we actually know about the universe. Everything that is known about the universe nowadays can be provided by two theories. The first one in chronological order is General Theory of Relativity (GTR or GR) <cit.> and the second one is the Standard Model (SM) <cit.> of particle physics.
GR constitutes the modern theory of gravity and it was formulated by Albert Einstein in 1915 <cit.>. The mathematical framework of the theory is Differential Geometry and, in this context, gravity emerges as a geometric property of spacetime[More precisely, the curvature of the spacetime and the gravitational field are the two sides of the same coin.], which is a four dimensional (4-D) manifold. GR provides a clear description and explanation for a number of astronomical and cosmological observations. However, only a year after its publication, Karl Schwarzschild showed that GR breaks down in high energies by proving that the spherically symmetric vacuum solution of the gravitational field equations[In 1923, George David Birkhoff proved that any spherically symmetric solution of the vacuum gravitational field equations must be static and asymptotically flat, which means that the spacetime outside of a spherical, non-rotating body must be described by the Schwarzschild metric.] leads to a singularity at r=0 <cit.>. For obvious reasons, infinities (or singularities) are undesirable features for a physical theory. Consequently, GR cannot be considered as the ultimate theory of gravity. When quantum effects become important GR fails to describe them.
R0.4

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Particle interactions as described by Standard Model.

SM on the other hand, constitutes the theory of the quantum world. It successfully describes the remaining three of the four known fundamental forces of nature (electromagnetism, weak interaction and strong interaction) and additionally accommodates all the known elementary particles (see Figure 1.1). SM is the most experimentally tested theory and its predictions are confirmed with extreme accuracy. Although its phenomenal success to explain the vast majority of processes in particle physics, SM is still not capable of being the fundamental theory that describes all four fundamental forces (including gravity). Moreover, SM does not provide any insight for the nature of dark energy and dark matter, which as we already mentioned are the biggest mysteries nowadays.
It is crystal clear from the aforementioned problems that a new and more fundamental theory is needed, or a deeper understanding of the already established ideas. Such a theory should encompass both GR and SM, namely a theory of everything, but it is not necessary to be based on either of them. However, this hypothetical theory should reproduce both GR and SM in the appropriate limit. Currently, the most promising theory of unification is string theory, which replaces elementary particles with one-dimensional strings, instead of being accounted for as point-like objects. The fatal flaw of string theory is its complexity; it requires at least 10 dimensions in order to be consistent and also it is beyond experimental verification because of the extremely high energies that are needed for such a purpose.
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The 5-D spacetime in Kaluza Klein theory.

Historically, the first theory that introduced extra dimensions was Kaluza-Klein theory. The primary idea came from Theodor Kaluza (1921) <cit.>, who extended general theory of relativity by allowing the existence of another spatial dimension in addition to the four dimensional spacetime fabric of GR. Furthermore, he constrained the five-dimensional metric tensor by demanding none of its components to be dependent on the extra dimension. This condition is also known as the “cylinder condition". His motivation for a higher dimensional spacetime was the unification of electromagnetism and gravity. The contribution of Oscar Klein (1926) <cit.> was his quantum interpretation of Kaluza's theory. He hypothesized that the extra dimension is curled up and tiny (see Figure 1.2) in order to explain the cylinder condition and also evaluated the scale of the extra dimension by taking into consideration the quantum nature of the electric charge.

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In the last twenty years, two additional extra dimensional theories was added to the literature, ADD model (1998) <cit.>  and RS models (1999) <cit.>. Both models <cit.> were motivated by the Hierarchy Problem[Hierarchy problem is the vast discrepancy between the electroweak scale m_EW∼ 1  TeV and Planck scale m_P∼ 10^19 GeV. (m_EW/m_P∼ 10^-16. m_P=√(ħ c/G), while M_P=m_P/√(8π)∼ 10^18 GeV).] and also both of them were based on the concept of braneworld, which simply indicates that Standard Model particles are confined on our 4-D spacetime (3-brane) while gravity can freely propagate in the bulk, i.e the entire (4+n)-dimensional spacetime, where n is the number of the extra dimensions.
Particularly, the ADD model allows the existence of n compact extra spatial dimensions of the same radius R. Thus, the total number of dimensions that gravity encounters are (4+n). Subsequently, the fundamental Planck scale corresponds to the higher dimensional one, namely M_P(4+n), while the Planck scale M_P which we experience is an effective one. Hence, it is possible the fundamental Planck scale M_P(4+n) to be equal to the electroweak scale m_EW and simultaneously the effective Planck scale to maintain its huge value. As it is indicated in <cit.>, the gravitational potential between two masses m_1 and m_2 which are separated by a distance r≪ R in (4+n)-dimensions is given by the following expression
V(r)∼m_1 m_2/M_P(4+n)^n+21/r^n+1
Equation (<ref>) can be derived by the gravitational Gauss's law in (4+n) dimensions <cit.>. Assuming now that r≫ R we obtain
V(r)∼m_1m_2/M^n+2_P(4+n)R^n1/r
Therefore, the effective Planck scale should fulfil the following equation 
M_P^2∼ M_P(4+n)^n+2R^n
Solving the above equation with respect to R and setting M_P(4+n)=m_EW, we obtain

    R∼ 10^30/n-19(1 TeV/m_EW)^1+2/n m m_EW∼ 1 TeV

R∼ 10^30/n-19 m
Equation (<ref>) associates the number of the extra dimensions n with the size of their radius R. It is obvious that for n=1 and R∼ 10^11 m Newton's gravitational law would differ from the law that we are all used to. Deviations from the conventional Newton's law have not been measured, thus, n≥ 2. The upper bound for the size of the extra dimensions is R∼ 10^-4 m <cit.> and it results for n=2. The figures below depict schematically the two different types of topologies of the extra dimensions that was taken into consideration in <cit.>.


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ADD model is elegant from a mathematical point of view, but only seemingly solves the Hierarchy Problem. In particular, ADD model simply replaces the problem of hierarchy with that of the size of the extra dimensions R, because now it is requisite to explain the reason why R can be so much larger than the length 10^-19 m, which is associated with the fundamental Planck scale M_P(4+n)=m_EW∼ 1 TeV. Although it is difficult to measure gravitational deviations of the Newton's law in sub-millimeter distances, the fact that the fundamental gravitational scale M_P(4+n) can be equal to 1 TeV gives us the opportunity to detect (indirectly) the existence of extra dimensions in collider experiments through the formation of tiny black holes from highly energetic particles <cit.>.

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On the other extreme, the RS models <cit.> -which were published only one year after the ADD model- examine the case of only but one curved extra dimension in the bulk. The special characteristic of these models is that now the 3-brane itself possesses tension, therefore it interacts gravitationally with the bulk. Particularly, the first RS model (RS1) assumes the existence of two 3-branes in the bulk and achieves to generate the electroweak scale from the Planck scale through an exponential hierarchy, which arises purely from the geometry of the 5-D spacetime. RS2 model assumes the existence of only one 3-brane embedded in an infinitely long extra dimension but nevertheless manages to reproduce 4-dimensional gravity on the brane. Both RS models will be discussed in Chapter 2, thus, any further detail will be postponed for later.



§ MOTIVATION AND THESIS OUTLINE

In the context of GR, all the information that is needed to describe a black hole comes only from three classical parameters: mass (M), electric charge (Q) and angular momentum (J). It is impossible to distinguish two black holes if they are characterized by the same aforementioned parameters and these parameters have the same value. This is an incredible characteristic of the 4-D black holes and it is rarely encountered in other objects in nature. Given these parameters, there are only four different black hole solutions[Schwarzschild metric (M≠ 0, Q=0, J=0) <cit.>, Reissner-Nordström metric (M≠ 0, Q≠ 0, J=0) <cit.>, Kerr metric (M≠ 0, Q=0, J≠ 0) <cit.>, Kerr-Newman metric (M≠ 0, Q≠ 0, J≠ 0) <cit.> (See also [bhs]Appendix A).] <cit.> that can be derived by the gravitational field equations of GR; this particular statement is also known as the no-hair theorem <cit.>. On the other hand, braneworld black holes are not so easily manageable and most importantly there is not a corresponding no-hair theorem for higher dimensional spacetimes. Therefore, the “families" of higher dimensional black holes solutions are not yet known. This thesis is entirely focused on braneworld black holes and more specifically on the existence of localized black hole solutions in an RS2-type braneworld model. 
The outline of the thesis is as follows. Chapter 2 -as it was mentioned previously- constitutes a detailed analysis of the RS models. The existence of black string solutions in the context of RS models and the difficulties in finding a localized 5-D black hole solution on our 3-brane are reviewed in Chapter 3. In the same Chapter, the theoretical framework of the thesis, namely the geometrical background that the thesis is based on, and the scalar field theory (two non-minimally-coupled and interacting scalar fields with a general coupling to the Ricci scalar) are also presented. This scalar field theory constitutes a more general theory compared to the scalar field theory that is presented in <cit.>. This extra degree of freedom -that the additional scalar field offers- may lead to a solution to the problem of localizing a 5-dimensional black hole on a 3-brane. Finally, in Chapter 3, the field equations of this particular ansatz are derived. Subsequently, the various cases[Different cases vary on the spacetime coordinates from which the scalar fields and the coupling function depend on.] which are studied in this particular scenario are presented in a series of paragraphs of increasing complexity in Chapter 4. Finally, our results and our conclusions are discussed in Chapter 5.
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CHAPTER: RANDALL-SUNDRUM BRANE-WORLD MODELS

In their first model <cit.> (RS1), Lisa Randall and Raman Sundrum introduced a compact extra dimension, which is finite and bounded by two 3-branes. This specific type of spacetime enriched with some additional properties, which are going to be discussed in the following section, manages to address the hierarchy problem. Amazingly, they proved that the 4-D gravity can also be recovered on the brane even if the extra dimension has infinite size, this constitutes the RS2 model <cit.>. Let us now proceed to the analysis of these models.



§ RS1 MODEL

The RS model postulates the existence of one extra dimension which is compactified on a circle S^1 (one-dimensional sphere) and also possesses a ℤ_2 symmetry, which simply means that the points (x^μ,y) and (x^μ,-y) are identified. Hence, the extra dimension is an S^1/ℤ_2 orbifold. As it is illustrated in the figure below, this type of compactification contains two fixed points, at y=0 and y=π r_c≡ L. The range of y is from -L to L, but the metric is completely specified by the values in the range 0≤ y≤ L. These fixed points of the extra dimension (y=0 and y=L) host the two 3-branes, which essentially are two separated 4-D worlds.



The action of the model is as follows:

    S=S_grav+S_1+S_2
    
    
    S_grav=∫ d^4x∫_-L^Ldy √(-g^(5))(R/2κ_(5)-Λ_5)
    
    
    S_1=∫ d^4x√(-g_1) (_1-_1)
    
    
    S_2=∫ d^4x√(-g_2) (_2-_2)

where

    =[g_MN(x^,y)]
    
    
    g_1=[g^1_μν(x^)]
    
    
    g_2=[g^2_μν(x^)]

The determinants g_1 and g_2 are derived by the metrics g^1_μν and g_μν^2 of the two 3-branes which are located at y=0 and y=L respectively. R is the 5-D Ricci scalar, Λ_5 is the higher dimensional cosmological constant and κ_(5)=8π G_(5)=M_P(5)^-3 where M_P(5) is the fundamental Planck scale of the 5-D spacetime. The quantities _1 and _2 represent the energy densities of the 3-branes, while _1 and _2 are the Lagrangians on each 3-brane. 
The variation of Eq.(<ref>) with respect to the components of the 5-D metric tensor g_MN provides us with the field equations (see [var]Appendix F for more details about the variation of a general action). The field equations, which are depicted below, were deduced under the assumption that _1=_2=0. This particular assumption was made in order to determine the geometrical background of the model. The field equations have the following form:

    √(-) G_MN=-κ_(5)[√(-) g_MN Λ_5.   +√(-g_1) _1 g^1_μν ^μ_M ^ν_N (y)
       .+√(-g_2) _2 g^2_μν ^μ_M ^ν_N (y-L)]

In order to continue the analysis of the model it is necessary to introduce an appropriate metric for such a setup. A property that we need to impose on the 5-D metric is to respect the Poincaré symmetry on the two 3-branes. This property comes naturally from the fact that the 4-D induced metrics on the 3-branes should describe the real world and therefore they should respect the same symmetries as the physical world. The general form of a five-dimensional line-element that includes an extra dimension with S^1/ℤ_2 compactification and also respects Poincaré symmetry has the following form

ds^2=e^2A(y)η_μνdx^μ dx^ν+dy^2
The function A(y) is called warp factor and it will be evaluated subsequently by the gravitational field equations of this theory. It is obvious that the metric tensor that results from this particular line-element is of the form

    (g_MN)=([ -e^2A(y)        0        0        0        0;        0  e^2A(y)        0        0        0;        0        0  e^2A(y)        0        0;        0        0        0  e^2A(y)        0;        0        0        0        0        1 ])
    
    
    g_MN=e^2A(y)η_μν_g_μν^μ_M^ν_N+^4_M^4_N=g_μν^μ_M^ν_N+^4_M^4_N
    
    
    (g^MN)=(
    [ -e^-2 A(y)          0          0          0          0;          0  e^-2 A(y)          0          0          0;          0          0  e^-2 A(y)          0          0;          0          0          0  e^-2 A(y)          0;          0          0          0          0          1;            ])
    
    
    g^MN=e^-2A(y)η^μν_g^μν_μ^M_ν^N+_4^M_4^N=g^μν_μ^M_ν^N+_4^M_4^N

Consequently, the induced metric tensors on the 3-branes at the points y=0 and y=L are given by

    g^1_μν=e^2A(0)η_μν
    
    
    g^2_μν=e^2A(L)η_μν

The non-zero and two-times covariant components of the Einstein tensor G_MN of the aforementioned ansatz are depicted below:

{[     G_00=G_tt=-3e^2A(y)[2A'^2(y)+A”(y)];                                        ; G_11=G_22=G_33=3e^2A(y)[2A'^2(y)+A”(y)];                                        ;                      G_44=G_yy=6A'^2(y) ]}{[ G_μν=3[2A'^2(y)+A”(y)]g_μν;                           ;              G_44=6A'^2(y) ]}

Combining equations (<ref>), (<ref>) and (<ref>) for M=N=4 we get:

6A'^2(y)=-κ_(5)Λ_5 A'^2(y)=-Λ_5/6

while for M=μ and N=ν we obtain:

    6A'^2(y)+3A”(y)=-Λ_5- _1 (y)- _2 (y-L)(<ref>)
    
    
    3A”(y)=-[_1 (y)+_2 (y-L)]
    
    
    A”(y)=-/3[_1 (y)+_2 (y-L)]

The five-dimensional cosmological constant Λ_5 must be negative in order to have a real solution for the function A(y). A negative 5-D cosmological constant affects decisively the geometry of the spacetime between the two 3-branes. Particularly, the bulk spacetime of this scenario leads to an anti-de Sitter spacetime between the two branes, which is also denoted as AdS_5. Subsequently, we define the following constant
k^2≡ -Λ_5/6
The specific function of A(y) can be easily deduced from Eq.(<ref>). Combining equations (<ref>) and (<ref>), we get
A'^2(y)=k^2 A'(y)=± k A(y)=± kyA(y)=-k|y|
The reason that we kept the minus sign in Eq.(<ref>) will be understood later. In addition, it is necessary to preserve the orbifold symmetry S_1/ℤ_2 for the extra dimension y. Hence, we are obliged to express the function A(y) in terms of |y|. Substituting Eq.(<ref>) into Eq.(<ref>) we obtain the geometrical background of the RS model:
ds^2=e^-2k|y|η_μνdx^μ dx^ν+dy^2
We can relate the constants _1 and _2 with k by evaluating the second derivative of A(y) as given by Eq.(<ref>) and then equate the result with Eq.(<ref>).
A'(y)=-k(|y|)'=-k sgn(y)=-k[H(y)-H(-y)]
where
H(y)={[       0, -L≤ y< 0;         ;       1, 0≤ y ≤ L ]}, H(-y)={[       1, -L≤ y< 0;         ;       0, 0≤ y ≤ L ]}

A”(y)=-k[H'(y)-H'(-y)]=-2k[(y)-(y-L)]
where

{
H'(y)=(y)-(y-L)


H'(-y)=[-(y-L)]-(-y)=-(y)+(y-L)
}


    (<ref>)(<ref>)-/3[_1 (y)+_2 (y-L)]=-2k[(y)-(y-L)]
    _1=-_2=6k/


Let us now focus on the 3-brane at y=L and include the Higgs field in the 4-dimensional action. We are going to evaluate the Vacuum Expectation Value (VEV) of the Higgs field on the brane which determines the physical masses in the Standard Model. The action will have the form:

    S_H=∫ d^4x∫_-L^L dy√(-)[g^MND_M H^† D_N H-(H^† H-v_0^2)^2](y-L)


    S_H=∫ d^4x√(-g_2)[g_2^μν
    D_μ H^† D_ν H-(|H|^2-v_0^2)^2][√(-g_2)=e^4A(L)=e^-4kL]g_2^μν=e^-2A(L)η^μν=e^2kLη^μν


    S_H=∫ d^4x e^-4kL[e^2kLη^μνD_μ H^† D_ν H-(|H|^2-v_0^2)^2]

S_H=∫ d^4x[η^μνD_μH̃^† D_νH̃-(|H̃|^2-v^2)^2]
In order to obtain a canonically normalized action we defined

    H̃≡ e^-kLH
    
    
    v≡ e^-kLv_0

The action of Eq.(<ref>) depicts the ordinary action of the Higgs field. The corresponding VEV of the renormalized Higgs scalar H̃ is v and is given by Eq.(<ref>), while the Higgs scalar H is the bare Higgs with VEV v_0. As we mentioned earlier, the VEV of the Higgs field determines all the mass parameters in the context of SM, thus we can safely conclude that
m=e^-kLm_0
where m constitutes the physical mass as it is measured on the 3-brane at y=L. Eq.(<ref>) is a simple and “powerful" result. It can be considered as a necessary condition for the solution of the hierarchy problem, because it does not demand a huge discrepancy between parameters k and L in order to achieve that. It is easy to verify the previous statement by setting m_0 equal to the Planck mass M_P∼ 10^18 GeV and m equal to the electroweak scale m_EW∼ 1 TeV. Then:

    10^3 GeV=e^-kL 10^18 GeV e^-kL=10^-15 kL=15ln(10)
    kL≈ 35

In order to confidently state that the type of exponential suppression of Eq.(<ref>) successfully addresses the hierarchy problem, it is also necessary to examine the dependence of the effective scale (4-D scale) of gravity on the size of the extra dimension y. For this purpose, we need to perturb the four-dimensional part of the five-dimensional line-element of Eq.(<ref>) and then extract the 4-D gravitational action from the original 5-D action which is shown at Eq.(<ref>). The form of the perturbative line-element is given by the following relation:

    ds^2   =e^-2k|y|[η_μν+h_μν(x)]dx^μ dx^ν+dy^2
       =e^-2k|y|g^per_μνdx^μ dx^ν+dy^2

where |h_μν|≪ 1. Using the metric that follows from Eq.(<ref>) into Eq.(<ref>) and focusing on the term which includes the Ricci scalar, we obtain
S_eff= ∫ d^4x ∫_-L^Ldy e^-2k|y|/2√(-g_per) R^4D(g^per_μν)
The S_eff should be also equal to
S_eff=∫ d^4x √(-g_per) R^4D/2κ
where κ=8π G=M_P^-2. Consequently, by equating the last two relations we obtain the following expression for the effective scale of gravity.

    1/κ=1/∫_-L^L dy e^-2k|y| M_P^2=M_P(5)^3(∫_-L^0 dy e^2ky+∫_0^Ldy e^-2ky)


    M_P^2=M_P(5)^3(1/2k-e^-2kL/2k-e^-2kL/2k+1/2k) M_P^2=M_P(5)^3/k(1-e^-2kL)

M_P^2=M_P(5)^3/k(1-e^-2kL)
Substituting now the value of the product kL given by Eq.(<ref>) into Eq.(<ref>) we get
M_P^2=M_P(5)^3/k(1-e^-70)≃M_P(5)^3/k
It is clear from Eq.(<ref>) and Eq.(<ref>) that gravity is essentially independent of the size of the extra dimension. Surprisingly, even if we infinitely extend the length L of the extra dimension in Eq.(<ref>), the four-dimensional Planck scale M_P remains finite. This particular observation was the central point of the RS2 model which is going to be analyzed in the section that follows.
Conclusively, it was shown that in the context of the RS model the hierarchy problem has an extremely simple and clear solution. Simultaneously, the RS model does not introduce new huge hierarchies (in contrast with the ADD model) between its fundamental parameters (k,  or M_P(5), v_0, L or r_c). The only constraint that is required by the model is that kL≈ 35. Of course, a stabilizing mechanism (Goldberger-Wise mechanism <cit.>) must be included in the model for this purpose but this is beyond the context of the present analysis.



§ RS2 MODEL

In their second paper, Lisa Randall and Raman Sundrum proved that is in fact possible to extend the length L of the extra dimension to an infinite value and nevertheless get an effectively four-dimensional gravity. The verification of the previous statement is derived from the fact that the 5-D graviton is localized near the 3-brane at y=0. The setup of the RS2 model is similar to the RS1 model with the difference that the 3-brane at y=L is practically removed from the original picture, by taking L to infinity. Thus, the action of the RS2 model is

    S=S_grav+S_1
    
    
    S_grav=∫ d^4x∫ dy √(-)(R/2-Λ_5)
    
    
    S_1=∫ d^4x√(-g_1)(_1-_1)

The line-element of the RS2 model is given by Eq.(<ref>) as well. Let us now proceed to the derivation of the graviton modes.
Gravitons correspond to small fluctuations in the spacetime “fabric". Therefore, in the context of RS2 model we have
ds^2=e^-2k|y|[η_μν+h_μν(x,y)]dx^μ dx^ν+dy^2
In the last equation it was chosen h_M4=0. It is always possible to find a set of coordinates with this particular property in the region of the 3-brane. We now seek a new variable z for the description of the extra dimension in order to construct a metric tensor which will be more convenient for future calculations. For this purpose, we demand from the new variable z to satisfy the following relation:
dy^2≡ e^-2k|y|dz^2
Performing the integration of the previous equation we find
k|z|=e^k|y|-1
where we chose the integration constant appropriately in order to get z=0 when y=0. Eq.(<ref>) leads to
e^-2k|y|=1/(k|z|+1)^2
Combining equations (<ref>) and (<ref>) we obtain

    ds^2=1/(k|z|+1)^2{[η_μν+h_μν(x,z)]dx^μ dx^ν+dz^2}h_4M=0
    
    
    ds^2=e^2A(z)[η_MN+h_MN(x,z)]dx^Mdx^N
    
    ds^2=e^2A(z)g̅_MN(x,z)dx^Mdx^N

where

    e^2A(z)≡1/(k|z|+1)^2
    
    g̅_MN(x,z)≡η_MN+h_MN(x,z)

From Eq.(<ref>) we get

    A(z)=-ln(k|z|+1)
    
    
    A'(z)=-k sgn(z)/k|z|+1=-k[H(z)-H(-z)]/k|z|+1
    
    
    A”(z)=-2k(z)/k|z|+1+k^2/(k|z|+1)^2

Under the above transformation of the extra dimension (from y to z), Eq.(<ref>) is also going to be transformed into
S_grav=∫ d^4x∫_-L_z^L_z dz√(-)(R/2-Λ_5)
where L_z=(e^kL-1)/k. The field equations which are yielded by the variation of Eq.(<ref>) with respect to the components of the 5-D metric are depicted below (_1 in Eq.(<ref>) is set to be 0). 
√(-) G_MN=-[√(-) g_MN Λ_5+√(-g_1) g^1_μν ^μ_M ^ν_N (z) _1]
where

    g^1_μν (z)=e^2A(z)(η_μν+h_μν) (z)
    
    
    g_1 (z)= e^-2A(z) (z)

Hence, the combination of equations (<ref>)-(<ref>) results to
G_MN=-[g_MN Λ_5+e^A(z)(η_μν+h_μν)^μ_M ^ν_N (z) _1]
The calculation of the Einstein tensor components G_MN using “brute force" is a difficult task in this model. Hence, we are going to use a conformal transformation[For more details about conformal transformations see <cit.>.] in order to obtain the components of the Einstein tensor more easily. Particularly, we mention (without proof) that if g̃_MN is the conformally transformed metric of the metric g_MN and the two metrics are connected through the relation
g̃_MN=Ω^2(x,z) g_MN
then the corresponding components of the Einstein tensor are connected as follows:
G̃_MN=G_MN+D-2/2Ω^2[4Ω_,MΩ_,N+(D-5)Ω_,KΩ^,K g_MN]-D-2/Ω(Ω_;MN-g_MN□Ω)
where D indicates the total number of spacetime's dimensions. The adjustment of equations (<ref>) and (<ref>) in our case is straightforward. We simply execute the following substitutions: {g̃_MN g_MN, Ω(x,z) e^A(z), g_MNg̅_MN and D=5}. Hence, we obtain the expression:

    G_MN=G̅_MN+3[_M A _N A-∇_M∇_NA+g̅_MN(□ A+_LA ^LA)]


    G_MN=G̅_MN+3[_M A _N A   -_M_NA+^L_MN_LA
       +g̅_MN(_L^LA+^L_LK^KA+_LA ^LA)]

For all the subsequent calculations in this section, the terms which contain fluctuations (h_MN) of order higher than the first will be neglected. Additionally, it is possible and extremely convenient to perform appropriate coordinate transformations[A complete analysis of the legitimacy and derivation of this particular gauge is presented in <cit.>. See [gnc]Appendix B as well.] in which the fluctuations satisfy the following properties:
{[     h_4M=0;           ;  h=h^μ_μ=0;           ; _μ h^μ_ν=0 ]}
In this gauge and with the use of equations (<ref>) and (<ref>) it is quite trivial to show that the Christoffel symbols ^L_MN and the components of Einstein tensor G̅_MN are expressed through the following equations: 
^L_MN=1/2(_Nh^L_M+_Mh^L_N-^Lh_MN)
G̅_MN=-1/2_L^Lh_MN
Using equations (<ref>)-(<ref>) we are led to
{[                                            G_44=6A'^2;                                                      ;                                                G_4μ=0;                                                      ; G_μν=-1/2_L^Lh_μν-3/2 _4h_μν A'+3(η_μν+h_μν)(A”+A'^2) ]}

The combination of equations (<ref>) and (<ref>) for M=N=4 results to Eq.(<ref>), namely

    6A'^2=-Λ_5 g_44=-Λ_5 e^2A(z)(η_44_1+h_44_0)(<ref>) A'^2(z)=k^2e^2A(z)


    A(z)=-ln(k|z|+1)

where we fixed again the integration constant appropriately (the reason has already been discussed in the sentence before equation (<ref>)). Correspondingly, for M=μ and N=ν we have:
-1/2_L^Lh_μν-3/2 _4h_μν A'+3(η_μν+h_μν)(A”+A'^2)=-(η_μν+h_μν)[e^2AΛ_5+e^A(z) _1]
It is not hard to combine equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) in order to prove the following expressions (these are going to simplify Eq.(<ref>) afterwards):
-Λ_5 e^2A(z)=-Λ_5/6 6 e^2A(z)=6 k^2 e^2A(z)=6A'^2(z)
- _1 e^A(z) (z)=-6 k e^A(z) (z)=3[A”(z)-A'^2(z)]
Consequently, from equations (<ref>)-(<ref>) we get
_L^Lh_μν+3 _4h_μν A'=0
We now rescale the fluctuations h_μν as follows.
h_μν e^ A(z)h_μν
Then, Eq.(<ref>) takes the form:
h_μν[(3+^2)A'^2+ A”]+(2+3)A' _4h_μν+_L^Lh_μν=0
It is now obvious that we can nullify the term which contains the quantity _4h_μν by choosing =-3/2. Furthermore, we perform a Kaluza-Klein decomposition on the fluctuations h_μν(x,z):
h_μν(x,z)=∑_n=0^∞ h_μν^n(x)ψ_n(z)
where
{[ h^n_μν(x)=e^i p_n  x=e^i p^n_ x^;                                 ;     _^ h_μν^n(x)=m_n^2 h_μν^n(x) ]}

Substituting equations (<ref>) and (<ref>) into Eq.(<ref>) and setting =-3/2, we obtain

    _L^L[∑_n=0^∞ h^n_μν(x)ψ_n(z)]-(3/2A”+9/4A'^2)∑_n=0^∞ h^n_μν(x)ψ_n(z)=0


    _4^4[∑_n=0^∞ h^n_μν(x)ψ_n(z)]+_^[∑_n=0^∞ h^n_μν(x)ψ_n(z)]-(3/2A”+9/4A'^2)∑_n=0^∞ h^n_μν(x)ψ_n(z)=0


    ∑_n=0^∞{h^n_μν(x)[ψ_n”(z)+m_n^2 ψ_n(z)-(3/2A”+9/4A'^2)ψ_n(z)]}=0

-ψ_n”(z)+V(z) ψ_n(z)=m_n^2 ψ_n(z)
where

V(z)=3/2A”+9/4A'^2=-3k(z)/k|z|+1+15k^2/4(k|z|+1)^2=15k^2/4(k|z|+1)^2-3k(z)
The boundary condition at z=0 can be found as follows:

    (<ref>)∫_0^-^0^+dz[-ψ_n”(z)+V(z)ψ(z)]=∫_0^-^0^+dz m_n^2 ψ_n(z)


    -ψ_n'(0^+)+ψ_n'(0^-)-3kψ_n(0)=0[ψ'_n(z)=-ψ'_n(-z)]ψ_n(z)=ψ_n(-z)

ψ'_n(0^+)=-3/2 k ψ_n(0)
Although we have focused on the RS2 model, we may develop a unified analysis for the study of gravitons in both RS models. Thus, if we reintroduce the second brane at y=L=π r_c or at z=(e^kπ r_c-1)/k≡ L_z (as it is indicated by Eq.(<ref>)), then the potential of Eq.(<ref>) will be modified as follows and a new boundary condition at z=L_z will be added as well:
V_new(z)=15k^2/4(k|z|+1)^2-3k[(z)-(z-L_z)]/k|z|+1
The boundary condition at z=L_z can be deduced with the use of equations (<ref>) and (<ref>).

    ∫_L_z^-^L_z^+dz[-ψ_n”(z)+V_new(z)ψ_n(z)]=∫_L_z^-^L_z^+dz m_n^2 ψ_n(z)


    -ψ_n'(L_z^+)+ψ_n(L_z^-)+3kψ_n(L_z)/k|L_z|+1=0ψ_n(L_z^+)=-ψ_n(L_z^-)

ψ_n'(L_z)=-3kψ_n(L_z)/2(kL_z+1)




 §.§ Kaluza-Klein Modes


The zero-mode ψ_0(z) for m_0=0 can be easily evaluated from Eq.(<ref>). Thus, it is

    -ψ”_0(z)+[3/2A”(z)+9/4A'^2(z)]ψ_0(z)=0ψ_0(z)=N_0 e^3/2A(z)(<ref>)

ψ_0(z)=N_0(k|z|+1)^-3/2
where N_0 is a normalization constant which is going to be evaluated followingly.

    ∫_-L_z^L_zdz |ψ_0(z)|^2=1 (N_0)^2∫^L_z_-L_zdz(k|z|+1)^-3=2(N_0)^2∫_0^L_zdz(kz+1)^-3=1

(N_0)^2[-1/k(kL_z+1)^2+1/k]=1kL_z=e^kL-1≫ 1(N_0)^21/k=1 N_0=√(k)
From equations (<ref>) and (<ref>) we are led to
ψ_0(z)=1/k(|z|+1/k)^-3/2
The figure above depicts the plot of the potential V(z) as it is given by equation (<ref>), where it was used the approximation (z)∼(1/a√(π))e^-(z/a)^2 with a=0.05. This particular approximation helps us to visualize better the behaviour of the potential.
The boundary conditions are both satisfied. The figure in the next page depicts the plot of the graviton zero-mode at the region of the 3-brane which is located at z=0.

The Kaluza-Klein (KK) modes for n>0 can be provided from the general equation (<ref>).
ψ_n”(z)+[m_n^2-15k^2/4(k|z|+1)^2]ψ_n(z)=0
while for z=0 and z=L_z equations (<ref>) and (<ref>) should be satisfied respectively. The above differential equation has the following general solution:
ψ_n(z)=(|z|+1/k)^1/2{a_n J_2[m_n(|z|+1/k)]+b_nY_2[m_n(|z|+1/k)]}
where a_n, b_n are constant coefficients and J_2(z), Y_2(z) are the Bessel functions of the first and second kind respectively. The derivative of the function ψ_n(z) is given by

    ψ_n'(z)=   m_n(|z|+1/k)^1/2{a_n J_1[m_n(|z|+1/k)]+b_nY_1[m_n(|z|+1/k)]}
       -3/2(|z|+1/k)^-1/2{a_n J_2[m_n(|z|+1/k)]+b_nY_2[m_n(|z|+1/k)]}

We can relate the constants a_n and b_n by simply applying the boundary condition at z=0. Thus, from equations (<ref>), (<ref>) and (<ref>) we get:
a_n=-b_nY_1(m_n/k)/J_1(m_n/k)
Hence, it is:
ψ_n(z)=N_n(|z|+1/k)^1/2{-Y_1(m_n/k)J_2[m_n(|z|+1/k)]+J_1(m_n/k)Y_2[m_n(|z|+1/k)]}
where we defined	
N_n=b_n/J_1(m_n/k)
Since the massless graviton mode
is accompanied by a tower of massive Kaluza-Klein states, one may wonder about the gravitational potential that an observer on the visible brane will feel. In order to answer this, we will now consider each RS model separately.



The limit m_n/k≫ 1



Combining equations (<ref>), (<ref>), (<ref>) and (<ref>) we get

    a_n J_1(m_n/k)+b_nY_1(m_n/k)=0
    
    
    a_n J_1[m_n(L_z+1/k)]+b_nY_1[m_n(L_z+1/k)]=0

It is well-known that a homogeneous system of equations in order to have a non-trivial solution it is necessary to have a vanishing determinant. Thus, for the previous system of equations we demand that


    J_1(m_n/k)Y_1[m_n(L_z+1/k)]-J_1[m_n(L_z+1/k)]Y_1(m_n/k)=0kL_z=e^kL-1

J_1(m_n/k)Y_1(m_n/ke^kL)-J_1(m_n/ke^kL)Y_1(m_n/k)=0

In this limit, the large value of the quantity m_n/k results to the following expressions for the Bessel functions:
{[             J_1(m_n/k)∼√(2k/π m_n)cos(m_n/k-3π/4), J_1(m_n/ke^kL)∼√(2k/π m_ne^-kL)cos(m_n/ke^kL-3π/4);                                                   ;             Y_1(m_n/k)∼√(2k/π m_n)sin(m_n/k-3π/4), Y_1(m_n/ke^kL)∼√(2k/π m_ne^-kL)sin(m_n/ke^kL-3π/4) ]}
Therefore, Eq.(<ref>) takes the form

    cos(m_n/k-3π/4)sin(m_n/ke^kL-3π/4)-cos(m_n/ke^kL-3π/4)sin(m_n/k-3π/4)=0


    sin[m_n/k(e^kL-1)]=0kL_z=e^kL-1sin(m_n L_z)=0

m_n=sπ/L_z,s=1,2,…
Correspondingly, Eq.(<ref>) is given by

    ψ_n(z)∼ N_n√(2k/π m_n)√(|z|+1/k)   {-sin(m_n/k-3π/4)J_2[m_n(|z|+1/k)].
       .+cos(m_n/k-3π/4)Y_2[m_n(|z|+1/k)]}m_n(|z|+1/k)≫ 1


    ψ_n(z)∼ N_n2√(k)/π m_n   {-sin(m_n/k-3π/4)cos[m_n(|z|+1/k)-5π/4].
       .+cos(m_n/k-3π/4)sin[m_n(|z|+1/k)-5π/4]}

ψ_n(z)∼ N_n2√(k)/π m_nsin(m_n|z|-π/2)
where we have used the following expressions for the Bessel function at the limit m_n(|z|+1/k)≫ 1.
(|z|+1/k)^1/2J_2[m_n(|z|+1/k)]∼√(2/π m_n)cos[m_n(|z|+1/k)-5π/4]
(|z|+1/k)^1/2Y_2[m_n(|z|+1/k)]∼√(2/π m_n)sin[m_n(|z|+1/k)-5π/4]
It is now trivial to evaluate the normalization constant N_n.

    ∫_-L_z^L_zdz ψ_n^2(z)=1 N_n^24k/π^2m_n^2∫_-L_z^L_zdz sin^2(m_n|z|-π/2)=1

N_n^24k/π^2m_n^2 2 ∫_0^L_zdz sin^2(m_nz-π/2)_L_z/2 (for L_z≫ 1)=1N_n=π m_n/2√(kL_z)

(<ref>)(<ref>)ψ_n(z)∼1/√(L_z)sin(m_n|z|-π/2)
For the higher graviton modes to be very heavy, as assumed above, we should have a very small inter-brane distance according to Eq. (<ref>). Then, this case applies to the RS1 model. Each graviton mode will contribute to the gravitational potential on the brane through a Yukawa-like potential, thus

    U(r) ∝∑_n G_(4) m_1 m_2 e^-m_n r/r

But since all the KK masses are very heavy, it is only the zero graviton that will create a 1/r potential while the contributions of the higher modes will be very much suppressed.



The limit m_n/k≪ 1



At the approximation of m_n/k≪ 1, it is
J_1(m_n/k)∼m_n/2k,
Y_1(m_n/k)∼ -2k/π m_n
Hence, we can safely state that -Y_1(m_n/k)≫ J_1(m_n/k). Subsequently, Eq.(<ref>) results to the following simple requirement in order to have a vanishing determinant:
J_1(m_n/k e^kL)=0m_n=k e^-kL R^J_1_n≃R^J_1_n/L_z
where R^J_1_n are the roots of the function J_1(x), namely J_1(R^J_1_n)=0. The masses m_n of the massive graviton modes constitute the graviton spectrum (or Kaluza Klein spectrum). The substitution of the approximate relations from equation (<ref>) into (<ref>) leads to

    ψ_n(z)∼ N_n(|z|+1/k)^1/2{2k/π m_nJ_2[m_n(|z|+1/k)]+m_n/2kY_2[m_n(|z|+1/k)]}N_n N_nm_n/2k

ψ_n(z)∼ N_n(|z|+1/k)^1/2{4k^2/π m_n^2J_2[m_n(|z|+1/k)]+Y_2[m_n(|z|+1/k)]}

The normalization constant N_n can be determined by the integral

    ∫_-L_z^L_zdz ψ_n^2(z)=1

The fact that L_z∞ allows us to perform the calculation of N_n in the approximation of m_n|z|≫ 1. Using also the fact that m_n/k≪ 1k/m_n≫ 1, we can ignore the second term of Eq.(<ref>) as negligible. Thus, the substitution of Eq.(<ref>) into Eq.(<ref>) leads to
ψ_n(z)∼ N_n4k^2/π m_n^2√(2/π m_n)cos(m_n|z|-5π/4)
Thus, from Eq.(<ref>) we have:

    ∫_-L_z^L_zdz ψ_n^2(z)=1 N^2_n 32k^4/π^3m^5_n∫^L_z_-L_zcos^2(m_n|z|-5π/4)=1

N^2_n 32k^4/π^3m^5_n 2 ∫^L_z_0cos^2(m_n  z-5π/4)_L_z/2  (for L_z≫ 1)=1N_n=π^3/2m_n^5/2/4k^2√(2L_z)
Consequently, the graviton modes ψ_n(z) for n>0 and m_n|z|≫ 1 are given by 
ψ_n(z)=1/√(L_z) cos(m_n|z|-5π/4)

In this case, the KK masses are assumed to be small and this can be realised only for a large inter-brane distance. Therefore, this analysis applies to the RS1 model, where resolving the hierarchy problem is not an objective any more, or to the RS2 model. Especially, in the latter case where L∞, even the higher graviton modes become massless. It is thus even more important to verify if the Newtonian limit of gravity is indeed recovered on the visible brane. Here, the sum over the KK graviton states changes to an integral that must be carefully evaluated - we refer the interested reader to <cit.> for more details on this. At the end, we present the result from <cit.> which is related to the gravitational potential that is generated by a massive object with mass M on the 3-brane at z=0. The energy-momentum tensor for a point-mass at rest on the brane at r⃗=0 is the following:

    T_μν=M^(3)(r⃗) ^0_μ ^0_ν

For r=|r⃗|≫ 1/k we get:
h_μν=2G_(5)kM/r[(1+1/3k^2r^2)η_μν+(2+1/k^2r^2)^0_μ^0_ν]
Consequently, from equations (<ref>) and (<ref>) we obtain

    h_00=2G_(4)M/r(1+2/3k^2r^2)
    
    
    h_ij=2G_(4)M/r(1+1/3k^2r^2)η_ij

The gravitational potential V(r) is given by h_00 as follows:
V(r)=1/2h_00=G_(4)M/r(1+2/3k^2r^2)V(r)∼G_(4)M/r
Obviously, the desirable behaviour of an effective 4-dimensional gravity at the brane is produced by the model as kr≫ 1. On the other hand, if we consider the case of kr≪ 1, then the gravitational potential is proportional to 1/r^2, i.e. V(r)∝ 1/r^2. This simply means that at short distances -compared to the scale k which is also associated with the AdS curvature- gravity becomes 5-dimensional. The possibility to observe deviations from the Newton's law of gravitation depends entirely on the value of k.







CHAPTER: LOCALIZATION OF A 5-DIMENSIONAL BRANE-WORLD BLACK HOLE


The preceding analysis of RS models made clear that there is indeed a possibility to live at the boundaries of a higher dimensional cosmos (specifically a five-dimensional one) and the effective gravitational interaction between massive objects on our four-dimensional brane can be the same (in some limit) as the gravitational interaction which is provided by the established and well-tested gravitational theory of General Relativity. Hence, it is absolutely natural to wonder about the behaviour of 5-dimensional black holes in the context of RS models. 
The first attempt to find a black hole solution in the aforementioned scenario was in <cit.>. The line-element which was considered is of the following form:
ds^2=e^2A(y)[-(1-2M/r)dt^2+(1-2M/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2)]+dy^2
where M is a constant quantity and represents the mass of the black hole. As stated in the previous chapter, the coordinate y is used for the extra dimension and A(y) denotes the warp factor. For A(y)=-k|y| we get the warp factor of the RS model. It is quite obvious that for y=0 the induced 4-dimensional line-element is identified with the well-known Schwarzschild spacetime geometry. However, a 5-dimensional observer will not be able to associate the complete 5-dimensional line-element of Eq.(<ref>) with a regular black hole. The previous statement can be easily verified by the Kretschmann scalar K≡ R_MNKLR^MNKL that emanates from  the line-element (<ref>).
K≡ R_MNKLR^MNKL=8 [2 A”(y)^2+5 A'(y)^4+4 A'(y)^2 A”(y)+6 M^2 e^-4 A(y)/r^6]
The last term of Eq.(<ref>) reveals the existence of a singularity at r=0 which extends from -∞ to +∞ along the extra dimension y. Hence, the line-element of Eq.(<ref>) generates a black string rather than a black hole. Of course, for any descending function A(y) the last term of Eq.(<ref>) becomes even more problematic, because when y∞ the quantity e^-4A(y) goes to infinity as well. This particular behaviour is in complete contrast to the purpose of the RS model, which intends to keep gravity localized near the brane at y=0.
There is a plethora of attempts in the literature that try to derive a 5-dimensional black hole which is localized close to the brane (we mention some of them: <cit.>. Additionally, there are numerical solutions which describe both mini and large brane-world black holes in the context of RS models and beyond (i.e. 6-dimensional black holes) <cit.>. However, an analytical solution which can be written in closed form has not been found, thus, the investigation of such a solution is still incomplete. Finally, it is important to refer that in <cit.> black hole solutions were found which are localized on a 2-brane embedded in a (3+1)-dimensional bulk. Therefore, the motivation to search for a corresponding solution for the higher dimensional problem (3-brane in a (4+1)-dimensional bulk) is completely justified.
This Chapter and the rest of the thesis is entirely focused on the examination of the existence of a localized black hole solution in the context of a brane-world model which is similar to the RS2 model. The word “similar" was used because the warp factor A(y) is allowed to differ from the expression -k|y| (but it is necessary to be a descending function, as it is in the RS models) in order to increase the chances of finding a localized 5-dimensional black hole solution. In the following section we present the geometrical background that is going to be used in the framework of this thesis.



§ THE GEOMETRY OF THE 5-DIMENSIONAL SPACETIME

We assume that the 4-dimensional part of the general 5-dimensional geometrical background is a Vaidya metric[See [vaid]Appendix D for more information about Vaidya metric.]. We also consider that mass is a function of three variables m=m(v,r,y) (motivated by <cit.>). The non-trivial dependence of the mass from the extra dimension y is needed in order to counter the anomalous effect of the quantity e^-4A(y) in the last term of equation (<ref>). An appropriate mass function that decreases faster than e^-4A(y) increases and has also the necessary r-dependence in order to describe a black hole, would solve the localization problem. The drawback of this non-trivial assumption of the mass parameter is that it demands the existence of a bulk matter distribution in order to get consistent field equations. The bulk matter distribution that it is going to be considered in the context of this thesis will be presented later in this Chapter. Thus, the line element is of the form
ds^2=e^2A(y)[-(1-2m(v,r,y)/r)dv^2+2dvdr+r^2(dθ^2+sin^2θ  d^2)]+dy^2
It is easily deducible that the covariant components of the metric tensor in matrix form are:
(g_MN)=(
[ -e^2 A(y)(1-2 m(u,r,y)/r)                  e^2 A(y)                         0                         0                         0;                  e^2 A(y)                         0                         0                         0                         0;                         0                         0              e^2 A(y) r^2                         0                         0;                         0                         0                         0   e^2 A(y) r^2 sin ^2(θ )                         0;                         0                         0                         0                         0                         1;                           ])
Thus, the contravariant components of the metric tensor are of the following form:
(g^MN)=(
[                         0                 e^-2 A(y)                         0                         0                         0;                 e^-2 A(y) e^-2 A(y)(1-2 m(u,r,y)/r)                         0                         0                         0;                         0                         0             e^-2 A(y)/r^2                         0                         0;                         0                         0                         0       e^-2 A(y)/r^2sin^2θ                         0;                         0                         0                         0                         0                         1;                           ]) 
For the metric ansatz (<ref>), the quantities that are invariant and also contain the complete information of the 5-dimensional curvature, are R=g_MNR^MN, R_MNR^MN and R_ABCDR^ABCD. As it is mentioned earlier, an acceptable solution for the mass function m=m(v,r,y) should have an appropriate dependence on the y-coordinate in order to expunge any singularities in the bulk. Hence, the desirable solution should yield to a regular Anti-de Sitter space-time at a finite distance from the brane and a 5-dimensional black hole localized on the brane, otherwise the solution is rejected.
The Christoffel symbols can be evaluated with the use of equations (<ref>), (<ref>) and the following relation:
^K_MN=1/2g^KL(g_ML,N+g_NL,M-g_MN,L)
In the table that follows are depicted the non-zero Christoffel symbols.
[box=]equation
[                    ^0_00=m-r _r m/r^2                        ^0_04=^0_40=A'                              ^0_22=-r;                                      ;                       ^0_33=-r sin^2θ ^1_00=r^2 _v m-(r-2 m) (r _r m-m)/r^3               ^1_01=^1_10=r _rm-m/r^2;                                      ;                     ^1_04=^1_40=_ym/r                        ^1_14=^1_41=A'                           ^1_22=2 m-r;                                      ;                   ^1_33=(2 m-r)sin^2θ                       ^2_12=^2_21=1/r                        ^2_24=^2_42=A';                                      ;                       ^2_33=-sinθcosθ                       ^3_13=^3_31=1/r                         ^3_23=^3_32=θ;                                      ;                        ^3_34=^3_43=A'        ^4_00=e^2 A (A' (r-2 m)-_ym)/r                 ^4_01=^4_10=-e^2 A A';                                      ;                   ^4_22=-r^2 e^2 A A'             ^4_33=-r^2A' e^2 A sin^2θ                                       ]

Subsequently, we present the 5-dimensional curvature invariant quantities.
[box=]equation
R=R_MNg^MN=-20 A'^2-8 A”+2 e^-2 A/r (_r^2m+2_rm/r)

[box=]equation
[ R_MNR^MN=80A'^2+64A'^2 A”+20A”^2-4 e^-2 A/r(_r^2m+2_rm/r)(4A'^2+A”);                                +2e^-4A/r^2[(_r^2m)^2+4 (_rm)^2/r^2] ]

[box=]equation
[ R^ABCDR_ABCD=40A'^4+32A'^2A”+16A”^2+48e^-4Am^2/r^6-8e^-2AA'^2/r(_r^2m+2_rm/r);                                                                              ;           +4e^-4A/r^2[(_r^2m)^2+4m/r^2(_r^2m-4_rm/r)-4_rm_r^2m/r+8(_r m)^2/r] ]


The non-zero covariant components of the Einstein tensor G_MN are:

[box=]equation
[ G_00=-e^2 A [3 (r-2 m) (2 A'^2+A”)+4 A' _ym+_y^2m] r^2+2 r_vm +2(r-2 m)_r m/r^3;                                                                                ;                                         G_01=G_10=3 e^2 A (2 A'^2+A”)-2 _rm/r^2;                                                                                ;                                                       G_04=G_40=_ym+r_r_y m/r^2;                                                                                ;                                            G_22=r [3 e^2 A r (2 A'^2+A”)-_r^2m];                                                                                ;                                      G_33=r sin^2θ[3 e^2 A r (2 A'^2+A”)-_r^2m];                                                                                ;                                          G_44=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]

while the mixed components of the Einstein tensor G^M_N=g^MKG_KN are the following:
[box=]equation
[         G^0_0=6 A'^2+3 A”-2 e^-2 A _rm/r^2;                                           ; G^1_0=-4 r A' _ym+r _y^2m-2 e^-2 A _vm/r^2;                                           ;         G^1_1=6 A'^2+3 A”-2 e^-2 A_r m/r^2;                                           ;             G^1_4=e^-2 A (_ym+r _r_ym)/r^2;                                           ;           G^2_2=6 A'^2+3 A”-e^-2 A_r^2 m/r;                                           ;           G^3_3=6 A'^2+3 A”-e^-2 A _r^2m/r;                                           ;                      G^4_0=_ym+r _r_ym/r^2;                                           ;    G^4_4=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]

A detailed presentation of all the geometrical quantities i.e. Christoffel symbols, Riemann tensor, Ricci tensor and Einstein tensor, takes place in [geom]Appendix E. Moreover, in the same Chapter of Appendices, one can find the proper mathematica commands in order to verify the validity of all the aforementioned quantities.
In the following section, we introduce the field theory model in the context of which the existence of a viable solution to the localization problem of a 5-dimensional black hole will be examined.



§ TWO NON-MINIMALLY COUPLED AND INTERACTING SCALAR FIELDS


We consider the most general case of two interacting scalar fields ϕ,χ which are not minimally coupled to gravity. These scalar fields can freely propagate into the bulk. Subsequently, the action of this model is the following:
S=∫ d^4xdy√(-)[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B]
where =[(g_MN)]. The scalar fields ϕ, χ are both functions of the variables (v,r,y) and Λ_B denotes the cosmological constant of the bulk, it is the same constant as Λ_5 which was used in Chapter 2. The field equations of this model can be derived by varying the action with respect to the metric tensor and also with respect to the fields ϕ,χ. The steps that are going to be followed for the variation of the above action are the same as the steps which are followed in [var]Appendix F, where it is thoroughly presented the variation of a single scalar field which is non-minimally coupled to gravity. Thus, we have:

    S=0   =∫ d^4xdy (√(-))[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B]
           +∫ d^4xdy√(-){f(ϕ,χ)/2 R-1/2[(∇ϕ)^2]-1/2[(∇χ)^2]}

Moreover, it is
[(∇ϕ)^2]=(∇^Mϕ∇_Mϕ)=(g^MN∇_Mϕ∇_Nϕ)=∇_Mϕ∇_Nϕ  g^MN
Using equations (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>) we get:

    0   =∫ d^4xdy√(-)  g^MN{-1/2g_MN[f(ϕ,χ)/2R-1/2(∇ϕ)^2-1/2(∇χ)^2-V(ϕ,χ)-Λ_B].
           .+1/2f(ϕ,χ)R_MN-1/2∇_M∇_Nf(ϕ,χ)+1/2g_MN□ f(ϕ,χ)-1/2∇_Mϕ∇_Nϕ-1/2∇_Mχ∇_Nχ}
    
    
    0   =f(ϕ,χ)/(R_MN-1/2g_MNR)+g_MN[(∇ϕ)^2/2+(∇χ)^2/2+V(ϕ,χ)]+g_MNΛ_B-∇_Mϕ∇_Nϕ
           -∇_Mχ∇_Nχ-∇_M∇_Nf(ϕ,χ)+g_MN□ f(ϕ,χ)


f(ϕ,χ)(R_MN-1/2g_MNR)=f(ϕ,χ) G_MN=(T_MN-g_MNΛ_B)
where
T_MN=∇_Mϕ∇_Nϕ+∇_Mχ∇_Nχ-g_MN[(∇ϕ)^2/2+(∇χ)^2/2+V]+∇_M∇_Nf-g_MN□ f
The above field equations are more convenient for calculations when they are expressed in terms of mixed tensor components. Hence, we have
[box=]equation
f(ϕ,χ) G^M_N=T^M_N-^M_NΛ_B

where we have absorbed the constant  inside the function f(ϕ,χ) and
[box=]equation
[ T^M_N=∇^Mϕ∇_Nϕ+∇^Mχ∇_Nχ-^M_N[(∇ϕ)^2/2+(∇χ)^2/2+V]+∇^M∇_Nf-^M_N□f;                                                                 ;            =^Mϕ_Nϕ+^Mχ_Nχ-^M_N[(ϕ)^2/2+(χ)^2/2+V]+∇^M∇_Nf-^M_N□f ]

The variation of the action (<ref>) with respect to the fields ϕ, χ (as it is done in the second section of [var]Appendix F) provides us with two additional equations that should be satisfied in order to have an acceptable solution for the mass function and the scalar fields. In complete analogy to the variation method of section F.2, it is straightforward to derive the following expressions:
[box=]equation
√(-)(1/2f/ϕR-V/ϕ)=-_M(√(-) g^MN_Nϕ)

[box=]equation
√(-)(1/2f/χR-V/χ)=-_M(√(-) g^MN_Nχ)


In order to derive the independent field equations that are resulting from relations (<ref>) and (<ref>) we firstly need to evaluate the mixed components of the energy-momentum tensor T^M_N. In [enmom]Appendix G there is a comprehensive evaluation of the components of the energy-momentum tensor given by Eq.(<ref>), thus, it is redundant to repeat any of these calculations. Subsequently, the non-zero components of the energy-momentum tensor are depicted in their compact form.
[box=]align
T^0_1   =e^-2A[(_1ϕ)^2+(_1χ)^2+_1^2f]
 

T^1_0   =e^-2A[(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1_0ϕ+_1χ_0χ+_1_0f).

   . +_1(m/r)_0f-_0(m/r)_1f+e^2A_4(m/r)_4f]
 

T^4_0   =_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1f
 

T^0_4   =e^-2A(_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f)
 

T^4_1   =e^2AT^0_4
 

T^1_4   =e^-2AT^4_0+(1-2m/r)T^0_4
 

T^0_0   =e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□f
 

T^1_1   =T^0_0+(1-2m/r)T^0_1
 

T^2_2   =e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□f
 

T^3_3   =T^2_2
 

T^4_4   =(_4ϕ)^2+(_4χ)^2+^2_4f+-□f
 

   =-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]}

   -1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ)
 

□f   =e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f)



§ THE FIELD EQUATIONS OF THE THEORY

Having in our disposal both the components of the Einstein tensor from equation (<ref>) and the components of the energy-momentum tensor given by equation (<ref>), the independent field equations can immediately ensue with the use of equation (<ref>). It is clear that the independent field equations are the following.


Equation (^0_1):



(<ref>)[(<ref>)](<ref>)(_1ϕ)^2+(_1χ)^2+_1^2f=0
Substituting _1^2f from equation (<ref>) into (<ref>) we get

(1+_ϕ^2f)(_1ϕ)^2+(1+_χ^2f)(_1χ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ f_1^2χ=0



Equation (^1_0):




    (<ref>)[(<ref>)](<ref>)(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1ϕ_0ϕ+_1χ_0χ+_1_0f)+_1(m/r)_0f
    
    -_0(m/r)_1f+e^2A_4(m/r)_4f=f[2/r^2_0m-e^2A/r(_4^2m+4A'_4m)]

The substitution of quantities _0^2f and _1_0f from equations (<ref>) and (<ref>) respectively into equation (<ref>) yields to

    e^-2A{(1+_ϕ^2f)(_0ϕ)^2+(1+_χ^2f)(_0χ)^2+2_χ_ϕ f_0ϕ_0χ+_ϕ f_0^2ϕ+_χ f_0^2χ.
    +(1-2m/r)[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1ϕ_0χ+_1χ_0ϕ)+_ϕ f_1_0ϕ+_χ f_1_0χ]
    .+(_1m/r-m/r^2)(_ϕ f_0ϕ+_χ f_0χ)-_0m/r(_ϕ f_1ϕ+_χ f_1χ)+e^2A_4m/r(_ϕ f_4ϕ+_χ f_4χ)}=
    
    =f[2/r^2_0m-e^2A/r(_4^2m+4A'_4m)]





Equation (^4_0):



(<ref>)[(<ref>)](<ref>)_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1f=f/r(_4m/r+_1_4m)
The last equation in its extended form is:

    (1+_ϕ^2f)_4ϕ_0ϕ+(1+_χ^2f)_4χ_0χ+_χ_ϕ f(_4χ_0ϕ+_4ϕ_0χ)+_ϕ f_4_0ϕ+_χ f_4_0χ
    
    -A'_0f-_4m/r(_ϕ f_1ϕ+_χ f_1χ)=f/r(_4m/r+_1_4m)





Equation (^0_4):



(<ref>)[(<ref>)](<ref>)_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f=0
Using equation (<ref>) into equation (<ref>) we obtain

    (1+_ϕ^2f)_1ϕ_4ϕ+(1+_χ^2f)_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ)+_ϕ f_1_4ϕ
    +_χ f_1_4χ-A'(_ϕ f_1ϕ+_χ f_1χ)=0




Equation (^0_0):




    (<ref>)[(<ref>)](<ref>)e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□ f-Λ_B=
    
    =f(6A'^2+3A”-2e^-2A/r^2_1m)

Equation (<ref>) is equivalent to

    e^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ.
    .+_ϕ f_1ϕ+_χ f_1χ/r(m/r-_1m)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f-Λ_B=
    
    =f(6A'^2+3A”-2e^-2A/r^2_1m)




Equation (^2_2):




    (<ref>)[(<ref>)](<ref>)e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ f-Λ_B=
    
    =f(6A'^2+3A”-e^-2A/r_1^2m)

Expanding the partial derivatives of the function f=f(ϕ,χ) we get

    e^-2A/r[(_ϕ f_0ϕ+_χ f_0χ)+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)
    
    +-□ f-Λ_B=f(6A'^2+3A”-e^-2A/r_1^2m)




Equation (^4_4):



(<ref>)[(<ref>)](<ref>)(_4ϕ)^2+(_4χ)^2+^2_4f+-□ f-Λ_B=f(6A'^2-e^-2A/r_1^2m-2e^-2A/r^2_1m)

The combination of equations (<ref>) and (<ref>) gives

    (1+_ϕ^2f)(_4ϕ)^2+(1+_χ^2f)(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χ+-□ f-Λ_B=
    
    =f(6A'^2-e^-2A/r_1^2m-2e^-2A/r^2_1m)

Instead of using equations (<ref>), (<ref>), (<ref>) and the corresponding extended equations (<ref>), (<ref>) and (<ref>), which contain the terms  and □ f that add extra complexity, we can be exempted from these terms by combining the aforementioned equations with each other.



Equation (^0_0)-Equation (^2_2):



Subtracting equation (<ref>) from (<ref>) we get:

r(_1ϕ_0ϕ+_1χ_0χ+_1_0f)-_0f-_1f(_1m+1-3m/r)=f(_1^2m-2/r_1m)
Correspondingly, from equations (<ref>) and (<ref>) one obtains

    r[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ] 
    
    -(_ϕ f_0ϕ+_χ f_0χ)-(_1m+1-3m/r)(_ϕ f_1ϕ+_χ f_1χ)=f(_1^2m-2/r_1m)




Equation (^0_0)-Equation (^4_4):



In the same way, subtracting equation (<ref>) from (<ref>) and (<ref>) from (<ref>) we respectively have:

    e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f-(_4ϕ)^2-(_4χ)^2-_4^2f=
    
    =f(3A”+e^-2A/r_1^2m)


    e^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ
    
    -_1(m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)-(1+_ϕ^2f)(_4ϕ)^2-(1+_χ^2f)(_4χ)^2
    
    -2_ϕ_χ f_4ϕ_4χ-_ϕ f_4^2ϕ-_χ f_4^2χ=f(3A”+e^-2A/r_1^2m)




Equation (^2_2)-Equation (^4_4):



Finally, the last field equation that has no dependence on the quantities  and □ f can be provided by equations (<ref>) and (<ref>) or equations (<ref>) and (<ref>). We respectively obtain
e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f-(_4ϕ)^2-(_4χ)^2-_4^2f=f(3A”+2e^-2A/r^2_1m)


    e^-2A/r[_ϕ f_0ϕ+_χ f_0χ+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]+A'(_ϕ f_4ϕ+_χ f_4χ)
    
    -(1+_ϕ^2f)(_4ϕ)^2-(1+_χ^2f)(_4χ)^2-2_ϕ_χ f_4ϕ_4χ-_ϕ f_4^2ϕ
    
    -_χ f_4^2χ=f(3A”+e^-2A/r_1^2m)


Of course, equation (<ref>) is not independent from equations (<ref>) and (<ref>), the subtraction of Eq.(<ref>) from (<ref>) gives Eq.(<ref>). Correspondingly, the subtraction of Eq.(<ref>) from Eq.(<ref>) gives Eq.(<ref>). The reason that equations (<ref>) and (<ref>) were presented is that in some of the cases -which are going to be investigated in the next Chapter- these equations might be easier to be solved or they might provide more directly useful information about the mass function m(v,r,y). In the context of the scalar field theory model, which was introduced at the beginning of this section, a solution to the localization problem -except from an appropriate expression for the mass function- requires also appropriate expressions for the scalar fields. A solution of the field equations that achieves to produce a mass function m(v,r,y) which describes a 5-dimensional black hole and it is also localized close to the 3-brane at y=0, with the cost of producing functions for the scalar fields ϕ(v,r,y), χ(v,r,y) that have an infinite value at y∞, cannot be accepted. Subsequently, we sum up the field equations. The field equations in their compact form are given by equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) (the last equation is not independent), while the field equations in their extended form are given by equations (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>) (of course, the last equation is also not independent).
[box=]gather
Compact Form
 
 

(_rϕ)^2+(_rχ)^2+_r^2f=0
 
 

(_vϕ)^2+(_vχ)^2+_v^2f+(1-2m/r)(_rϕ_vϕ+_rχ_vχ+_r_vf)+_r(m/r)_vf


-_v(m/r)_rf+e^2A_y(m/r)_yf=f[2/r^2_vm-e^2A/r(_y^2m+4A'_ym)]
 
 


_yϕ_vϕ+_yχ_vχ+_y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)
 
  


_rϕ_yϕ+_rχ_yχ+_r_yf-A'_rf=0
 
  


(_yϕ)^2+(_yχ)^2+^2_yf+-□f-Λ_B=f(6A'^2-e^-2A/r_r^2m-2e^-2A/r^2_rm)
 
  


r(_rϕ_vϕ+_rχ_vχ+_r_vf)-_vf-_rf(_rm+1-3m/r)=f(_r^2m-2/r_rm)
 
  

e^-2A[_rϕ_vϕ+_rχ_vχ+_r_vf-_r(m/r)_rf]+A'_yf-(_yϕ)^2
 

-(_yχ)^2-_y^2f=f(3A”+e^-2A/r_r^2m)
 
  

e^-2A/r[_vf+(1-2m/r)_rf]+A'_yf-(_yϕ)^2-(_yχ)^2


-_y^2f=f(3A”+2e^-2A/r^2_rm)

Last but not least, there are also the following two equations for the scalar fields that are necessary to be satisfied:
[box=]equation
√(-)(1/2f/ϕR-V/ϕ)=-_M(√(-) g^MN_Nϕ)

[box=]equation
√(-)(1/2f/χR-V/χ)=-_M(√(-) g^MN_Nχ)


The last two equations are going to bother us only in the case that we find appropriate functions for the quantities A(y), m(v,r,y), ϕ(v,r,y) and χ(v,r,y) which satisfy equations (<ref>)-(<ref>). In this hypothetical scenario it would be necessary to verify if these two equations are satisfied as well.







CHAPTER: SOLVING THE FIELD EQUATIONS


In this Chapter, we will try to solve the field equations, meaning that we will try to determine from the field equations the mass function m=m(v,r,y) and then the functions of the scalar fields ϕ(v,r,y), χ(v,r,y). As we already mentioned in the previous Chapter, the mass function should have a suitable dependence on the extra dimension y in order to be able to constitute a 5-dimensional black hole that is localized on our 4-dimensional 3-brane (namely our universe). In order to achieve that, we are going to use equations (<ref>)-(<ref>).
The aforementioned field equations resulted from the assumption that the scalar fields ϕ, χ and consequently the coupling function f=f(ϕ,χ) depend on the coordinates (v,r,y). However, it is possible to consider simpler cases where one or both of the scalar fields depend on just one or two of the (v,r,y) coordinates. It is also possible to consider cases in which either _ϕ f or _χ f equals to zero but not both of them simultaneously, because it is important to preserve the non-minimal coupling. As will be clear from the next pages of this Chapter, it is extremely difficult to find a suitable solution of the field equations within the framework of our field theory which could yield to a mass function m=m(v,r,y) that has the desirable dependence on the extra dimension y. On the contrary, in most of these cases the field equations are not consistent with our assumptions.



§ ALL POSSIBLE CASES

We now present all the possible cases that were mentioned previously, starting from the simplest cases and ending to the most complicated ones.


  * 
Both scalar fields depend on one coordinate 


    [ 1)  {ϕ=ϕ(v), χ=χ(v)};                     ; 2)  {ϕ=ϕ(v), χ=χ(r)};                     ; 3)  {ϕ=ϕ(v), χ=χ(y)};                     ; 4)  {ϕ=ϕ(r), χ=χ(r)};                     ; 5)  {ϕ=ϕ(r), χ=χ(y)};                     ; 6)  {ϕ=ϕ(y), χ=χ(y)};                     ;                      ]

The field equations are manifestly symmetrical under the exchange of ϕ and χ. Therefore, the case {ϕ=ϕ(v), χ=χ(r)} is the same as {ϕ=ϕ(r), χ=χ(v)}. This property reduces significantly the number of independent cases.


  * 
One scalar field depends on two coordinates and the other one depends on one  


    [  7)  {ϕ=ϕ(v,r), χ=χ(v)};                        ;  8)  {ϕ=ϕ(v,r), χ=χ(r)};                        ;  9)  {ϕ=ϕ(v,r), χ=χ(y)};                        ; 10)  {ϕ=ϕ(v,y), χ=χ(v)};                        ; 11)  {ϕ=ϕ(v,y), χ=χ(r)};                        ; 12)  {ϕ=ϕ(v,y), χ=χ(y)};                        ; 13)  {ϕ=ϕ(r,y), χ=χ(v)};                        ; 14)  {ϕ=ϕ(r,y), χ=χ(r)};                        ; 15)  {ϕ=ϕ(r,y), χ=χ(y)};                        ;                         ]



  * 
Both scalar fields depend on two coordinates


    [ 16)  {ϕ=ϕ(v,r), χ=χ(v,r)};                          ; 17)  {ϕ=ϕ(v,r), χ=χ(v,y)};                          ; 18)  {ϕ=ϕ(v,r), χ=χ(r,y)};                          ; 19)  {ϕ=ϕ(v,y), χ=χ(v,y)};                          ; 20)  {ϕ=ϕ(v,y), χ=χ(r,y)};                          ; 21)  {ϕ=ϕ(r,y), χ=χ(r,y)};                          ;                           ]


  * 
One scalar field depends on all three coordinates and the other one depends on one


    [ 22)  {ϕ=ϕ(v,r,y), χ=χ(v)};                          ; 23)  {ϕ=ϕ(v,r,y), χ=χ(r)};                          ; 24)  {ϕ=ϕ(v,r,y), χ=χ(y)};                          ;                           ]


  * 
One scalar field depends on all three coordinates and the other one depends on two


    [ 25)  {ϕ=ϕ(v,r,y), χ=χ(v,r)};                            ; 26)  {ϕ=ϕ(v,r,y), χ=χ(v,y)};                            ; 27)  {ϕ=ϕ(v,r,y), χ=χ(r,y)};                            ;                             ]



  * 
Both scalar fields depend on all three coordinates


    28)  {ϕ=ϕ(v,r,y), χ=χ(v,r,y)}




We henceforth start our quest for a valid solution of the field equations examining one by one the cases that were presented previously. The examination of the aforementioned cases will be split in two large categories. The first category includes the cases from 1 to 21 which can be studied (and excluded as it is shown    below) without further assumptions. The cases from 22 to 28 belong to the second category in which it is necessary to introduce an expression for the coupling function f(ϕ,χ) in order to be able to proceed to the solution of the field equations. In these last cases, if we do not fix the coupling function f(ϕ,χ) the field equations are unapproachable.



§ EXPLICITLY REJECTED CASES





1){ϕ=ϕ(v), χ=χ(v), f=f(ϕ,χ)=f(v)}



    (<ref>) 0=f(3A”+e^-2A/r_r^2m)_r^2m=-3A”e^2A r

_rm=-3/2A”e^2Ar^2+m_0(v,y)

    (<ref>)    -_vf=f(_r^2m-2/r_rm)f≠0 -_vf/f=_r^2m-2/r_rm(<ref>)
        -_vf/f=-3A”e^2Ar-2/r(-3/2A”e^2Ar^2+m_0(v,y))
        -_vf/f=-3A”e^2Ar+3A”e^2Ar-2/rm_0(v,y)

_vf/f=2/rm_0(v,y)

Obviously, the last equation is totally inconsistent, the left hand side (LHS) depends only on the v-coordinate while the right hand side (RHS) depends on the coordinates (v,r,y). Even if we demand _ym_0=0 in order to eliminate the y-dependence, we definitely cannot cancel the factor 1/r in the RHS. Therefore, this case is rejected.



2){ϕ=ϕ(v), χ=χ(r), f=f(ϕ,χ)=f(v,r)}


(<ref>) -A'_rf=0 A'_rf=0{[                 A'=_yA=0;                       or; _rf=_χ f_rχ=0_rχ≠0_χ f=0 ]}

From equation (<ref>) it is easily deducible that this case is rejected as well. The possibility A'(y)=0 is immediately rejected, while the constraint _χ f=0 leads to _rχ=0 if we use equation (<ref>). Hence, none of the possibilities (A'=0 or _χ f=0) can be valid because they contradict with our assumptions about the functions A(y) and χ(r).



3){ϕ=ϕ(v), χ=χ(y), f=f(ϕ,χ)=f(v,y)}


(<ref>) A'_yf-(_yχ)^2-_y^2f=f(3A”+e^-2A/r_r^2m)
The LHS of equation (<ref>) has (v,y)-dependence, while the RHS of the same equation is depended on (v,r,y), hence, we are led to the following constraint:

    _r[f(3A”+e^-2A/r_r^2m)]=0_rf_0(3A”+e^-2A/r_r^2m)+fe^-2A_r(_r^2m/r)=0f≠0

_r(_r^2m/r)=0_r^3m/r-_r^2m/r^2=0_r^3m-_r^2m/r=0


The differential equation (<ref>) can be solved easily assuming that the function of mass m=m(v,r,y) can be written as a power series expansion with respect to r-coordinate, containing either positive or negative powers of r. The same assumption has been used as well in <cit.>. Thus, it is
m=m(v,r,y)=∑_n a_n(v,y)r^n
The combination of equations (<ref>) and (<ref>) lead to

    ∑_n n(n-1)(n-2)a_n(v,y)r^n-3-∑_n n(n-1)a_n(v,y)r^n-3=0


    ∑_nn(n-1)(n-3)a_n(v,y)r^n-3=0{[                a_n=0  ∀ n≠{0,1,3};                                  ; a_0,a_1,a_3  arbitrary  functions ]}


m(v,r,y)=a_0(v,y)+a_1(v,y)r+a_3(v,y)r^3


    (<ref>)    -_vf=f(_r^2m-2/r_rm)f≠0-_vf/f=_r^2m-2/r_rm(<ref>)
        -_vf/f=6a_3(v,y)r-2/r[a_1(v,y)+3a_3(v,y)r^2]=6a_3(v,y)r-6a_3(v,y)r-2/ra_1(v,y)

_vf/f=2/ra_1(v,y)

Similarly to the case 1, the LHS of equation (<ref>) depends on (v,y)-coordinates while the RHS has the factor 1/r which cannot be cancelled, therefore this case is inconsistent as well.



4){ϕ=ϕ(r), χ=χ(r), f=f(ϕ,χ)=f(r)}



    (<ref>)-A'_rf=0 A'_rf=0{[ A'=_yA=0;       or;    _rf=0 ]}

In this case, we are also led to a contradiction to our primary assumption. Either A'(y)=0 or _r f=0 cannot be true. Especially, the constraint _r f=0 implies that the coupling function f(ϕ,χ) is completely independent from both scalar fields ϕ and χ, thus, the non-minimal coupling of the scalar fields to gravity is entirely vanished. Therefore, this case is also excluded from the list of possible solutions.



5){ϕ=ϕ(r), χ=χ(y), f=f(ϕ,χ)=f(r,y)}



    (<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2(_rf)/ r=-(_rϕ)^2


    ∫ dr(_rf)/ r=-∫ (_rϕ)^2dr_rf-f_0(y)=-∫ (_rϕ)^2dr

_rf=-∫ (_rϕ)^2dr+f_0(y)


    (<ref>)_y(_rf)-A'_rf=0(<ref>)_y[-∫ (_rϕ)^2dr+f_0(y)]-A'[-∫ (_rϕ)^2dr+f_0(y)]=0


    -_y[∫(_rϕ)^2dr]_0+_yf_0+A'∫(_rϕ)^2dr-A'f_0=0

A'f_0-_yf_0=A'∫(_rϕ)^2dr
The LHS of equation (<ref>) depends only on y-coordinate, while the RHS depends on (r,y)-co­or­di­nates. Hence, the constraint that is derived is the following:
_r[A'∫ (_rϕ)^2dr]=0 A'(_rϕ)^2=0{[ A'=_yA=0;       or;    _rϕ=0 ]}
According to our assumptions, both functions A' and _rϕ cannot be zero, so none of the constraints which are demanded by equation (<ref>) is able to be fulfilled. Neither _ϕ f=0 nor _χ f=0 are able to be assumed in the context of this case. The constraint _ϕ f=0 leads to _rϕ=0 through equation (<ref>) and _χ f=0 leads to the same analysis and the same negative result as the original case 5 with the only difference that in the case of _χ f=0 f_0 is a constant and not a function of y.



6){ϕ=ϕ(y), χ=χ(y), f=f(ϕ,χ)=f(y)}



    (<ref>)     0=f(_r^2m-2/r_rm)f≠0_r^2m-2/r_rm=0(_rm)/ r=2/r_rm
       1/_rm(_rm)/ r=2/r[ln(_rm)]/ r=2/r∫ dr[ln(_rm)]/ r=∫2/rdr
       ln(_rm)-m_0(v,y)=2ln r_rm=e^m_0(v,y)+ln r^2e^m_0(v,y) m_0(v,y)
       _rm=m_0(v,y)r^2∫ dr m/ r=m_0(v,y)∫ r^2dr

m(v,r,y)=m_0(v,y)r^3/3+m_1(v,y)


    (<ref>)     0=f/r(_ym/r+_r_ym)f≠0_ym/r+_r_ym=0(<ref>)_ym_0r^2/3+_ym_1+_ym_0r^2=0
        2r^2_ym_0+_ym_1=0{[ _ym_0=0 m_0=m_0(v);                and; _ym_1=0 m_1=m_1(v) ]}


(<ref>)(<ref>)m=m(v,r)=m_0(v)r^3/3+m_1(v)

It is clear from equation (<ref>) that in order to satisfy simultaneously both field equations (<ref>) and (<ref>), we are led to a mass function which does not depend on the y-coordinate. A mass function which is y-independent is impossible to describe a localized black hole. Therefore, this case is rejected without second thought.


7){ϕ=ϕ(v,r), χ=χ(v), f=f(ϕ,χ)=f(v,r)}



    (<ref>)-A'_rf=0{[             A'=_yA=0;                   or; _rf=0_ϕ f_rϕ=0_ϕ f=0 ]}

Of course, A'(y)≠ 0. Thus, if _ϕ f=0 then equation (<ref>) leads to _r ϕ=0. Hence, we reject this case as well.



8){ϕ=ϕ(v,r), χ=χ(r), f=f(ϕ,χ)=f(v,r)}



    (<ref>)-A'_rf=0{[         A'=_yA=0;               or; _rf=0_ϕ f=_χ f=0 ]}

Clearly, this case is also excluded.



9){ϕ=ϕ(v,r), χ=χ(y), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2

_rf=-∫(_rϕ)^2dr+f_0(v,y)


    (<ref>)_y_rf-A'_rf=0(<ref>)_y[-∫(_rϕ)^2dr+f_0]-A'[-∫(_rϕ)^2dr+f_0]=0


    -_y∫(_rϕ)^2dr_0+_yf_0+A'∫(_rϕ)^2dr-A'f_0=0

A'f_0-_yf_0=A'∫(_rϕ)^2dr
The LHS of equation (<ref>) depends on (v,y)-coordinates while the RHS depends on (v,r,y)-co­or­di­nates. Hence, equation (<ref>) leads to the following constraint.
_r[A'∫(_rϕ)^2dr]=0 A'(_rϕ)^2=0{[ A'=_yA=0;       or;    _rϕ=0 ]}
Equation (<ref>) is not possible to be satisfied because it contradicts with the original assumptions. We cannot assume _ϕ f=0 because equation (<ref>) results to _rϕ=0 as well. Moreover, _χ f=0 is not helpful either, because the only difference with the previous analysis is that f_0(v,y) f_0(v). 


10){ϕ=ϕ(v,y), χ=χ(v), f=f(ϕ,χ)=f(v,y)}



    (<ref>)A'_yf-(_yϕ)^2-_y^2f_(v,y)-dependent=f(3A”+e^-2A/r_r^2m)_(v,r,y)-dependent_r(RHS)=0


    _r[f(3A”+e^-2A/r_r^2m)]=0 fe^-2A_r(_r^2m/r)=0f≠0_r(_r^2m/r)=0

_r^3m-_r^2m/r=0


Equation (<ref>) is identical to equation (<ref>). Hence, combining (<ref>) with (<ref>) as we did in case 3, we obtain the mass function which is given by equation (<ref>). Subsequently, substituting equation (<ref>) into (<ref>) we are led to the same result as in case 3, namely equation (<ref>) which is inconsistent in this case as well.



11){ϕ=ϕ(v,y), χ=χ(r), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rχ)^2+_r^2f=0_r^2f=-(_rχ)^2

_rf=-∫ (_rχ)^2dr+f_0(v,y)


    (<ref>)_r_yf-A'_rf=0(<ref>)_y[-∫ (_rχ)^2dr+f_0(v,y)]-A'[-∫ (_rχ)^2dr+f_0(v,y)]=0


    _yf_0+A'∫ (_rχ)^2dr-A'f_0=0A'f_0-_yf_0_(v,y)-dependent=A'∫ (_rχ)^2dr_(r,y)-dependent_r(RHS)=0

_r[A'∫ (_rχ)^2dr]=0 A'(_rχ)^2=0{[ A'=_yA=0;       or;    _rχ=0 ]}
None of the two choices of equation (<ref>) can be satisfied. In addition, demanding either the constraint _ϕ f=0 or _χ f=0, nothing changes. Both sub-cases lead to the inconsistent result _rχ=0 as well.



12){ϕ=ϕ(v,y), χ=χ(y), f=f(ϕ,χ)=f(v,y)}




Using equations (<ref>), (<ref>) and the expansion of the mass function given by equation (<ref>) we obtain the same differential equations and constraints as in case 10. Therefore, this case results to an inconsistency as well.



13){ϕ=ϕ(r,y), χ=χ(v), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rϕ)^2+_r^2f=0_r^2f=-(_rϕ)^2_rf=-∫(_rϕ)^2dr+B(v,y)

f(v,r,y)=-∫[∫(_rϕ)^2dr]dr+B(v,y)r+C(v,y)


    (<ref>)    _yϕ_rϕ+_r_yf-A'_rf=0(<ref>)
        _rϕ_yϕ+_y[-∫(_rϕ)^2dr+B(v,y)]-A'[-∫(_rϕ)^2dr+B(v,y)]=0
        _rϕ_yϕ-2∫_rϕ _y_rϕ dr+_yB+A'∫(_rϕ)^2dr-A'B=0
        A'B-_yB_(v,y)-dependent=_rϕ_yϕ+∫_rϕ(A'_rϕ-2_y_rϕ)dr_(r,y)-dependent


As it is indicated by the last equation (<ref>), its LHS depends on (v,y)-coordinates, while its RHS depends on (r,y)-coordinates. Consequently, we can derive two new constrains by demanding the partial derivative of the LHS with respect to v to be zero and likewise the partial derivative of the RHS with respect to r to be zero. Thus, we have:
(<ref>)_v(LHS)=0_v(A'B-_yB)=0_v_yB=A'_vB


    (<ref>)   _r(RHS)=0_r[_rϕ_yϕ+∫_rϕ(A'_rϕ-2_y_rϕ)dr]=0
       _r^2ϕ_yϕ+_rϕ_r_yϕ+_rϕ(A'_rϕ-2_r_yϕ)=0
       _r^2ϕ_yϕ-_rϕ_r_yϕ+A'(_rϕ)^2=0
       -(_rϕ_r_yϕ-_r^2ϕ_yϕ)+A'(_rϕ)^2=0_rϕ≠0
       -_rϕ_r_yϕ-_r^2ϕ_yϕ/(_rϕ)^2(_rϕ)^2+A'(_rϕ)^2=0
       -_r(_yϕ/_rϕ)(_rϕ)^2+A'(_rϕ)^2=0 A'=_r(_yϕ/_rϕ)
       _yϕ/_rϕ=A'r+F(y)_yϕ=_rϕ[A'(y)r+F(y)]



    (<ref>)   _y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)
       _y_vf-A'_vf=f_ym/r^2+f_r_ym/r+_rf_ym/r(<ref>)
        r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)
       rA'_vB+_v_yC-rA'_vB-A'_vC=f_ym/r^2+_r(f_ym)/r


    _v_yC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r[f_ym/r^2+_r(f_ym)/r]=0
    
       _r(f_ym)/r^2-2f_ym/r^3+_r^2(f_ym)/r-_r(f_ym)/r^2=0_r^2(f_ym)-2f_ym/r^2=0


The last differential equation, with respect to the function f_ym, leads us to seek out the solution in the following form:

f_ym=∑_n b_n(v,y)r^n
Substituting equation (<ref>) into equation (<ref>) we get


    ∑_n n(n-1)b_n(v,y)r^n-2-2∑_n b_n(v,y)r^n-2=0


    ∑_n (n^2-n-2)_roots: {-1,2}b_n(v,y)r^n-2=0{[              b_n=0  ∀ n≠{-1,2};                               ; b_-1,b_2  arbitrary  functions ]}


    f_ym=b_-1(v,y)/r+b_2(v,y)r^2[b_-1(v,y) D(v,y)]b_2(v,y) E(v,y)f_ym=D(v,y)/r+E(v,y)r^2[we need the dependence
    on the extra dimension]_ym≠ 0


f(v,r,y)=1/_y[m(v,r,y)][D(v,y)/r+E(v,y)r^2]

(<ref>)_vf=-_v_ym/(_ym)^2(D/r+E r^2)+1/_ym(_vD/r+_vE r^2)

(<ref>)_vf=_vB r+_vC


    (<ref>),(<ref>)   _vB r+_vC=-_v_ym/(_ym)^2(D/r+E r^2)+1/_ym(_vD/r+_vE r^2)
    
       r(_ym)^2(r_vB+_vC)=-_v_ym(D+E r^3)+_ym(_vD+_vE r^3)

The combination of equations (<ref>) and (<ref>) leads to the relation


    r(∑_n_ya_n r^n)^2(r_vB+_vC)=-(∑_n_v_ya_n r^n)(D+E r^3)+(∑_n_ya_n r^n)(_vD+_vE r^3)

(∑_n_ya_n r^n+1)^2_vB+r(∑_n_ya_n r^n)^2_vC=∑_n_v(D/_ya_n)(_ya_n)^2r^n+∑_n_v(E/_ya_n)(_ya_n)^2r^n+3
Equation (<ref>) can be mathematically consistent if and only if each one of its terms vanish. Concentrating our attention on the LHS, we can immediately deduce that in order to nullify both terms, it should be either {_ya_n=0  ∀ n} or {_vB=0 and _vC=0}. If {_ya_n=0  ∀ n}, then we lose the desirable dependence of the mass function on the extra dimension. On the other hand, if {_vB=0 and _vC=0} then equation (<ref>) becomes


    f=-∫[∫(_rϕ)^2dr]dr+rB(y)+C(y)_vf=0_χ f_≠0_vχ=0_vχ=0


which is inconsistent with our assumption about the field χ=χ(v). Therefore, this case also fails to offer us a solution to the problem. If we consider _ϕ f=0 then equation (<ref>) gives _rϕ=0, which is inconsistent with the assumption about ϕ=ϕ(r,y). Finally, we consider the sub-case in which _χ f=0. Hence, we have:
_χ f=0_vf=0(<ref>)_v B=_C=0
Moreover, the substitution of equation (<ref>) into (<ref>) leads to
0=f_ym/r^2+_r(f_ym)/r_r(f_ym)/f_ym=-1/rf_ym=D(v,y)/r
From equation (<ref>) and (<ref>) we get

    -_rf(_rm+1-3m/r)=f(_r^2m-2_rm/r)(<ref>)
    D(_r_ym r+_ym)/(r_ym)^2(_rm+1-3m/r)=D/r_ym(_r^2m-2_rm/r)
    
    (r_r_ym+_ym)(_rm+1-3m/r)=r_ym(_r^2m-2_rm/r)(<ref>)
    ∑_n(_ya_n)(n+1)r^n(1+∑_ℓ a_ℓ(ℓ-3)r^ℓ-1)=∑_n(_ya_n)r^n∑_ℓ a_ℓ[ℓ(ℓ-1)-2ℓ]_ℓ(ℓ-3)r^ℓ-1
    ∑_n(_ya_n)r^n[(n+1)+∑_ℓ a_ℓ(n+1)(ℓ-3)r^ℓ-1-∑_ℓ a_ℓℓ(ℓ-3)r^ℓ-1]
    ∑_n(_ya_n)r^n[(n+1)+∑_ℓ a_ℓ(ℓ-3)(n+1-ℓ)r^ℓ-1]=0

We demand that _ya_n≠0 ∀ n, thus equation (<ref>) can only hold if n,ℓ take the values 1 and 3. Consequently, we obtain:

    ∑_n=1,3(_ya_n)r^n[(n+1)+a_1(-2)(n+2)]=0
    
    (_ya_1)r(2-6a_1)+(_ya_3)r^3(4-10a_1)=0{[ a_1=1/3;     and; a_1=5/2 ]} rejected

If on the other hand, we assume that n,ℓ take only the value 1. Then we have:

    (_ya_1)r(2-6a_1)=0 a_1=1/3_ya_1=0

which is also an undesirable result.



14){ϕ=ϕ(r,y), χ=χ(r), f=f(ϕ,χ)=f(r,y)}



    (<ref>)    -_ym/r_rf=f/r(_ym/r+_r_ym)f_ym/r^2+_r(f_ym)/r=0
       _r(f_ym)=-f_ym/r[_ym≠ 0]f≠01/f_ym(f_ym)/ r=-1/r
       [ln(f_ym)]/ r=-1/rln(f_ym)=-ln r+B(v,y)e^B(v,y) C(v,y)
        f_ym=C(v,y)/r_ym≠ 0f=f(r,y)=C(v,y)/_ym(v,r,y) r

Using now equation (<ref>) together with (<ref>) we are led to the same result as in the sub-case _χ f=0 of case 13, which was presented before.



  * 

_ϕ f=0:


From equation (<ref>) we obtain:

    -A'_r f=0 -A'_χ f_rχ_≠ 0=0{[         A'=0_yA=0;                or; _χ f=0_χ f=_ϕ f=0 ]}rejected


  * 

_χ f=0:


Using the same equations and performing the same steps as in the original case, we can show that this sub-case provides us with exactly the same negative result as the initial case 14.





15){ϕ=ϕ(r,y), χ=χ(y), f=f(ϕ,χ)=f(r,y)}


In this case, similarly to the case 14, the coupling function f depends on (r,y)-coordinates. Therefore, it is straightforward to verify that equations (<ref>) and (<ref>) result exactly to the same differential equations as in case 14 and subsequently to the same inappropriate form of the mass function. Consequently, case 15 is rejected as well.



16){ϕ=ϕ(v,r), χ=χ(v,r), f=f(ϕ,χ)=f(v,r)}



    (<ref>) -A'_rf=0{[         A'=_yA=0;               or; _rf=0_ϕ f=_χ f=0 ]}

Obviously, this is also not a viable solution to the problem.



17){ϕ=ϕ(v,r), χ=χ(v,y), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rϕ)^2+_r^2f=0_rf=-∫(_rϕ)^2 dr+B(v,y)

f=-∫[∫(_rϕ)^2 dr]dr+r B(v,y)+C(v,y)

    (<ref>)   _y_rf-A'_rf=0(<ref>)_y[-∫(_rϕ)^2 dr+B]-A'[-∫(_rϕ)^2 dr+B]=0
        -2∫_rϕ _y_rϕ_0 dr+_yB+A'∫(_rϕ)^2 dr-A'B=0
       A'B-_yB_(v,y)-dependent=A'∫(_rϕ)^2 dr_(v,r,y)-dependent_r(RHS)=0_r[A'∫(_rϕ)^2 dr]=0
        A'(_rϕ)^2=0{[  A'=0;    or; _rϕ=0 ]}

None of the choices which are depicted above is in agreement with our assumptions, hence, in this case the field equation (<ref>) is inconsistent.



  * 

_χ f=0:



    (<ref>) -A'_r f=0{[                            A'=0;                              or; _r f=0_ϕ f _rϕ_≠ 0=0_ϕ f=0=_χ f ]}rejected


  * 

_ϕ f=0:



    (<ref>)-_v f=f(_r^2m-2/r_rm)f≠ 0-_vf/f_(v,y)-dependent=_r^2m-2/r_rm_(v,r,y)-dependent_r(RHS)=0
    _r(_r^2m-2/r_rm)=0_r^3m-2/r_r^2m+2/r^2_rm=0(<ref>)
    ∑_n a_nn(n-1)(n-2)r^n-3-2∑_na_nn(n-1)r^n-3+2∑_na_nnr^n-3=0
    ∑_na_n[n(n-1)(n-2)+2n-2n(n-1)]_n(n-2)(n-3)r^n-3=0∑_na_nn(n-2)(n-3)r^n-3=0
    m(v,,r,y)=a_0(v,y)+a_2(v,y)r^2+a_3(v,y)r^3

Combining equations (<ref>) and (<ref>) we have:

    _yχ_vχ+_y_vf-A'_vf=f/r(_ya_0/r+_ya_2 r+_ya_3 r^2+2r_ya_2+3r^2_ya_3)
    _yχ_vχ+_y_vf-A'_vf/f_(v,y)-dependent=_ya_0/r^2+3_ya_2+4r_ya_3_(v,r,y)-dependent[consistency]for
    {[ _ya_0=0; _ya_2=0; _ya_3=0 ]}_ym=0

This sub-case should also be excluded, because with _ym=0 the localization is not possible.




18){ϕ=ϕ(v,r), χ=χ(r,y), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rϕ)^2+(_rχ)^2+_r^2f=0_rf=-∫[(_rϕ)^2+(_rχ)^2]dr+B(v,y)

f=-∫{∫[(_rϕ)^2+(_rχ)^2]dr}dr+rB(v,y)+C(v,y)


    (<ref>)   _rχ_yχ+_y_rf-A'_rf=0
       _rχ_yχ+_y{-∫[(_rϕ)^2+(_rχ)^2]dr+B}-A'{-∫[(_rϕ)^2+(_rχ)^2]dr+B}=0
       _rχ_yχ-∫(2 _rϕ _y_rϕ_0+2 _rχ _y_rχ)dr+_yB+A'∫[(_rϕ)^2+(_rχ)^2]dr-A'B=0
       A'B-_yB_(v,y)-dependent=_rχ_yχ+∫[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2]dr_(v,r,y)-dependent


    (<ref>)   _r(RHS)=0_r{_rχ_yχ+∫[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2]dr}=0
       _r^2χ _yχ+_rχ _r_yχ+A'(_rχ)^2-2 _rχ _y_rχ+A'(_rϕ)^2=0
       _r^2χ _yχ-_rχ _r_yχ+A'(_rχ)^2_(r,y)-dependent=-A'(_rϕ)^2_(v,r,y)-dependent_v(RHS)=0_v[A'(_rϕ)^2]=0
        2A'_rϕ _v_rϕ=0[_rϕ≠0]A'≠0_v_rϕ=0_v[_rϕ(v,r)]=0_rϕ(v,r)=C_1(r)
       ϕ(v,r)=∫ C_1(r) dr+C_2(v)ϕ(v,r)=ϕ_1(r)+ϕ_2(v)



    (<ref>)_vf=[(2 _rϕ _v_rϕ_0+2 _rχ _v_rχ_0)dr]dr+r_vB+_vC

_vf=r_vB+_vC


    (<ref>)(<ref>)A'B-_yB_(v,y)-dependent=_rχ_yχ+[_rχ(A'_rχ-2 _y_rχ)+A'(_rϕ)^2^r-dependent]dr_(r,y)-dependent_v(LHS)=0

_v[A'B-_yB]=0A'_vB=_v_yB


    (<ref>)   _y_vf-A'_vf-_ym/r_rf=f/r(_ym/r+_r_ym)(<ref>)
        r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)
       _y_vC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent



    (<ref>)_r(RHS)=0_r[f_ym/r^2+_r(f_ym)/r]=0_r(f_ym)/r^2-2f_ym/r^3+_r^2(f_ym)/r-_r(f_ym)/r^2=0

_r^2(f_ym)-2/r^2(f_ym)=0

Equation (<ref>) is identical to equation (<ref>) and equation (<ref>) is identical to (<ref>) of case 13; we can henceforth follow the same steps as in case 13 in order to show that this case should also be rejected.



  * 

_ϕ f=0:



    (<ref>)(_rϕ)^2+(_rχ)^2+_r^2f=0_rf=-∫[(_rϕ)^2+(_rχ)^2]dr+B(y)

f(r,y)=-∫{∫[(_rϕ)^2+(_rχ)^2]dr}dr+r B(y)+C(y)


    (<ref>)_vf=0-∫(∫ 2_rϕ_v_rϕ dr)dr=0_v_rϕ=0

ϕ(v,r)=ϕ_1(v)+ϕ_2(r)

(<ref>) 0=f_ym/r^2+_r(f_ym)/r=0_ym≠0f(r,y)=D(v,y)/r _y[m(v,r,y)]
where the following constraint is necessary to be satisfied:
_vf=0_v(D/_ym)=0


    (<ref>)e^-2A/r(1-2m/r)_rf+A'_yf-(_yχ)^2-_y^2f=f(3A”+2e^-2A/r^2_rm)_v( )

-2e^-2A/r^2_vm _rf=f2e^-2A/r^2_v_rm-_vm _rf=f_v_rm


    (<ref>)[(<ref>),(<ref>)](<ref>)-_vm[-D(_r_ym r+_ym)/(r_ym)^2]=D/r_ym_v_rm
    
    _vm(r_r_ym+_ym)=r_ym _v_rm
    
    (∑_n(_va_n)r^n)(∑_ℓ(_ya_ℓ)(ℓ+1)r^ℓ)=(∑_n(_ya_n)r^n)(∑_ℓ(_va_ℓ) ℓ r^ℓ)
    
    ∑_n,ℓ(_va_n)(_ya_ℓ)(ℓ-n+1)r^n+ℓ=0{_va_n=0 ∀ n}_vm=0

Thus, the combination of equation (<ref>) and (<ref>) results to
_vD=0
Hence, we obtain:
(<ref>)[(<ref>)](<ref>)f(r,y)=D(y)/r _y[m(r,y)]


    (<ref>) r_rϕ_vϕ-_rf(_rm-3m/r+1)=f(_r^2m-2/r_rm)_v( )r_v_rϕ_0_vϕ+r_rϕ_v^2ϕ=0

_v^2ϕ=0_v^2ϕ_1=0ϕ_1(v)= v+ξ
where  and ξ are constants.
_rϕ_vϕ=_rf/r(_rm-3m/r+1)+f/r(_r^2m-2/r_rm)

    (<ref>)(_vϕ)^2+(1-2m/r)_rϕ_vϕ   =-e^-2A/r(_ym _yf+f_y^2m+4A' f_ym)
       =-e^-2A/r(e^4A_ym _yf+e^4Af_y^2m+4A'e^4Af_ym)
       =-e^-2A/r_y(e^4Af_ym)=-e^-2A/r_y(e^4AD/r)
       =-e^-2A/r^2_y(De^4A)

Substituting equation (<ref>) into (<ref>) and using also equations (<ref>), (<ref>) and (<ref>), we obtain:

    ^2+e^-2A/r^2_y(De^4A)+(1-2m/r)[-D(r _r_ym+_ym/(r_ym)^2)(_rm+1-3m/r).
    .+D/r^2_ym(_r^2m-2/r_rm)]=0


    ^2+e^-2A/r^2_y(De^4A)+(1-2m/r)D/(r_ym)^2[-(r _r_ym+_ym)(_rm+1-3m/r).
    ._ym(_r^2m-2/r_rm)]=0
    
    ^2(r_ym)^2+e^-2A_y(De^4A)(_ym)^2-(_r_ym r+_ym)(_rm+1-3m/r)
    
    +2m/r(_r_ym r+_ym)(_rm+1-3m/r)
    _ym(_r^2m-2/r_rm)
    -2m/r_ym(_r^2m-2/r_rm)=0


    ∑_n,ℓ,k_ya_n{(n+1)+[(_ya_ℓ)(^2 r^2+e^-2A_y(e^4AD))+a_ℓ((n+1)(5-ℓ)r^-1+ℓ(ℓ-3)r^-2)..
    ..+2a_ℓ a_k(ℓ-3)((n+1)r^k-2-ℓ r^k-3)]r^ℓ}r^n=0

Demanding _ya_n≠0 the above equation is impossible to be satisfied. Therefore, in order to have a consistent field equation we need to allow _ya_n=0 at least for some values of n, but this is catastrophic for the localization of the 5-dimensional black hole.


  * 

_χ f=0:



    (<ref>) (_rϕ)^2+(_rχ)^2+_r^2f=0_y( )2_rϕ_y_rϕ_0+2_rχ _y_rχ+_y_r^2f_0=0_y_rχ=0

χ(r,y)=χ_1(r)+χ(y)


    (<ref>)_rχ_yχ-A'_rf=0_v( )-A'_v_rf=0

_v_rf=0 f(v,r)=f_1(v)+f_2(r)


    (<ref>)_rχ_yχ-A'_rf=0χ_1'(r)χ_2'(y)-A'(y)f_2'(r)=0χ_2'(y)/A'(y)=f_2'(r)/χ_1'(r)=


    χ_2(y)= A(y)+κ
    
    f_2(r)= χ_1(r)+κ̃


(<ref>)[(<ref>)]_v( )2_rϕ _v_rϕ=0_v_rϕ=0ϕ(v,r)=ϕ_1(v)+ϕ_2(r)


    (<ref>)-A'_vf_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0_r^2(f_ym)-2f_ym/r^2=0_ym≠0

f(v,r)=1/_y[m(v,r,y)][B(v,y)/r+C(v,y) r^2]

(<ref>)_yf=0{[ _y(B/_ym)=0;            ; _y(C/_ym)=0 ]}


    _y(B/_ym)=0_B_ym-B_y^2m=0∑_n(_yB_ya_n-B_y^2a_n)r^n=0

_yB/B=_y(_ya_n)/_ya_n_y[a_n(v,y)]/B(v,y)=b_n(v)
Similarly, we have:
_y[a_n(v,y)]/C(v,y)=c_n(v)
Equation (<ref>) with the use of equations (<ref>) and (<ref>) gets the following form:
f(v,r)=1/∑_n d_n(v)r^n+1+1/∑_n c_n(v)r^n-2
Finally, from equation (<ref>) the relation _v_rf=0 should be satisfied. Thus, it is:

    _v_rf=0(<ref>)    ∑_n d_n'(n+1)r^n/(∑_n d_nr^n)^2-2∑_n d_n(n+1)r^n∑_ℓ d_ℓ' r^ℓ+1/(∑_n d_nr^n+1)^3
       +∑_n c_n'(n-2)r^n-3/(∑_n c_nr^n-2)^2-2∑_n c_n(n-2)r^n-3∑_ℓ c_ℓ' r^ℓ-2/(∑_n c_nr^n-2)^3=0


{[ d_n'(v)=0 ∀ n;           and; c_n'(v)=0 ∀ n ]}_vf=0
Therefore, this sub-case should be rejected as well. In the previous analysis we silently assumed that both B(v,y) and C(v,y) were not equal to zero. In case that one of these functions is zero, we are led to exactly the same result, namely _vf=0. This happens because even if we nullify one of the functions B(v,y) or C(v,y) in equation (<ref>), none of the following steps is going to be changed.




19){ϕ=ϕ(v,y), χ=χ(v,y), f=f(ϕ,χ)=f(v,y)}



    (<ref>)A'_yf-(_yϕ)^2-(_yχ)^2+_y^2f_(u,y)-dependent=f(3A”+e^-2A/r_r^2m)_(v,r,y)-dependent_r(RHS)=0fe^-2A_r(_r^2m/r)=0

_r^3m-_r^2m/r=0m(v,r,y)=a_0(v,y)+a_1(v,y)r+a_3(v,y)r^3


    (<ref>)-_vf=f(_r^2m-2/r_rm)(<ref>)-_vf=-f2a_1/r_v[f(v,y)]/f(v,y)=2a_1(v,y)/r

It is clear that the last equation cannot be true, because of the factor 1/r in the RHS. For the sub-cases _χ f=0 and _ϕ f=0 nothing changes. The previous equations have exactly the same form, thus, the sub-cases are rejected as well.



20){ϕ=ϕ(v,y), χ=χ(r,y), f=f(ϕ,χ)=f(v,r,y)}



    (<ref>) (_rχ)^2+_r^2f=0_rf=-∫ (_rχ)^2dr+B(v,y)

f=-∫[∫(_rχ)^2dr]dr+rB(v,y)+C(v,y)


    (<ref>)_rχ_yχ+_r_yf-A'_rf=0(<ref>)


    _rχ_rχ-2∫_rχ _y_rχ dr+B-A'[-∫(_rχ)^2dr+_yB]=0

A'B-_yB_(v,y)-dependent=_rχ_yχ+∫_rχ(A'_rχ-2 _y_rχ)dr_(r,y)-dependent

(<ref>)_v(LHS)=0_v(A'B-_yB)=0A'_vB=_v_yB


    (<ref>)_r(RHS)=0_r^2χ_yχ+_rχ _r_yχ+_rχ(A'_rχ-2 _r_yχ)=0


    _r^2χ_yχ-_rχ _r_yχ+A'(_rχ)^2=0_rϕχ0-_r(_yχ/_rχ)(_rχ)^2+A'(_rχ)^2=0

_r(_yχ/_rχ)=A'_yχ=_rχ[A' r+F(y)]



    (<ref>)   _vϕ _yϕ+_y_vf-A'_vf-_ym _rf/r=f/r(_ym/r+_r_ym)
       _vϕ _yϕ+r_y_vB+_y_vC-A'(r_vB+_vC)=f_ym/r^2+_r(f_ym)/r(<ref>)
       _vϕ _yϕ+_y_vC-A'_vC_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0
       _r[f_ym/r^2+_r(f_ym)/r]=0_r^2(f_ym)-2/r^2(f_ym)=0
       f(v,r,y)=1/_y[m(v,r,y)][D(v,y)/r+E(v,y)r^2]

We have already shown in previous cases (see case 13) that the last related combined with equation _vf=r_vB+_vC and (<ref>) leads to an inconsistency. If we assume _ϕ f=0 then equation (<ref>) gives _rχ=0 which is not accepted. If on the other hand assume _χ f=0 then using equations (<ref>), (<ref>) and (<ref>) as before, we obtain again the relation (<ref>) (but now _vf=0). Finally, with the use of (<ref>) it is:

    -_rf(_rm+1-3m/r)=f(_r^2m-2/r_rm)

Using equations (<ref>), (<ref>) into the previous expressions and after some algebra we get

    ∑_n(_ya_n){D/r^2(n+1)+E r(n-2)+∑_ℓ a_ℓ(ℓ-3)[D/r^3r^ℓ(ℓ+n+1)+E r^ℓ(ℓ+n-2)]}=0

In the above equation, if we demand _ya_n≠0 then it is also restrictive to nullify both functions D(y) and E(y) in order for the relation to be consistent. However, there is the possibility to fix n,ℓ=1 and E(y)=0. Then, we can calculate the value of the function D(y) that manages to make the LHS of the above expression to vanish. The problem in this sub-sub-case is that for n=1 the metric tensor does not describe a black hole. Hence, this complete case does not give a viable solution to the problem.



21){ϕ=ϕ(r,y), χ=χ(r,y), f=f(ϕ,χ)=f(r,y)}


The field equation (<ref>), in this case, leads to the same expression for the coupling function f(r,y) as in case 14, namely (<ref>). Additionally, if we combine equations (<ref>), (<ref>) and (<ref>), as it has already been done in case 13, we are led to the same negative results. The sub-cases _ϕ f=0 and _χ f=0 do not solve the problem either. They produce the same results as the original case 21.




§ F(Φ,Χ)=A Φ+B Χ

The rest of the cases are extremely difficult to be checked and even more difficult to be solved by using the field equations (<ref>)-(<ref>); the complexity of the differential equations that emerge by the field equations increases dramatically with each variable that we add into the fields. Hence, we are going to examine the behaviour of some special coupling functions f(ϕ,χ)=aϕ+bχ in each one of the remaining cases (22-28).



22){ϕ=ϕ(v,r,y), χ=χ(v)}



We consider the general case where a,b∈∧ a≠0∧ b≠0.


    (<ref>)    (_rϕ)^2+a _r^2ϕ=0 -(_rϕ)/(_rϕ)^2=dr/a (_rϕ)^-1=r/a+B(v,y)
       _rϕ=1/r/a+B(v,y)_rϕ=a/r+a B(v,y)a B(v,y) B(v,y)
       _rϕ=a/r+B(v,y)ϕ(v,r,y)=aln[r+B(v,y)]+C(v,y)


    (<ref>)   _rϕ _yϕ+a _r_yϕ-A'a _rϕ=0(<ref>)
       a/r+B(_yB/r+B+_yC)-a^2_yB/(r+B)^2-A' a^2/r+B=0
       a_yC-A' a^2/r+B=0_yC=a A'C(v,y)=a A(y)+D(v)


(<ref>)(<ref>)ϕ(v,r,y)=a{ln[r+B(v,y)]+A(y)}+D(v)

    (<ref>)   _yϕ_vϕ+a _y_vϕ-A'( a_vϕ+b_vχ)-_ym/ra _rϕ=f/r(_ym/r+_r_ym)(<ref>)
       (a_yB/r+B+aA')(a _vB/r+B+_vD)+a(_y_vB/r+B-a_vB_yB/(r+B)^2)
       -A'(a^2_vB/r+B+aD'+bχ')-_ym/ra^2/r+B=aϕ+bχ/r(_ym/r+_r_ym)×(r+B)

aD'_yB+a^2_v_yB-bBA'χ'_(v,y)-dependent=(aϕ+bχ)(r+B)(_ym/r^2+_r_ym/r)+A'bχ'r+a^2_ym/r_(v,r,y)-dependent

Demanding the consistency of the previous equation, we are led to demand _r(RHS)=0. Thus, it is:

    (<ref>)_r(RHS)=0A'bχ'+a^2(_r_ym/r-_ym/r^2)+a^2(_ym/r^2+_r_ym/r)+(aϕ+bχ)(_ym/r^2+_r_ym/r)
    
    +(aϕ+bχ)(r+B)(-2_ym/r^3+_r^2_ym/r)=0
    
    
    A'bχ'+2a^2_r_ym/r+(aϕ+bχ)(_r_ym/r-_ym/r^2+_r^2_ym-2B_ym/r^3+B_r^2_ym/r)=0(<ref>)
    
    
    A'bχ'+2a^2∑_n(_ya_n)nr^n-2+(aϕ+bχ)[∑_n(_ya_n)nr^n-2-∑_n(_ya_n)r^n-2+.
    .+∑_n(_ya_n)n(n-1)r^n-2-2B∑_n(_ya_n)r^n-3+B∑_n(_ya_n)n(n-1)r^n-3]=0
    
    A'bχ'+∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3

Both for b=0 and b≠0 the above equation cannot have an overall value of zero, without allowing some of the function a_n(v,y) to be y-independent. This is easy to be understood by writing the equation in the following expanded form.

    A'bχ'+∑_n(_ya_n)[2a^2n+bχ(n+1)]r^n-2+Bbχ∑_n(_ya_n)(n+1)(n-2)r^n-3+
    
    +aϕ∑_n(_ya_n)(n+1)(n-1)r^n-2+Baϕ∑_n(_ya_n)(n+1)(n-2)r^n-3=0

Demanding finite number of values for n[The parameter n is directly connected to the behaviour of the mass function m(v,r,y) with respect to the variable r via equation (<ref>). Thus, the parameter n should take some specific values in order to lead to a mass function that describes a black hole, or even a modified black hole. In any case, we cannot allow n to take infinite number of values.], it is impossible to find a set of values that could allow the last two terms of equation (<ref>) to cancel each other (any other combination of terms in order to have a vanishing LHS either results to _rϕ=0 or to an inconsistency in the expression of ϕ with respect to r). Hence, the consistency of the equation can only be achieved by setting _ya_n=0 for some values of n. However, this would be devastating for the localization. Consequently, this case should also be excluded. Lastly, for a=0, equation (<ref>) results to _rϕ=0, meaning that we need to investigate the case {ϕ(v,y), χ=χ(v)}, but this case has already been excluded.



23){ϕ=ϕ(v,r,y), χ=χ(r)}



    (<ref>)(_rϕ)^2+(χ')^2+a _r^2ϕ+b χ”=0 (_rϕ)^2+a _r^2ϕ=-(χ')^2-b χ”

(_rϕ)^2+a _r^2ϕ=h(r)=-(χ')^2-b χ”

With the use of mathematica or any other software that can solve differential equations, one can verify that the above differential equations (with respect to ϕ and χ) have extremely complicated solutions even in case that h(r)=r. One can also examine different functions for h(r) in order to be completely convinced. As an example we present the solutions of the differential equations with respect to the scalar fields ϕ, χ that emanate from the relation (<ref>) in case of h(r)=r.

    ϕ(v,r,y)=   -_1^r a {√(-1/a)√(1/a) q^3/2[-c_1 J_-4/3(2/3√(-1/a)√(1/a) q^3/2)+c_1
       J_2/3(2/3√(-1/a)√(1/a)
       q^3/2)]}/2 q (c_1
       J_-1/3(2/3√(-1/a)√(1/a)
       q^3/2)+J_1/3(2/3√(-1/a)√(1/a)
       q^3/2))  dq
       +_1^r a[2 √(-1/a)√(1/a) q^3/2J_2/3(2/3√(-1/a)√(1/a) q^3/2)+c_1 J_-1/3(2/3√(-1/a)√(1/a) q^3/2)]/2 q (c_1
       J_-1/3(2/3√(-1/a)√(1/a)
       q^3/2)+J_1/3(2/3√(-1/a)√(1/a)
       q^3/2))  dq+c_2



    χ(r)=_1^r  q^3/2[-c_1
       J_-4/3(-2 q^3/2/3 b)+c_1 J_2/3(-2
       q^3/2/3 b)-2 J_-2/3(-2 q^3/2/3
       b)]+b c_1 J_-1/3(-2 q^3/2/3 b)/2
       q (c_1 J_-1/3(-2 q^3/2/3
       b)+J_1/3(-2 q^3/2/3 b)) 
       dq+c_2

Thus, it would be practically impossible to use these complicated expression of the fields ϕ, χ into the other field equations in order to examine their consistency. Consequently, we are essentially obliged to consider only the cases in which the function h(r) is simply a constant (negative or positive) and zero.


  * h(r)=C=Q^2>0∧ a≠ 0∧ b≠0:
(<ref>)(_rϕ)^2+a _r^2ϕ=Q^2=-(χ')^2-b χ”{[ (_rϕ)^2+a _r^2ϕ=Q^2;                    ;    (χ')^2+b χ”=-Q^2 ]}
From equation (<ref>) we obtain the following functions for the scalar fields:

    ϕ(v,r,y)=a ln[cosh(Q/ar+B(v,y))]+D(v,y)
    
    χ(r)=b ln[cos(Q/br+E)]+F

We have assumed that C>0. In case that C<0, we would get the same expressions for the fields with the difference that now the hyperbolic cosine would describe the field χ, while the cosine would correspond to the field ϕ. The subsequent analysis in each case is the same.

    (<ref>)_rϕ_yϕ+a_r_yϕ-A'(a_rϕ+b_rχ)=0[(<ref>)](<ref>)


    Qtanh(Q/ar+B)[atanh(Q/ar+B)_yB+_yD]+aQ_yB/cosh^2(Q/ar+B)
    
    -A'aQtanh(Q/ar+B)+A'bQtan(Q/br+E)=0
    
    
    a_yB+tanh(Q/ar+B)(_yD-aA')+A'btan(Q/br+E)=0{[     _yB=0;       and; _yD-aA'=0;       and;      A'=0 ]}A'≠0rejected


  * h(r)=C=-Q^2<0∧ a≠ 0∧ b=0:

(<ref>)(_r ϕ)^2+a _r^2ϕ=-Q^2=-(χ')^2{[ (_rϕ)^2+a _r^2ϕ=-Q^2;                     ;           (χ')^2=Q^2 ]}
From equation (<ref>) we get:
ϕ(v,r,y)=a ln[cos(Q/ar+B(v,y))]+D(v,y)
χ(r)=± Qr+E

    (<ref>)_rϕ_yϕ+a_r_yϕ-A'a_rϕ=0(<ref>)
    
     
    -Qtan(Q/ar+B)[-atan(Q/ar+B)_yB+_yD]-aQ_yB/cos^2(Q/ar+B)+A'aQtan(Q/ar+B)=0
    
    
    -aQ_yB-Qtan(Q/ar+B)(_yD-aA')=0
    
    {[              _yB=0B=B(v);                      and; _yD=aA'D(v,y)=aA(y)+F(v) ]}

Thus, equations (<ref>) and (<ref>) lead to
ϕ(v,r,y)=a ln[cos(Q/ar+B(v))]+a A(y)+F(v)

    (<ref>)_yϕ_vϕ+a_y_vϕ_0-A'a_vϕ=aϕ/r(_ym/r+_r_ym)+_ym/ra_rϕ
    
    
    aA'[-tan(Q/ar+B)_vB+_vF]-A'a[-atan(Q/ar+B)_vB+_vF]=aϕ_ym/r^2+_r(aϕ_ym)/r
    
    aϕ_ym/r^2+_r(aϕ_ym)/r=0(ϕ_ym)/ϕ_ym=-dr/r_ym=G(v,y)/rϕ

The substitution of equation (<ref>) into (<ref>) leads to
_ym=G(v,y)r^-1{a ln[cos(Q/ar+B(v))]a A(y)+F(v)}^-1
An expansion of the form m(v,r,y)=∑_na_n(v,y)r^n and a finite number of values for the index n cannot produce the above equation for the mass function. Hence, this sub-case does not result to a viable solution for the localization problem.


  * h(r)=C=Q^2>0∧ a=0∧ b≠0:

Using again equation (<ref>) in the same way as in the previous sub-case, we obtain:
ϕ(v,r,y)=± Qr+B(v,y)
χ(r)=b ln[cos(Q/br+D)]+E

    (<ref>)_rϕ_yϕ-A'b _rχ=0[(<ref>)](<ref>)± Q_yB+A'Qtan(Q/br+D)=0
    
    ±_yB+A'tan(Q/br+D)=0{[ _yB=0;   and;  A'=0 ]}A'≠0rejected


  * h(r)=0∧ a≠ 0∧ b≠0:

In this sub-case, it is not possible to assume either a=0 or b=0, because from equation (<ref>) we are led to _rϕ=0 or _rχ=0 respectively, but these results are inconsistent with our primary assumption for the fields. Consequently, from equation (<ref>) we get:

    (_rϕ)^2+a _r^2ϕ=0=(χ')^2+b χ”{[ (_rϕ)^2+a _r^2ϕ=0;                  ;     (χ')^2+b χ”=0 ]}

The solutions that emanate from the above differential equations are:
ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y)
χ(r)=b ln(r+D)+E

    (<ref>)_rϕ_yϕ+a _r_yϕ-A'(a _rϕ+b _rχ)=0[(<ref>)](<ref>)
    
    a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-A'(a^2/r+B+b^2/r+D)=0
    
    a_yC/r+B-A'a^2/r+B-b^2A'/r+D=0_y[C(v,y)]-aA'(y)_(v,y)-dependent=A'(y)b^2/a r+B(v,y)/r+D_(v,r,y)-dependent

Demanding the consistency of equation (<ref>), we get the following constraint.
(<ref>)_r(RHS)=0_r(r+B(v,y)/r+D)=0B(v,y)=D
Substituting equation (<ref>) into (<ref>), we obtain:
_y[C(v,y)]-aA'(y)=A'(y)b^2/aC(v,y)=a^2+b^2/aA(y)+F(v)
Hence, from equations (<ref>), (<ref>) and (<ref>) it is:
ϕ(v,r,y)=a ln(r+D)+a^2+b^2/aA(y)+F(v)

    (<ref>)_yϕ_vϕ+a_y_vϕ_0-A'a _vϕ-_ym/r_rf=f/r(_ym/r+_r_ym)
    
    a^2+b^2/aA'F'-aA'F'_(v,y)-dependent=f_ym/r^2+_r(f_ym)/r_(v,r,y)-dependent_r(RHS)=0
    
    _r^2(f_ym)-2f_ym/r^2=0_ym≠0f(v,r,y)=1/_y[m(v,r,y)][G(v,y)/r+H(v,y)r^2]

However, the coupling function f(v,r,y) is also given by the primary expression f=a ϕ+b χ. Thus, the substitution of equations (<ref>) and (<ref>)
into the previous expression and then equating the result with the RHS of equation (<ref>), we obtain:

    (a^2+b^2)ln(r+D)+(a^2+b^2)A(y)+aF(v)+bE=1/_ym[G(v,y)/r+H(v,y)r^2]

_ym=G(v,y)/r+H(v,y)r^2/(a^2+b^2)ln(r+D)+(a^2+b^2)A(y)+aF(v)+bE
Similarly to sub-case (ii), the above equation cannot be consistent with the expansion of equation (<ref>) and the condition of having a finite number of values for the index n. Hence, this sub-case is also rejected.





24){ϕ=ϕ(v,r,y), χ=χ(y)}


(<ref>) (_rϕ)^2+a _r^2ϕ=0ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y)


    (<ref>)_rϕ_yϕ+a _r_yϕ-A'a _rϕ=0a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-a^2A'/r+B=0

_yC=aA'C(v,y)=aA(y)+D(v)

(<ref>)(<ref>)ϕ(v,r,y)=a ln[r+B(v,y)]+aA(y)+D(v)

    (<ref>)_yϕ_vϕ+a _v_yϕ-A'a _vϕ-_ym/ra _rϕ=aϕ+bχ/r(_ym/r+_r_ym)(<ref>)
    
    a _yB D'+a^2_v_yB_(v,y)-dependent=(aϕ+bχ)(r+B)(_ym/r+_r_ym)+a^2_ym/r_(v,r,y)-dependent

The RHS of equation (<ref>) is similar to the RHS of equation (<ref>) except for the missing term A'bχ'r. Thus, demanding _r(RHS)=0 of equation (<ref>) (as in case 22) and also using equation (<ref>), we get:
∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3=0
The above equation cannot be consistent if we demand a finite number of values for n. The arguments here are the same as the arguments which were presented in case 22.




25){ϕ=ϕ(v,r,y), χ=χ(v,r)}



    (<ref>) (_rϕ)^2+(_rχ)^2+a _r^2ϕ+b _r^2χ=0 (_rϕ)^2+a _r^2ϕ=-(_rχ)^2-b _r^2χ

(_rϕ)^2+a _r^2ϕ=h(v,r)=-(_rχ)^2-b _r^2χ

As it was justified in case 23, the field equation are essentially unapproachable if we do not demand _r[h(v,r)]=0. Therefore, in this case, we are going to consider that h=h(v). In complete analogy to the case 23, four different sub-cases are going to be investigated in the context of the original case.



  * h(v,r)=[C(v)]^2>0∧ a≠ 0∧ b≠0:

From equation (<ref>) we are led to the following functions:
ϕ(v,r,y)=a ln[cosh(C(v)/ar+B(v,y))]+D(v,y)
χ(v,r)=b ln[cos(C(v)/br+E(v))]+F(v)


    (<ref>)_rϕ_yϕ+a _r_yϕ-A'(a _rϕ+b _rχ)=0[(<ref>)](<ref>)


    a_yB+_yDtanh(C/ar+B)-A'atanh(C/ar+B)+A'btan(C/br+E)=0


    {[     _yB=0;       and; _yD-aA'=0;       and;      A'=0 ]}A'≠0rejected



  * h(v,r)=-[C(v)]^2<0∧ a≠ 0∧ b=0:

In this sub-case equation (<ref>) leads to
ϕ(v,r,y)=a ln[cos(C(v)/ar+B(v,y))]+D(v,y)
χ(v,r)=± C(v)r+E(v)


    (<ref>)_rϕ_yϕ+a _r_yϕ-A'a_rϕ=0(<ref>)


    -a_yB-_yDtan(C/ar+B)+A'atan(C/ar+B)=0

{[              _yB=0B=B(v);                      and; _yD=aA'D(v,y)=aA(y)+F(v) ]}

Thus, equations (<ref>) and (<ref>) give:
ϕ(v,r,y)=a ln[cos(C(v)/ar+B(v))]+aA(y)+E(v)
Consequently, we have:

    (<ref>)_yϕ_vϕ+a _v_yϕ_0-A'a_vϕ=aϕ_ym/r^2+_r(aϕ_ym)/r


    ϕ_ym/r+_r(ϕ_ym)=0ϕ_ym=G(v,y)/rϕ≠0_ym=G(v,y)/rϕ

_ym=G(v,y)r^-1{a ln[cos(C(v)/ar+B(v))]+aA(y)+E(v)}^-1

The reason that we reject this sub-case is exactly the same as in sub-case (ii) of case 23.

One can easily verify that the remaining two sub-cases of case 25, namely h(v,r)=[C(v)]^2>0∧ a=0∧ b≠0 and h(v,r)=0∧ a≠ 0∧ b≠0, can be excluded as well. The reason is that the complete analysis of case 25 can be simply deduced by the analysis of case 23 if we just replace the constant Q with the function C(v) (To clarify this statement, compare the analysis of the previous two sub-cases -which were performed extensively- to the corresponding analysis of case 23). This particular substitution does not change the negative results that was found in each sub-case of case 23. Therefore, we proceed to the following case.




26){ϕ=ϕ(v,r,y), χ=χ(v,y)}


(<ref>) (_rϕ)^2+a _r^2ϕ=0ϕ(v,r,y)=a ln[r+B(v,y)]+C(v,y)


    (<ref>)_rϕ_yϕ+a _r_yϕ-A'a _rϕ=0a/r+B(a_yB/r+B+_yC)-a^2_yB/(r+B)^2-a^2A'/r+B=0

_yC=aA'C(v,y)=aA(y)+D(v)

(<ref>)(<ref>)ϕ(v,r,y)=a ln[r+B(v,y)]+aA(y)+D(v)

    (<ref>)_yϕ_vϕ+_yχ_vχ+a _v_yϕ+b_v_yχ-A'(a _vϕ+b_vχ)=f_ym/r^2+_r(f_ym)/r(<ref>)
    
    a _yB D'+_yχ_vχ B+a^2_v_yB-A'bB_vχ_(v,y)-dependent=
    
    =(aϕ+bχ)(r+B)(_ym/r^2+_r_ym/r)-_yχ_vχ r-b_y_vχ r+A'b_vχ r+a^2_ym/r_(v,r,y)-dependent

The RHS of equation (<ref>) is similar to the RHS of equation (<ref>) except from the additional terms -_yχ_vχ r and -b_y_vχ r. Thus, demanding _r(RHS)=0 of equation (<ref>) (as in case 22) and also using equation (<ref>), we get:
A'b_vχ-_yχ_vχ-b_v_yχ+∑_n(_ya_n){2a^2nr+(aϕ+bχ)(n+1)[(n-1)r+B(n-2)]}r^n-3=0

The above equation cannot be consistent if we demand a finite number of values for n. The arguments here are the same as the arguments which were presented in case 22 and also used in case 24.




27){ϕ=ϕ(v,r,y), χ=χ(r,y)}



    (<ref>) (_rϕ)^2+(_rχ)^2+a _r^2ϕ+b _r^2χ=0 (_rϕ)^2+a _r^2ϕ=-(_rχ)^2-b _r^2χ

(_rϕ)^2+a _r^2ϕ=h(r,y)=-(_rχ)^2-b _r^2χ

The same argument as it was presented in case 23 leads us to the following sub-cases:


  * h(r,y)=C(y)>0∧ a≠ 0∧ b≠0:

From euqation (<ref>) we obtain:
ϕ(v,r,y)=a ln[cosh(√(C(y))/ar+B(v,y))]+D(v,y)
χ(r,y)=b ln[cos(√(C(y))/br+E(y))]+F(y)


    (<ref>)_rϕ_yϕ+_rχ_yχ+a_r_yϕ+b_r_yχ-A'(a_rϕ+b_rχ)=0[(<ref>)](<ref>)


    a_yB-bE'+tanh(√(C)/ar+B)(_yD+aC'/2C-aA')-tan(√(C)/br+E)(F'+bC'/2C-bA')=0

{[                           _yB=b/aE'B(v,y)=b/aE(y)+G(v);                                                    and;  _yD-aA'+a/2_y[ln C(y)]=0D(v,y)=aA(y)-a/2ln[C(y)]+H(v);                                                    and; F'-bA'+b/2_y[ln C(y)]=0F(y)=bA(y)-b/2ln[C(y)]+J_const. ]}

Thus, equations (<ref>) and (<ref>) take the following form:
ϕ(v,r,y)=a ln[cosh(√(C(y))/ar+b/aE(y)+G(v))]+aA(y)-a/2ln[C(y)]+H(v)
χ(r,y)=b ln[cos(√(C(y))/br+E(y))]+bA(y)-b/2ln[C(y)]+J


    (<ref>)_yϕ_vϕ+a_v_yϕ-A'a_vϕ=f_ym/r^2+_r(f_ym)/r

f_ym/r^2+_r(f_ym)/r=arC'G'/2√(C)+abE'G'+tanh(√(C)/ar+b/aE+G)(C'H'r/2√(C)+bE'H'-aC'G'/2C)-aC'H'/2C

Moreover, it is:
f=aϕ+bχ=a^2 ln[cosh(√(C(y))/ar+b/aE(y)+G(v))]+b^2 ln[cos(√(C(y))/br+E(y))]+(a^2+b^2)A(y)-a^2+b^2/2ln[C(y)]+aH(v)+bJ

Obviously, the substitution of equation (<ref>) into (<ref>) leads to a very complicated equation which is not possible to be satisfied using equation (<ref>) for the mass function and demanding n to take a finite number of values. The previous statement holds even in case of C'=0.



  * h(r,y)=C(y)=-[K(y)]^2<0∧ a≠ 0∧ b=0:

Equation (<ref>) results to the following functions for the fields:
ϕ(v,r,y)=a ln[cos(√(C(y))/ar+B(v,y))]+D(v,y)
χ(r,y)=±√(C(y)) r+E(y)
Using equation (<ref>) we obtain:

    _rϕ_yϕ+_rχ_yχ+a_r_yϕ-A'a_rϕ=0


    -a_yB-tan(√(C)/ar+B)(_yD+aC'/2C-aA')± E'=0

{[                          _yB=±E'/aB(v,y)=±E(y)/a+F(v);                                                   and; _yD-aA'+a/2_y[ln C(y)]=0D(v,y)=aA(y)-a/2ln[C(y)]+G(v) ]}

Hence, we have:
ϕ(v,r,y)=a ln[cos(√(C(y))/ar±E(y)/a+F(v))]+aA(y)-a/2ln[C(y)]+G(v)


    (<ref>)_yϕ_vϕ+a_y_vϕ-A'a_vϕ=f_ym/r^2+_r(f_ym)/r

f_ym/r^2+_r(f_ym)/r=-a^2E'(C'r/2√(C)a±E'/a)-atan(√(C)/ar±E/a+F)(G'C'r/2√(C)a±G'E'/a-aF'C'/2C)-aC'G'/2C
For _ym≠0, it is necessary to nullify the quantity _ym/r^2+_r_ym/r which is the factor of the term which includes the coupling function f in the LHS of the above equation. This term cannot be equated with another term, so it must vanish, otherwise the field equation is not consistent. Thus, we get:
_ym/r^2+_r_ym/r=0m(v,r,y)=(v,y)/r
For the same reason the quantity _r^2m-2_rm/r should also be zero in the RHS of equation (<ref>). Hence, it is:
_r^2m-2/r_rm=0m(v,r,y)=κ(v,y)r^3+ξ(v,y)
It is clear that equations (<ref>) and (<ref>) cannot be simultaneously true. Consequently, we should reject this sub-case as well, because it leads to an inconsistency.



  * h(r,y)=C(y)>0∧ a=0∧ b≠0:

This sub-case is almost identical to the previous one, but the functions of ϕ and χ are alternated. Here, equation (<ref>) leads to
ϕ(v,r,y)=±√(C(y)) r+B(v,y)
χ(r,y)=bln[cos(√(C(y))/br+D(y))]+E(y)
Following the same steps as in sub-case (ii), we obtain again equations (<ref>) and (<ref>). Therefore, this sub-case is also not capable of providing a viable solution to the localization problem.


  * h(r,y)=0∧ a≠ 0∧ b≠0:

From equation (<ref>) we have:
ϕ(v,r,y)=aln[r+B(v,y)]+C(v,y)
χ(r,y)=bln[r+D(y)]+E(y)

(<ref>)[[(<ref>)](<ref>)_yC-aA'_(v,y)-dependent=-(b/aE'-b^2/aA')r+B/r+D_(v,r,y)-dependent
(<ref>)_r(RHS)=0B(v,y)=D(y)

Thus:
(<ref>)(<ref>)C(v,y)=a^2+b^2/aA(y)-b/aE(y)+F(v)(<ref>)[(<ref>)](<ref>)ϕ(v,r,y)=aln[r+D(y)]+a^2+b^2/aA(y)-b/aE(y)+F(v)

Using equations (<ref>) and (<ref>) as they were used in sub-cases (ii) and (iii), we obtain again the equations (<ref>) and (<ref>). Hence, the case 27 is complete excluded.




28){ϕ=ϕ(v,r,y), χ=χ(v,r,y)}


Finally, the most general case 28 can be investigated and excluded in exactly the same way as case 27. The only difference is that in this case, we have to replace the functions h(r,y) and C(y) with h(v,r,y) and C(v,y) respectively.






CHAPTER: CONCLUSIONS AND DISCUSSION


In summary, the study that was preceded in the framework of this thesis had the following structure. In Chapter 1 (Introduction) took place a concise presentation of the biggest mysteries in physics that remain unsolved until these days i.e. the nature of Dark Matter and Dark Energy and the existence of a Unified Theory. The attempts that were made in order to resolve these problems led to various extra-dimensional theories, in which String Theory plays a starring role. The Hierarchy Problem led to the formulation of the Randall-Sundrum models (RS1 and RS2), which were analyzed in detail in Chapter 2. Briefly, RS1 model manages to resolve the Hierarchy Problem by assuming the existence of a compact extra dimension, which is finite and also bounded by two 3-branes. However, in the context of the RS2 model, the extra dimension is allowed to be infinite and therefore the second brane is essentially removed from the model. The astonishing result of the RS2 model is that although we have an infinite extra dimension, the gravity on the remaining 3-brane is effectively 4-dimensional. Although very popular, these models face the problem of the absence of an analytic solution describing a regular, 5-dimensional black hole solution localised close to the brane. Therefore, in Chapter 3, the geometrical framework and the scalar field theory -which were used in the context of the thesis in an attempt to solve black hole localization problem- were presented and the field equations of our theory were derived. Subsequently, these field equations were used in Chapter 4 in order to find a localized black hole solution in the context of a generalized RS2 brane-world model.
The main purpose of this thesis was to find a localized 5-dimensional black hole solution close to our 3-brane (or our 4-dimensional universe) by using an RS2-type geometrical background and a scalar field theory which consists of two scalar fields ϕ, χ that interact with each other and they are also non-minimally coupled to gravity. This complicated choice of a scalar field theory was made because simpler types of scalar field theories failed to provide a localized black hole solution. Therefore, we assumed the existence of an extra scalar field χ. Despite the additional degree of freedom that was provided by the second scalar field χ, we were not able to find a viable configuration thus the analytical solution to the localization problem remains still an open problem. Although we were able to exclude mathematically all considered cases, the complexity of the equations did not allow us to formulate a no-go theorem that would exclude altogether the existence of a viable configuration. Our negative result is only one in a series of failed analytical attempts over a period of almost 20 years, and this certainly creates well-founded concerns about the compatibility of brane-world models with the predictions of General Relativity. It is also crucial to indicate that some of the solutions rejected may comprise instead novel black-string solutions, whose study we will undertake in the near future. If this proves to be true, it will refuel the existing debate in the literature regarding the question of why black-string solutions are so much easier to find in the context of brane-world models compared to black-hole solutions.
In the context of the scalar field theory that was considered in this thesis, there are some extra coupling functions that is possible to be examined, though they are more complicated:


f(ϕ,χ)=a ϕ^2+b χ^2+c ϕ χ
 

f(ϕ,χ)=a ϕ^κ+b χ^


f(ϕ,χ)=e^a ϕ+b χ
 



Except for the most complicated case where a≠0∧ b≠0∧ c≠0, it is possible to set a=0 and/or b=0 and/or c=0. The previous analysis for the coupling function f(ϕ,χ)=aϕ+bχ have made clear that the most valuable information about the form of the functions of the scalar fields ϕ and χ emanates from equation (<ref>). The reason is that compared to the other field equations, equation (<ref>) is the simplest one. If we therefore consider a coupling function, which is complicated enough to make the differential equation (<ref>) unsolvable, then we are doomed to give up the attempt to solve the localization problem. Consequently, one should be careful about the choice of the coupling function f(ϕ,χ). If the coupling function is too simple, then probably the localization of a 5-dimensional black hole would not be possible, because the simple cases have already been investigated and excluded. On the other hand, if the coupling function is too complicated, then, even if there is a solution to the problem, we might not be able to find it.
After about 20 years of research in the direction of finding a closed-form, analytical 5-dimensional localized braneworld black hole solution, there is still work to do. Fortunately, there are numerical solutions to the problem, thus, the research for an analytical solution is not completely in vain.






CHAPTER: BLACK HOLE SOLUTIONS IN GENERAL RELATIVITY



Schwarzschild Solution (M≠ 0, Q=0, J=0):
ds^2=-(1-r_S/r)(cdt)^2+(1-r_S/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2)
where

    r_S=2GM/c^2


Reissner-Nordström Solution (M≠0, Q≠0, J=0):
ds^2=-Δ (cdt)^2+Δ^-1 dr^2+r^2(dθ^2+sin^2θ φ^2)
where

    Δ=1-r_S/r+r_Q^2/r^2

and

    r_Q^2=GQ^2/4πϵ_0c^4


Kerr Solution (M≠0, Q=0, J≠0):
ds^2=-(1-r_S r/ρ^2)(cdt)^2-2 r_S r a sin^2θ/ρ^2cdtdφ+ρ^2/Δdr^2+ρ^2 dθ^2
+sin^2θ/ρ^2[(r^2+a^2)^2-a^2Δsin^2θ]dφ^2
where

    a=J/Mc
    
    Δ=r^2-r_S r+a^2
    
    ρ^2=r^2+a^2cos^2θ

The Kerr-Newman line element (M≠0,Q≠0,J≠0) results from Eq.(<ref>) by replacing  r_S r with r_S r-r_Q^2.







CHAPTER: GAUSSIAN NORMAL COORDINATES



Let us consider a Lorentzian (n+1)-manifold ℳ with coordinates {x^M}={x^0,x^1…, x^n}, where M runs from 0 to n. A (d+1)-submanifold 𝒮 of the original (n+1)-dimensional manifold ℳ with intrinsic coordinates {y^μ}={y^0,y^1,…,y^d} (d<n) can be defined via the following set of n+1 parametric equations:
x^M=x^M(y^0,y^1,…,x^d){[ x^0=x^0(y^0,y^1,…,x^d); x^1=x^1(y^0,y^1,…,x^d);                      ⋮; x^n=x^n(y^0,y^1,…,x^d) ]}
For d=n-1 we have a hypersurface. In this case, it is possible to define the hypersurface without using the above parametric equations. The way that this can be done, is the following:
F(x^0,x^1,…,x^n)=c
where c is a constant. In every point p of the hypersurface 𝒮  there is a tangent plane T̅_p which can be thought as an n-dimensional subspace of the tangent plane T_p of the (n+1)-dimensional manifold ℳ. It is intuitively clear that there is always an (n+1)-vector n⃗∈ T_p (unique up to scaling) which is orthogonal to all vectors in T̅_p. This vector n⃗ is said to be normal to the hypersurface 𝒮. The unit normal vector n̂ in any point on 𝒮 is given by
n̂=n^Me⃗_M=^M F/√(|g^AB_AF_BF|)e⃗_M
In order to clarify the unitarity of n̂ one can perform the scalar product of n̂ with itself and substitute e⃗_M·e⃗_N for g_MN. We silently assumed that g_AB n^An^B≠ 0, in the opposite case we have a null-hypersurface and therefore is not possible to normalize the normal vector as we did in Eq.(<ref>). Let us now consider two events on the hypersurface 𝒮 which are separated by the n-vector dr⃗=dx^Ae⃗_A. These events are so close that we can safely assume that the vector dr⃗ lies on the tangent plane T̅_p. Hence, the scalar product between dr⃗ and n̂ is zero.
n̂· dr⃗=0(n^Ae⃗_A)·(dx^Be⃗_B)=0 n^Adx^Bg_AB=n^A dx_A=n_A dx^A=0
The Gaussian Normal Coordinates for any non-null hypersurface can be constructed by using the following steps. Firstly, for each point p∈𝒮 we find the unique geodesic curve that passes by the point p and its tangent vector is n̂. Thereafter, we choose a coordinate system {x^0,x^1,…,x^n-1} on 𝒮 and then we characterize each point in a neighborhood of the hypersurface 𝒮 by using these coordinates and the parameter y which is along the geodesic curve that emanates from the point p∈𝒮. Therefore, it is always possible to find a local coordinate system {x^0,x^1,…,x^n,y} in a neighborhood of a point p∈𝒮 where a vector along coordinate y is perpendicular to the hypersurface 𝒮.







CHAPTER: LINEARIZED GRAVITY



Linearized gravity is simply an approximation which is used to describe weak gravitational fields. The spacetime in the context of this approximation is considered nearly flat, thus, the metric tensor is expressed by the following relation:
g_MN=η_MN+h_MN(x^,y)
where
|h_MN|≪ 1
The use of capital Latin characters indicates that we take into account extra spatial dimensions. In this chapter particularly, we consider a total of (4+1)-dimensions {x^0,x^1,…,x^3,y}. In the calculations that follow we ignore as negligible any term that contains non-linear orders of h_MN. Consequently, the Christoffel symbols are calculated as follows:

    ^L_MN=1/2g^LR(g_MR,N+g_NR,M-g_MN,R)
    ^L_MN=1/2(η^LR+h^LR)[_N(η_MR+h_MR)+_M(η_NR+h_NR)-_R(η_MN+h_MN)]
    ^L_MN=1/2η^LR(h_MR,N+h_NR,M-h_MN,R)=1/2(h^L_M,N+h^L_N,M-h_MN^,L)

The components of the Riemann tensor can be computed very easily as well.

    R^L_MRN=^L_MN,R-^L_MR,N+^L_RK^K_MN_h^20-^L_NK^K_MR_h^20
    
    R^L_MRN=1/2(h^L_M,NR+h^L_N,MR-h_MN,R^L-h^L_M,RN-h^L_R,MN+h_MR,N^L)
    R^L_MRN=1/2(h^L_N,MR-h_MN,R^L-h^L_R,MN+h_MR,N^L)

Using Eq.(<ref>) and contracting indices L and R we obtain the components of the Ricci tensor.

    R_MN=1/2(h^L_N,ML-h_MN,L^L-h^L_L,MN+h_ML,N^L)
    R_MN=1/2(-□ h_MN-h_,MN+h_ML,N^L+h^L_N,ML)

where 
h≡ h^L_L
Subsequently, the Ricci scalar is going to be evaluated, which emanates directly from Eq.(<ref>).

    R=g^MNR_MN=η^MNR_MN+h^MNR_MN_h^20=1/2(-□ h-□ h+h^ML_,ML+ h^ML_,ML)
    R=-□ h+h^ML_,ML

Finally, the components of the Einstein tensor are given by

    G_MN=R_MN-1/2R g_MN=R_MN-1/2R η_MN

G_MN=1/2(-□ h_MN-h_,MN+h_ML,N^L+h^L_N,ML-η_MN□ h+η_MNh^RL_,RL)







CHAPTER: FROM SCHWARZSCHILD TO VAIDYA



First of all, we clarify that for all the mathematical expressions that are depicted below, the Planck units have been used. The Schwarzschild metric as the static and spherically symmetric solution to ­Einstein's equations of gravity is given by the following line element
ds^2=-(1-2M/r)dt^2+(1-2M/r)^-1dr^2+r^2(dθ^2+sin^2θ dφ^2)
Someone could switch to Eddington-Finkelstein coordinates in order to remove ­the ­undesirable ­coordinate singularity of the metric at r=2M. The new coordinates are using the ­null ­coordinate v with the following definition.
t=v-r-2Mln(r/2M-1)  dt=dv-(1-2M/r)^-1dr
Subsequently, using equation (<ref>) into (<ref>) the line element takes the form
ds^2=-(1-2M/r)dv^2+2dvdr+r^2(dθ^2+sin^2θ dφ^2)
If we now extend the mass parameter M from a constant to a function of v we get the Vaidya metric.
ds^2=-(1-2M(v)/r)dv^2+2dvdr+r^2(dθ^2+sin^2θ dφ^2)







CHAPTER: COMPLETE INFORMATION ABOUT THE 5-D GEOMETRICAL BACKGROUND





§ CHRISTOFFEL SYMBOLS

Christoffel symbols can be calculated by equation
^K_MN=1/2g^KL(g_ML,N+g_NL,M-g_MN,L)
Hence, combining equations (<ref>), (<ref>) and (<ref>), we obtain

[box=]equation
[                    ^0_00=m-r _r m/r^2                        ^0_04=^0_40=A'                              ^0_22=-r;                                      ;                       ^0_33=-r sin^2θ ^1_00=r^2 _v m-(r-2 m) (r _r m-m)/r^3               ^1_01=^1_10=r _rm-m/r^2;                                      ;                     ^1_04=^1_40=_ym/r                        ^1_14=^1_41=A'                           ^1_22=2 m-r;                                      ;                   ^1_33=(2 m-r)sin^2θ                       ^2_12=^2_21=1/r                        ^2_24=^2_42=A';                                      ;                       ^2_33=-sinθcosθ                       ^3_13=^3_31=1/r                         ^3_23=^3_32=θ;                                      ;                        ^3_34=^3_43=A'        ^4_00=e^2 A (A' (r-2 m)-_ym)/r                 ^4_01=^4_10=-e^2 A A';                                      ;                   ^4_22=-r^2 e^2 A A'             ^4_33=-r^2A' e^2 A sin^2θ                                       ]




§ RIEMANN TENSOR

Riemann tensor's components are defined by
R^L_KMN=^L_NK,M-^L_MK,N+^L_MJ^J_NK-^L_NJ^J_MK
Thus, having in our disposal the Christoffel symbols from equation (<ref>) it is possible to compute the components of the Riemann tensor, but also one would need an eternity to make calculations. Therefore, it will be presented a less time-consuming method that provides the Riemann tensor's components.

First and foremost, the main ingredients of this method are the matrices _M and B_MN.


  * The _M matrices are resulting from Christoffel symbols ^K_ML as follows. The component ^K_ML is defined to be the element of the K-th row and L-th column of the matrix _M. So, there are 5 matrices _M of size 5× 5.

  * The B_MN matrices are resulting from the components of Riemann tensor R^L_KMN. The component R^L_KMN is defined to be the element of the L-th row and K-th column of the matrix B_MN. Thus, there are 25 matrices B_MN of size 5× 5.

It is now obvious that making use of the previous definitions, equation (<ref>) can be written in the following form.
B_MN=_N,M-_M,N+_M_N-_N_M
It is necessary to mention that the matrices B_MN have the useful property B_MN=-B_NM. This property arise from the antisymmetry of Riemann tensor R^L_KMN=-R^L_KNM. Therefore, this property ensures us that B_MM=0.
We can easily extract the Γ_M matrices from equation (<ref>). Hence, we have
_0=(
[                   m-r _rm/r^2                             0                             0                             0                            A'; r^2 _vm-(r-2 m) (r _rm-m)/r^3                   r _rm-m/r^2                             0                             0                         _ym/r;                             0                             0                             0                             0                             0;                             0                             0                             0                             0                             0;       e^2 A[(r-2 m) A'-_ym]/r                     -e^2 A A'                             0                             0                             0;                               ])
_1=(
[           0           0           0           0           0; r _rm-m/r^2           0           0           0          A';           0           0         1/r           0           0;           0           0           0         1/r           0;   -e^2 A A'           0           0           0           0;             ])
_2=(
[             0             0            -r             0             0;             0             0         2 m-r             0             0;             0           1/r             0             0            A';             0             0             0             θ             0;             0             0 -e^2 A r^2 A'             0             0;               ])
_3=(
[                      0                      0                      0             -r sin ^2θ                      0;                      0                      0                      0        (2 m-r) sin ^2θ                      0;                      0                      0                      0              -cosθsinθ                      0;                      0                    1/r                      θ                      0                     A';                      0                      0                      0 -e^2 A r^2 sin ^2θ  A'                      0;                        ])
_4=(
[    A'     0     0     0     0; _ym/r    A'     0     0     0;     0     0    A'     0     0;     0     0     0    A'     0;     0     0     0     0     0;       ])

The combination of equations (<ref>)-(<ref>) leads to the following B_MN matrices.


𝐁_01=-𝐁_10




    B_01=_1,0-_0,1+_0_1-_1_0

B_01=(
[                 -e^2 A A'^2+2 (m-r _rm)/r^3+_r^2m/r                                                   0                                                   0                                                   0                                                   0; -(r-2 m) [r (e^2 A r^2 A'^2+2 _rm-r _r^2m)-2 m]/r^4             e^2 A
   A'^2-r^2_r^2m -2 r_rm +2 m/r^3                                                   0                                                   0                                     _ym-r _r_ym/r^2;                                                   0                                                   0                                                   0                                                   0                                                   0;                                                   0                                                   0                                                   0                                                   0                                                   0;                              e^2 A(r _r_ym-_ym)/r^2                                                   0                                                   0                                                   0                                                   0;                                                     ])


𝐁_02=-𝐁_20




    B_02=_2,0-_0,2+_0_2-_2_0

B_02=(
[                                                               0                                                               0                                          _rm-e^2 A A'^2 r^3+m/r                                                               0                                                               0;                                                               0                                                               0                                              _vm-e^2 A r A' _ym                                                               0                                                               0; -e^2 A A' r^3[(r-2 m) A'-_ym] -r^2_vm +(r-2 m) (r
   _rm-m)/r^4                                      e^2 A A'^2 r^3-r_rm +m/r^3                                                               0                                                               0                                                        -_ym/r^2;                                                               0                                                               0                                                               0                                                               0                                                               0;                                                               0                                                               0                                                        e^2 A_ym                                                               0                                                               0;                                                                 ])


𝐁_03=-𝐁_30




    B_03=_3,0-_0,3+_0_3-_3_0

B_03=(
[                                                               0                                                               0                                                               0                              -sin ^2θ(e^2 A A'^2 r^3-r_rm +m)/r                                                               0;                                                               0                                                               0                                                               0                                    -sin ^2θ(e^2 A r A' _ym-_vm)                                                               0;                                                               0                                                               0                                                               0                                                               0                                                               0; -e^2 A A' r^3[(r-2 m) A'-_ym] -r^2_vm +(r-2 m) (r
   _rm-m)/r^4                                      e^2 A A'^2 r^3-r_rm +m/r^3                                                               0                                                               0                                                        -_ym/r^2;                                                               0                                                               0                                                               0                                                 e^2 Asin ^2θ_ym                                                               0;                                                                 ])


𝐁_04=-𝐁_40




    B_04=_4,0-_0,4+_0_4-_4_0

B_04=(
[                            r _r_ym-_ym/r^2                                          0                                          0                                          0                                   -A'^2-A”;                  (r-2 m) (r _r_ym-_ym)/r^3                            _ym-r _r_ym/r^2                                          0                                          0                           -2A' _ym+_y^2m/r;                                          0                                          0                                          0                                          0                                          0;                                          0                                          0                                          0                                          0                                          0; e^2 A[-(r-2 m) (A'^2+A”)+2 A' _ym+_y^2m]/r                             e^2 A(A'^2+A”)                                          0                                          0                                          0;                                            ])



𝐁_12=-𝐁_21


B_12=_2,1-_1,2+_1_2-_2_1=(
[                          0                          0                          0                          0                          0;                          0                          0     _rm-e^2 A A'^2 r^3+m/r                          0                          0; e^2 A A'^2 r^3-r_rm +m/r^3                          0                          0                          0                          0;                          0                          0                          0                          0                          0;                          0                          0                          0                          0                          0;                            ])


𝐁_13=-𝐁_31


B_13=_3,1-_1,3+_1_3-_3_1=(
[                                  0                                  0                                  0                                  0                                  0;                                  0                                  0                                  0 -sin ^2θ(e^2 A A'^2 r^3-r_rm +m)/r                                  0;                                  0                                  0                                  0                                  0                                  0;         e^2 A A'^2 r^3-r_rm +m/r^3                                  0                                  0                                  0                                  0;                                  0                                  0                                  0                                  0                                  0;                                    ])


𝐁_14=-𝐁_41


B_14=_4,1-_1,4+_1_4-_4_1=(
[              0              0              0              0              0;              0              0              0              0       -A'^2-A”;              0              0              0              0              0;              0              0              0              0              0; e^2 A(A'^2+A”)              0              0              0              0;                ])


𝐁_23=-𝐁_32


B_23=_3,2-_2,3+_2_3-_3_2=(
[                             0                             0                             0                             0                             0;                             0                             0                             0                             0                             0;                             0                             0                             0 sin ^2θ(2 m-e^2 A r^3 A'^2)/r                             0;                             0                             0          e^2 A r^2 A'^2-2 m/r                             0                             0;                             0                             0                             0                             0                             0;                               ])


𝐁_24=-𝐁_42


B_24=_4,2-_2,4+_2_4-_4_2=(
[                   0                   0                   0                   0                   0;                   0                   0                -_ym                   0                   0;             _ym/r^2                   0                   0                   0            -A'^2-A”;                   0                   0                   0                   0                   0;                   0                   0 e^2 A r^2 (A'^2+A”)                   0                   0;                     ])


𝐁_34=-𝐁_43


B_34=_4,3-_3,4+_3_4-_4_3=(
[                          0                          0                          0                          0                          0;                          0                          0                          0                -sin ^2θ_ym                          0;                          0                          0                          0                          0                          0;                    _ym/r^2                          0                          0                          0                   -A'^2-A”;                          0                          0                          0 e^2 A r^2 sin ^2θ(A'^2+A”)                          0;                            ])
Finally, knowing the matrices B_MN from equations (<ref>)-(<ref>) and using the fact that the L-th row and K-th column of the matrix B_MN gives the R^L_KMN component of the Riemann tensor, it is quite easy to extract the components of the Riemann tensor, all non-zero components are depicted below. 
[box=]equation
[                          R^0_001=-R^0_010=2 (m-r _rm)/r^3+^2_rm/r-e^2 A A'^2                                             R^0_004=-R^0_040=r _r_ym-_ym/r^2;                                                                             ;                                      R^0_202=-R^0_220=_rm-m+r^3 e^2 A A'^2/r                          R^0_303=-R^0_330=-sin^2θ(-r _rm+m+r^3 e^2 A A'^2)/r;                                                                             ;                                              R^0_404=-R^0_440=-A”(y)-A'(y)^2                                   R^1_004=-R^1_040=(r-2 m) (r _r_ym-_ym)/r^3;                                                                             ;         R^1_001=-R^1_010=-(r-2 m) [r (2 _rm-r ^2_rm+r^2 e^2 A A'^2)-2 m]/r^4                        R^1_101=-R^1_110=e^2 A A'^2-r^2 _r^2m-2 r _rm+2 m/r^3;                                                                             ;                                             R^1_104=-R^1_140=_ym-r _r_ym/r^2                                          R^1_202=-R^1_220=_vm-r e^2 A A' _ym;                                                                             ;                                      R^1_212=-R^1_221=_rm-m+r^3 e^2 A A'^2/r                                                        R^1_224=-R^1_242=-_ym;                                                                             ;                                 R^1_303=-R^1_330=-sin^2θ(r e^2 A A' _ym-_vm)                          R^1_313=-R^1_331=-sin^2θ(-r _rm+m+r^3 e^2 A A'^2)/r;                                                                             ;                                                  R^1_334=-R^1_343=-sin^2θ_ym                                             R^1_401=-R^1_410=_ym-r _r_ym/r^2;                                                                             ;                                           R^1_404=-R^1_440=-2 A' _ym+^2_ym/r                                                    R^1_414=-R^1_441=-A”-A'^2;                                                                             ; R^2_002=-R^2_020=-r^3e^2 A A' [A' (r-2 m)-_ym]-r^2 _vm+(r-2 m) (r _rm-m)/r^4                                 R^2_012=-R^2_021=-r _rm+m+r^3 e^2 A A'^2/r^3;                                                                             ;                                                     R^2_024=-R^2_042=_ym/r^2                                 R^2_102=-R^2_120=-r _rm+m+r^3 e^2 A A'^2/r^3;                                                                             ;                                R^2_323=-R^2_332=sin^2θ(2 m-r^3 e^2 A A'^2)/r                                                    R^2_402=-R^2_420=-_ym/r^2;                                                                             ;                                                    R^2_424=-R^2_442=-A”-A'^2                                 R^3_013=-R^3_031=-r _rm+m+r^3 e^2 A A'^2/r^3;                                                                             ;  R^3_003=-R^3_030=-r^3e^2 A A' [A' (r-2m)-_ym]-r^2 _vm+(r-2 m) (r _rm-m)/r^4                                                     R^3_034=-R^3_043=_ym/r^2;                                                                             ;                                 R^3_103=-R^3_130=-r _rm+m+r^3 e^2 A A'^2/r^3                                        R^3_223=-R^3_232=r^2 e^2 A A'^2-2 m/r;                                                                             ;                                                    R^3_403=-R^3_430=-_ym/r^2                                                    R^3_434=-R^3_443=-A”-A'^2;                                                                             ;                                     R^4_001=-R^4_010=e^2 A (r _r_ym-_ym)/r^2                                             R^4_014=-R^4_041=e^2 A (A”+A'^2);                                                                             ;                    R^4_004=-R^4_040=e^2 A [2 A'_ym+(A”+A'^2) (2m-r)+_y^2m]/r                                             R^4_104=-R^4_140=e^2 A (A”+A'^2);                                                                             ;                                                   R^4_202=-R^4_220=e^2 A _ym                                         R^4_224=-R^4_242=r^2 e^2 A (A”+A'^2);                                                                             ;                                             R^4_303=-R^4_330=e^2 A sin^2θ_ym                                   R^4_334=-R^4_343=r^2 e^2 A sin^2θ(A”+A'^2) ]



§ RIEMANN SCALAR

Riemann scalar is defined as
R^ABCDR_ABCD
Therefore, it is necessary to calculate the four-times contravariant and four-times covariant components of Riemann tensor in order to be able to execute the above contraction.
Contravariant and covariant components of Riemann tensor have some very useful properties that reduce the number of independent components. Hence, we present these properties below in order to use them afterwards.


  * R^ABCD=-R^BACD=-R^ABDC

  * R^ABCD=R^CDAB

  * R_ABCD=-R_BACD=-R_ABDC

  * R_ABCD=R_CDAB

Knowing the components R^L_KMN of the Riemann tensor, the four-times contravariant components are given by the equation
R^ABCD=R^A_KMNg^KBg^MCg^ND
and the non-zero four-times contravariant components are
[                             R^0101=-R^1001=-R^0110=R^1010=e^-6 A(e^2 A A'^2-r^2 _r^2m-2 r _rm+2 m/r^3);                                                                                                       ;                    R^0114=-R^1014=-R^0141=R^1401=-R^4101=-R^1410=R^1041=R^4110=e^-4 A(r _r_ym-_ym)/r^2;                                                                                                       ;       R^0212=-R^2012=-R^0221=R^1202=-R^2102=-R^1220=R^2021=R^2120=-e^-6 A(-r _rm+m+r^3 e^2 A A'^2)/r^5;                                                                                                       ;   R^0313=-R^3013=-R^0331=R^1303=-R^3103=-R^1330=R^3031=R^3130=-e^-6 A ^2θ(-r _rm+m+r^3 e^2 A A'^2)/r^5;                                                                                                       ;                           R^0414=-R^4014=-R^0441=R^1404=-R^4104=-R^1440=R^4041=R^4140=-e^-2 A(A”+A'^2);                                                                                                       ;       R^1212=-R^2112=-R^1221=R^2121=e^-6 A[(r-2 m) (_rm-m+r^3 e^2 A A'^2/r)+r (_vm-r e^2 A A'_ym)]/r^5;                                                                                                       ;                             R^1224=-R^2124=-R^1242=R^2412=-R^4212=-R^2421=R^2142=R^4241=-e^-4 A_ym/r^4;                                                                                                       ; R^1313=-R^3113=-R^1331=R^3131=e^-6 A ^2θ[r^2(_vm-r e^2 A A' _ym)-(r-2 m) (-r _rm+m+r^3 e^2A A'^2)]/r^6;                                                                                                       ;                         R^1334=-R^3134=-R^1343=R^3413=-R^4313=-R^3431=R^3143=R^4331=-e^-4 A ^2θ_ym/r^4;                                                                                                       ;                                        R^2323=-R^3223=-R^2332=R^3232=e^-6 A ^2θ(2 m-r^3 e^2 AA'^2)/r^7;                                                                                                       ;                                                     R^2424=-R^4224=-R^2442=R^4242=-e^-2 A(A”+A'^2)/r^2;                                                                                                       ;                                                 R^3434=-R^4334=-R^3443=R^4343=-e^-2 A ^2θ(A”+A'^2)/r^2;                                                                                                       ;                               R^1414=-R^4114=-R^1441=R^4141=-e^-2 A[2 A' _ym+(A”+A'^2) (r-2m)+_y^2m]/r;                                                                                                       ;                                                                                                        ]
The four-times covariant components are given by the equation
R_ABCD=g_ALR^L_BCD
and the non-zero four-times covariant components are
[                            R_0101=-R_1001=-R_0110=R_1010=e^4 A A'^2-e^2 A[r (r_r^2m-2 _rm)+2 m]/r^3;                                                                                                    ;                  R_0104=-R_1004=-R_0140=R_0401=-R_4001=-R_0410=R_1040=R_4010=e^2 A(_ym-r _r_ym)/r^2;                                                                                                    ;        R_0202=-R_2002=-R_0220=R_2020=e^2 A[(r-2 m) (-r _rm+m+r^3 e^2 A A'^2)/r^2-re^2 A A' _ym+_vm];                                                                                                    ;           R_0212=-R_2012=-R_0221=R_1202=-R_2102=-R_1220=R_2021=R_2120=e^2 A(_rm-m+r^3 e^2 A A'^2/r);                                                                                                    ;                               R_0224=-R_2024=-R_0242=R_2402=-R_4202=-R_2420=R_2042=R_4220=-e^2 A_ym;                                                                                                    ; R_0303=-R_3003=-R_0330=R_3030=e^2 Asin ^2θ[(r-2 m) (-r _rm+m+r^3 e^2 A A'^2)/r^2-re^2 A A' _ym+_vm];                                                                                                    ; R_0313=-R_3013=-R_0331=R_1303=-R_3103=-R_1330=R_3031=R_3130=-e^2 Asin ^2θ(-r_rm+m+r^3 e^2 A A'^2)/r;                                                                                                    ;                        R_0334=-R_3034=-R_0343=R_3403=-R_4303=-R_3430=R_3043=R_4330=-e^2 Asin ^2θ_ym;                                                                                                    ;                             R_0404=-R_4004=-R_0440=R_4040=e^2 A[-2 A'_ym+(A”+A'^2) (r-2 m)-_y^2m]/r;                                                                                                    ;                         R_0414=-R_4014=-R_0441=R_1404=-R_4104=-R_1440=R_4041=R_4140=-e^2 A(A”+A'^2);                                                                                                    ;                                    R_2323=-R_3223=-R_2332=R_3232=r e^2 Asin ^2θ(2 m-r^3 e^2 A A'^2);                                                                                                    ;                                                    R_2424=-R_4224=-R_2442=R_4242=-r^2e^2 A(A”+A'^2);                                                                                                    ;                                             R_3434=-R_4334=-R_3443=R_4343=-r^2e^2 Asin ^2θ(A”+A'^2);                                                                                                    ;                                                                                                     ]
Possessing all non-zero contravariant and covariant components of Riemann tensor from equations (<ref>) and (<ref>) respectively, we are now able to calculate the Riemann scalar.
R^ABCDR_ABCD   =   4R^0101R_0101+8R^0212R_0212+8R^0313R_0313+8R^0414R_0414
      +4R^2323R_2323+4R^2424R_2424+4R^3434R_3434
All the other combinations of contraction of the indices ABCD nullify either R^ABCD or R_ABCD.
[box=]equation
[ R^ABCDR_ABCD=40A'^4+32A'^2A”+16A”^2+48e^-4Am^2/r^6-8e^-2AA'^2/r(_r^2m+2_rm/r);                                                                              ;           +4e^-4A/r^2[(_r^2m)^2+4m/r^2(_r^2m-4_rm/r)-4_rm_r^2m/r+8(_r m)^2/r] ]



§ RICCI TENSOR AND RICCI SCALARS

The two-times covariant components of Ricci tensor is defined by
R_MN=R^L_MLN=R^0_M0N+R^1_M1N+R^2_M2N+R^3_M3N+R^4_M4N
Subsequently, the non-zero components of the Ricci tensor are
[box=]equation
[ R_00=e^2 A r [(r-2 m) (4 A'^2+A”)-4 A' _ym-_y^2m]-(r-2 m)_r^2m+2 _vm/r^2;                                                                         ;                                      R_01=R_10=_r^2m/r-e^2 A (4 A'^2+A”);                                                                         ;                                                R_04=R_40=_ym+r _r_ym/r^2;                                                                         ;                                         R_22=2 _rm-e^2 A r^2 (4 A'^2+A”);                                                                         ;                                R_33= sin^2θ[2 _rm-e^2 A r^2 (4 A'^2+A”)];                                                                         ;                                                        R_44=-4 (A'^2+A”) ]
There are two Ricci scalars that one can evaluate in order to extract information for the curvature of spacetime. The first one is defined as R=R_MNg^MN and the second one is defined as R_MNR^MN. It is obvious that for the evaluation of the latter scalar the two-times contravariant components of Ricci tensor are necessary. The following table depicts all the non-zero contravariant components of Ricci tensor. The evaluation is made by the following equation
R^MN=R_ABg^AMg^BN
and the in between analytical actions are skipped here as well. Therefore, we have
[box=]equation
[                                     R^01=R^10=e^-4 A [_r^2m/r-e^2 A (4 A'^2+A”)];                                                                                 ; R^11=e^-4 A [-e^2 A r ((r-2m) (4 A'^2+A”)+4 A'_ym+_y^2m)+(r-2 m) _r^2m+2_vm]/r^2;                                                                                 ;                                               R^14=R^41=e^-2 A (_ym+r _r_ym)/r^2;                                                                                 ;                                    R^22=e^-4 A [2 _rm-e^2 A r^2 (4 A'^2+A”)]/r^4;                                                                                 ;                                 R^33=e^-4 A ^2θ[2 _rm-e^2 A r^2 (4 A'^2+A”)]/r^4;                                                                                 ;                                                               R^44= -4 (A'^2+A”) ]
We now have everything that is needed for the evaluation of the two aforementioned Ricci scalars.
[box=]equation
R=R_MNg^MN=-20 A'^2-8 A”+2 e^-2 A/r (_r^2m+2 _rm/r)

[box=]equation
R_MNR^MN=80A'^2+64A'^2 A”+20A”^2-4 e^-2 A/r(_r^2m+2_rm/r)(4A'^2+A”)+ 2e^-4A/r^2[(_r^2m)^2+4 (_rm)^2/r^2]




§ EINSTEIN TENSOR

Einstein tensor G_MN is defined through Ricci tensor R_MN and Ricci scalar R as follows
G_MN=R_MN-1/2g_MNR 
The combination of equations (<ref>),(<ref>) and (<ref>) can be done easily, thus, after a bit of algebra we are led to the following non-zero components of Einstein tensor.
[box=]equation
[ G_00=-e^2 A [3 (r-2 m) (2 A'^2+A”)+4 A' _ym+_y^2m] r^2+2 r_vm +2(r-2 m)_r m/r^3;                                                                                ;                                         G_01=G_10=3 e^2 A (2 A'^2+A”)-2 _rm/r^2;                                                                                ;                                                       G_04=G_40=_ym+r_r_y m/r^2;                                                                                ;                                            G_22=r [3 e^2 A r (2 A'^2+A”)-_r^2m];                                                                                ;                                      G_33=r sin^2θ[3 e^2 A r (2 A'^2+A”)-_r^2m];                                                                                ;                                          G_44=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]

The mixed components of Einstein tensor G^M_N are given by
G^M_N=g^MAG_AN
Equations (<ref>),(<ref>) and (<ref>) yield to
[box=]equation
[         G^0_0=6 A'^2+3 A”-2 e^-2 A _rm/r^2;                                           ; G^1_0=-4 r A' _ym+r _y^2m-2 e^-2 A _vm/r^2;                                           ;         G^1_1=6 A'^2+3 A”-2 e^-2 A_r m/r^2;                                           ;             G^1_4=e^-2 A (_ym+r _r_ym)/r^2;                                           ;           G^2_2=6 A'^2+3 A”-e^-2 A_r^2 m/r;                                           ;           G^3_3=6 A'^2+3 A”-e^-2 A _r^2m/r;                                           ;                      G^4_0=_ym+r _r_ym/r^2;                                           ;    G^4_4=6 A'^2-e^-2 A (2 _rm+r_r^2 m)/r^2 ]




§ MATHEMATICA CODE

The mathematica code that is used to verify all the above calculations is illustrated below.



Definition of coordinates and metric tensor



n=5;

coord={v,r,θ ,φ ,y};

g={{-Exp[2A[y]](1-2m[v,r,y]/r),Exp[2A[y]],0,0,0},{Exp[2A[y]],0,0,0,0},
{0,0,Exp[2A[y]]*r^∧2,0,0},{0,0,0,Exp[2A[y]]*r^∧2*Sin[θ ]^∧2,0},{0,0,0,0,1}};

StringJoin[Characters[(g_MN)=]]MatrixForm[g]

invg=Simplify[Inverse[g]];

StringJoin[Characters[(g^MN)=]]MatrixForm[invg]



Christoffel Symbols:   ^𝐋_𝐌𝐍



christoffel=FullSimplify[Table[(1/2)*Sum[(invg[[λ ,ρ ]])*(D[g[[μ ,ρ ]],coord[[ν
]]]
+D[g[[ν ,ρ ]],coord[[μ ]]]-D[g[[μ ,ν ]],coord[[ρ ]]]),{ρ ,1,n}],{λ ,1,n},{μ ,1,n},{ν
,1,n}]];

chr=Table[If[UnsameQ[christoffel[[λ ,μ ,ν ]],0],{ToString[Γ [λ -1,μ -1,ν
-1]],
christoffel[[λ ,μ ,ν ]]}],{λ ,1,n},{μ ,1,n},{ν ,1,n}];

TableForm[Partition[DeleteCases[Flatten[chr],Null],2],TableSpacing→{2,2}]



Riemann Tensor's Components:    𝐑^𝐊_𝐋𝐌𝐍



riemann=FullSimplify[Table[D[christoffel[[α ,μ ,σ ]],coord[[ρ ]]]-D[christoffel[[α
,μ ,ρ ]],coord[[σ ]]]
+Sum[christoffel[[α ,ρ ,λ ]]*christoffel[[λ ,μ ,σ ]],{λ ,1,n}]-Sum[christoffel[[α
,σ ,λ ]]
*christoffel[[λ ,μ ,ρ ]],{λ ,1,n}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}]];

rie=Table[If[UnsameQ[riemann[[α ,μ ,ρ ,σ ]],0],{ToString[R[α -1,μ -1,ρ -1,σ
-1]],
riemann[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];

TableForm[Partition[DeleteCases[Flatten[rie],Null],2],TableSpacing→{2,2}]



Contravariant Components of Riemann Tensor:    𝐑^𝐀𝐁𝐂𝐃



riemanncon=FullSimplify[Table[Sum[Sum[Sum[riemann[[μ ,α ,β ,γ ]]*invg[[α
,ν ]],{α ,1,n}]*
invg[[β ,κ ]],{β ,1,n}]*invg[[γ ,λ ]],{γ ,1,n}],{μ ,1,n},{ν ,1,n},{κ
,1,n},{λ ,1,n}]];

riecon=Table[If[UnsameQ[riemanncon[[α ,μ ,ρ ,σ ]],0],{ToString[Rcon[α
-1,μ -1,ρ -1,σ -1]],
riemanncon[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];

TableForm[Partition[DeleteCases[Flatten[riecon],Null],2],TableSpacing→{2,2}]



Covariant Components of Riemann Tensor:    𝐑_𝐀𝐁𝐂𝐃



riemanncov=FullSimplify[Table[Sum[g[[μ ,δ ]]*riemann[[δ ,ν ,κ ,λ ]],
{δ ,1,n}],{μ ,1,n},{ν ,1,n},{κ ,1,n},{λ ,1,n}]];

riecov=Table[If[UnsameQ[riemanncov[[α ,μ ,ρ ,σ ]],0],{ToString[Rcov[α
-1,μ -1,ρ -1,σ -1]],
riemanncov[[α ,μ ,ρ ,σ ]]}],{α ,1,n},{μ ,1,n},{ρ ,1,n},{σ ,1,n}];

TableForm[Partition[DeleteCases[Flatten[riecov],Null],2],TableSpacing→{2,2}]



Covariant Components of Ricci Tensor:    𝐑_𝐌𝐍



ricci=FullSimplify[Table[Sum[riemann[[μ ,α ,μ ,β ]],{μ ,1,n}],{α ,1,n},{β
,1,n}]];

StringJoin[Characters[(R_MN)=]]MatrixForm[ricci]



Contravariant Components of Ricci Tensor:    𝐑^𝐌𝐍



riccicon=FullSimplify[Table[Sum[ricci[[α ,β ]]*invg[[α ,μ ]]*invg[[β
,ν ]],{α ,1,n},{β ,1,n}],{μ ,1,n},
{ν ,1,n}]];

StringJoin[Characters[(R^MN)=]]MatrixForm[riccicon]



Ricci and Riemann Scalars



scalarricci=FullSimplify[Sum[invg[[μ ,ν ]]*ricci[[μ ,ν ]],{μ ,1,n},{ν ,1,n}]];

StringJoin[Characters[R_MNg^MN=]].scalarricci

scalarricci2=FullSimplify[Sum[ricci[[μ ,ν ]]*riccicon[[μ ,ν ]],{μ ,1,n},{ν ,1,n}]];

StringJoin[Characters[R_MNR^MN=]] .scalarricci2

StringJoin[Characters[R_ABCDR^ABCD=]] .FullSimplify[Sum[riemanncov[[μ
,ν ,κ ,λ ]]*
riemanncon[[μ ,ν ,κ ,λ ]],{μ ,1,n},{ν ,1,n},{κ ,1,n},{λ ,1,n}]]



Covariant Components of Einstein Tensor:   𝐆_𝐌𝐍



einstein=FullSimplify[Table[ricci[[μ ,ν ]]-(1/2)g[[μ ,ν ]] *scalarricci,{μ ,1,n},{ν
,1,n}]];

StringJoin[Characters[(G_MN)=]]MatrixForm[einstein]



Mixed Components of Einstein Tensor:   𝐆^𝐌_𝐍



einsteinUD=FullSimplify[Table[Sum[invg[[μ ,λ ]]*einstein[[λ ,ν ]],{λ
,1,n}],{μ ,1,n},{ν ,1,n}]];

StringJoin[Characters[(G_N^M)=]]MatrixForm[einsteinUD]






CHAPTER: NON-MINIMAL COUPLING AND VARIATION OF THE ACTION





§ VARIATION WITH RESPECT TO THE METRIC TENSOR


We consider the following general action for a non-minimally coupled scalar field.
S=∫ d^4x√(-g)[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+∫ d^4x√(-g) _m
where _m describes matter and/or radiation. It is obvious from the above action that we are restricted in a 4-D spacetime. Every calculation and proof in this section is going to be done in 4-D spacetime. However, the generalization for extra spatial dimensions is instantaneous, the only thing that changes in the final field equations are the indices. For a 4-D spacetime the indices are Greek letters while for (4+n)-D spacetime the indices are capital Latin letters. 
In order to deduce the field equations of this general theory, we apply the principle of least action to Eq.(<ref>). We vary the action with respect to the metric tensor. Thus, we have:

    S=0   =∫ d^4x{(√(-g))[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+√(-g)[f(Φ)/2κ R-1/2(_μΦ^μΦ)]}
       +∫ d^4x (√(-g) _m)
    
    0   =∫ d^4x{(√(-g))[f(Φ)/2κ R-1/2_μΦ^μΦ-V(Φ)-Λ_B]+√(-g)[f(Φ)/2κ R-1/2(_μΦ_νΦ) g^μν]}
       +∫ d^4x (√(-g) _m)

We will calculate one by one the varying terms of the previous equation. First of all, we will prove the Jacobi's formula

    d/dt{det[A(t)]}=tr{adj[A(t)]dA(t)/dt}
 
which is necessary in order to derive the desirable field equations. We will prove Eq.(<ref>) in two steps. At first, we prove a preliminary lemma.
 Let A and B be a pair of square matrices of the same dimension n. Then

    ∑_i=1^n ∑_j=1^n A_ijB_ij=tr(A^TB)




    (AB)_jk   =∑_i=1^n A_jiB_ik
    
    
    (A^TB)_jk   =∑_i=1^n A^T_jiB_ik=∑_i=1^n A_ijB_ik
    
    
    tr(A^TB)   =∑_j=1^n (A^TB)_jj=∑_j=1^n∑_i=1^n A_ijB_ij=∑_i=1^n∑_j=1^nA_ijB_ij


(Jacobi's formula):

    d[det(A)]=tr[adj(A)dA]



Laplace's formula (or cofactor expansion) for the determinant of a matrix A can be stated as

    det(A)=∑^n_j=1 A_ij(-1)^(i+j)M_ij=∑^n_j=1 A_ijC_ij=∑^n_j=1 A_ij[adj(A)]_ji=∑^n_j=1 A_ij[adj^T(A)]_ij


where M_ij is the i, j minor matrix of A, that is, the determinant of the (n-1)×(n-1) matrix that results from deleting the i-th row and j-th column of A. The summation is performed over some arbitrary row i of the matrix. The i, j cofactor of A is the scalar C_ij defined by C_ij≡(-1)^i+jM_ij. We note as well that the adjugate matrix of A is the transpose of the cofactor matrix C of A, which means that adj(A)≡ C^T [adj(A)]_ij=C_ji.
The determinant of A can also be considered to be a function of elements of A:

    det(A)=F(A_11,A_12,...,A_21,...,A_nn)

so that, by the chain rule, its differential is

    d[det(A)]=∑_i,j∂ F/∂ A_ijdA_ij=∑_i,j∂ det(A)/∂ A_ijdA_ij


    ∂ det(A)/∂ A_ij   =∂/∂ A_ij(∑_k A_ik[adj^T(A)]_ik)=∑_k∂(A_ik[adj^T(A)]_ik)/∂ A_ij
       =∑_k∂ A_ik/∂ A_ij__kj[adj^T(A)]_ik+∑_k A_ik∂ [adj^T(A)]_ik/∂ A_ij_=0=∑_kδ_kj[adj^T(A)]_ik
       =[adj^T(A)]_ij

Hence, we have

    d[det(A)]=∑_i,j∂ det(A)/∂ A_ijdA_ij=∑_i,j [adj^T(A)]_ijdA_ij=tr[adj^TdA]=tr[adj(A)dA]

Consequently, Eq.(<ref>) is proved.

We are now ready to calculate the variations of Eq.(<ref>).


  * δ (√(-g)):

    δ (√(-g))=1/21/√(-g)δ (-g)=-1/21/√(-g)δ g


It is important to explain the following notation
g≡ det(g_μν)
where (g_μν) constitutes the matrix which depicts the components of metric tensor g_μν.
From equations (<ref>) and (<ref>) we obtain

    δ g=δ[det(g_μν)]=∑_,β[adj^T(g_μν)]_β δ g_β

In matrix calculus it is well known that A^-1=1/det(A)adj(A), hence we have

    (g_μν)^-1=(g^μν)=1/det(g_μν)adj(g_μν)

adj(g_μν)=g(g^μν)=adj^T(g_μν)⇒ [adj^T(g_μν)]_β=g g^β

Thus, from Eq.(<ref>) and Eq.(<ref>) we get

    δ g=∑_,βg g^βδ g_β⇒δ g=g g^μνδ g_μν

It also obvious that
δ(g_μνg^νλ)   =   δ(δ^λ_μ)=0 ⇒
δ(g_μν)g^νλ   =   -g_μνδ (g^νλ)⇒

    δ g^ρσ=-g^ρμg^σνδ g_μν ,  δ g_αβ=-g_αμg_βνδ g^μν


Combining now equations (<ref>) and (<ref>) we have

    δ g=g g^μνδ g_μν=-g g_αβδ g^αβ

Therefore, we can now easily evaluate the quantity (√(-g)).
δ (√(-g))=-1/21/√(-g)g g^μνδ g_μν=1/21/√(-g)g g_αβδ g^αβ

      δ (√(-g))=√(-g)/2 g^μνδ g_μν=-√(-g)/2 g_αβδ g^αβ


  * δ R:

 
In order to calculate δ R we need to prove first the Palatini's identity:

    δ R_μν=(δΓ^λ_μν)_;λ-(δΓ^λ_λμ)_;ν=∇_(δΓ^λ_μν)-∇_ν(δΓ^λ_λμ)



    R_μν=Γ^λ_μν ,λ-Γ^λ_μλ ,ν+Γ^α_μνΓ^λ_αλ-Γ^λ_ανΓ^α_μλ⇒


    δ R_μν=δΓ^λ_μν ,λ   -δΓ^λ_μλ,ν+δ (Γ^α_μν)Γ^λ_αλ+Γ^α_μνδ (Γ^λ_αλ)
       -δ (Γ^λ_αν)Γ^α_μλ-Γ^λ_ανδ (Γ^α_μλ)

Christoffel symbols of first kid _μν is defined by
Γ_μνσ   ≡    g_μλΓ^λ_νσ=1/2g_μg^ρ(g_νρ,+g_ρ,ν-g_ν,ρ)=1/2^ρ_μ(g_νρ,+g_ρ,ν-g_ν,ρ)
   =   1/2(g_μν ,σ+g_μσ ,ν-g_νσ ,μ)
where we have used the expression of Christoffel symbols of second kid (or simply Christoffel symbols) which are well known. Using Eq.(<ref>) we have

    δ(Γ^λ_νσ)   =δ(g^λμΓ_μνσ)=δ(g^λμ)Γ_μνσ+g^λμδΓ_μνσ=-g^λαg^μβδ (g_αβ)Γ_μνσ+g^λμδΓ_μνσ
       =-g^λαδ (g_αβ)Γ^β_νσ+g^λμ δ[1/2(g_μν ,σ+g_μσ ,ν-g_νσ ,μ)] 
       =-g^λμδ (g_μβ)Γ^β_νσ+g^λμ1/2(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ) 
       =1/2g^λμ(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ-2Γ^β_νσδ g_μβ) 
       =1/2g^λμ(δ g_μν ,σ+δ g_μσ ,ν-δ g_νσ ,μ-2Γ^β_νσδ g_μβ-Γ^β_νμδ g_σβ
       +Γ^β_νμδ g_σβ-Γ^β_σμδ g_νβ+Γ^β_σμδ g_νβ) 
       =1/2g^λμ[(δ g_μν ,σ-Γ^β_σμδ g_νβ-Γ^β_νσδ g_μβ)+(δ g_μσ ,ν-Γ^β_νμδ g_σβ-Γ^β_νσδ g_μβ)
       -(δ g_νσ ,μ-Γ^β_νμδ g_σβ-Γ^β_σμδ g_νβ)] 
       =1/2g^μ(∇_ g_μν+∇_ν g_μ-∇_μ g_ν)


It is clear that elements like g_μν and g_μν; are tensors. Thus, from Eq.(<ref>) we can easily deduce that ^_μν is also a tensor, because ^_μν is expressed as a linear combination of tensors. Moreover, the quantity (^_μν)_;-(^_μ)_;ν constitutes a tensor. Performing the expansion of the expression (^_μν)_;-(^_μ)_;ν we obtain

    (δΓ^λ_μν)_;λ-(δΓ^λ_μλ)_;ν=   +(δΓ^λ_μν ,λ+Γ^λ_λρδΓ^ρ_μν-Γ^ρ_μλδΓ^λ_ρν-Γ^ρ_νλδΓ^λ_ρμ) 
       -(δΓ^λ_μλ ,ν+Γ^λ_νρδΓ^ρ_μλ-Γ^ρ_μνδΓ^λ_ρλ-Γ^ρ_νλδΓ^λ_ρμ) 
    
    =   +(δΓ^λ_μν ,λ+Γ^λ_λρδΓ^ρ_μν-Γ^ρ_μλδΓ^λ_ρν) 
       -(δΓ^λ_μλ ,ν+Γ^λ_νρδΓ^ρ_μλ-Γ^ρ_μνδΓ^λ_ρλ)

The right hand side of Eq.(<ref>) is identical to the right hand side of Eq.(<ref>), so the left hand sides should be equal to each other as well. Hence, we derived the desirable Palatini's identity.

The calculation of the quantity R using the Eq.(<ref>) is very simple.

    δ R   =δ (g^μνR_μν)=(δ g^μν)R_μν+g^μνδ R_μν
       =(δ g^μν)R_μν+g^μν(∇_δΓ^λ_μν-∇_νδΓ^λ_μλ)

We have shown that 

    δΓ^λ_νσ   =1/2g^μ(∇_ g_μν+∇_ν g_μ-∇_μ g_ν)
       =1/2g^ρ(∇_ g_ρν+∇_ν g_ρ-∇_ρ g_ν)

Using Eq.(<ref>) we rename the index  to μ. Furthermore, using the symmetry ^_νμ=^_μν we get
δΓ^λ_μν=1/2g^ρ(∇_μ g_ρν+∇_ν g_ρμ-∇_ρ g_νμ)
Contracting the index  with ν we have
δΓ^λ_μ=δΓ^λ_μ   =   1/2g^ρ(∇_μ g_ρ+∇_ g_ρμ-∇_ρ g_μ)
   =   1/2g^ρ∇_μ g_ρ+1/2(g^ρ∇_ g_ρμ-g^ρ∇_ρ g_μ)
   =   1/2g^ρ∇_μ g_ρ+1/2(g^ρ∇_ρ g_μ-g^ρ∇_ρ g_μ_0)
   =   1/2g^ρ∇_μ g_ρ
We combine equations (<ref>), (<ref>) and (<ref>). Thus, we get
R_μν   =   ∇_[1/2g^ρ(∇_μ g_ρν+∇_ν g_ρμ-∇_ρ g_νμ)]-∇_ν(1/2g^ρ∇_μ g_ρ)
   =   1/2g^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ)-1/2g^ρ∇_ν∇_μ g_ρ
   =   1/2g^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ-∇_ν∇_μ g_ρ)
Previously, we used the property ∇_ g^μν=0. The proof of this property is presented below.



    ∇_ g^μν   =g^μν_,+^μ_ρ g^ρν+^ν_ρ g^ρμ
       =g^μν_,+1/2g^μκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρν+1/2g^νκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρμ
       =g^μν_,+1/2g^μκ(g_κ,ρ+g_ρκ,-g_ρ,κ)g^ρν+1/2g^νρ(g_ρ,κ+g_κρ,-g_κ,ρ)g^κμ
       =g^μν_,+g^μκ g_κρ, g^ρν=g^μν_,+(g^μκ g_κρ)_,_^μ_ρ,=0 g^ρν-g^μκ_, g_κρ g^ρν__κ^ν
       =g^μν_,-g^μν_,=0


g^μν R_μν   =   1/2g^μνg^ρ(∇_∇_μ g_ρν+∇_∇_ν g_ρμ-∇_∇_ρ g_νμ-∇_ν∇_μ g_ρ)
   =   1/2(g^μνg^ρ∇_∇_μ g_ρν+g^μνg^ρ∇_∇_ν g_ρμ-g^μνg^ρ∇_∇_ρ g_νμ-g^μνg^ρ∇_ν∇_μ g_ρ)
   =   1/2(g^νμg^ρ∇_∇_ν g_ρμ+g^μνg^ρ∇_∇_ν g_ρμ-2g^μνg^ρ∇_∇_ρ g_νμ)
   =   1/2(2g^ρg^νμ∇_∇_ν g_ρμ-2g^μνg^ρ∇_∇_ρ g_νμ)
   =   g^ρg^μν∇_∇_μ g_ρν-g^μνg^ρ∇_∇_ρ_□ g_μν=∇^ρ∇^ν g_ρν-g^μν□ g_μν
Using Eq.(<ref>) into Eq.(<ref>) we obtain
g^μν R_μν   =   g^ρg^μν∇_∇_μ(-g_ρg_νβ g^β)-g^μνg^ρ∇_∇_ρ(-g_μg_νβ g^β)
   =   -g^ρg^μνg_ρg_νβ∇_∇_μ g^β+g^μνg^ρg_μg_νβ∇_∇_ρ g^β
   =   -^_^μ_β∇_∇_μ g^β+g^ρ^ν_g_νβ∇_∇_ρ g^β
   =   -∇_∇_β g^β+g_βg^ρ∇_∇_ρ_□ g^β
   =   -∇_∇_β g^β+g_β□ g^β
Putting together Eq.(<ref>) and Eq.(<ref>) we have
g^μν R_μν=∇^μ∇^ν g_μν-g^μν□ g_μν=-∇_μ∇_ν g^μν+g_μν□ g^μν
From Eq.(<ref>) and Eq.(<ref>) we conclude that

    δ R=δ (g^μν)R_μν-∇_μ∇_ν g^μν+g_μν□ g^μν


  * (√(-g)_m):
 

The energy-momentum tensor is defined by
T^(m)_μν≡-2/√(-g)(√(-g)_m)/ g^μν
Thus, we have
(√(-g)_m)=-1/2 T^(m)_μν√(-g)  g^μν


Replacing the right hand sides from equations (<ref>), (<ref>), (<ref>) into Eq.(<ref>). We get


    0=    -1/2∫ d^4x √(-g) g_μν  g^μν[f(Φ)/2κ R-1/2_Φ^Φ-V(Φ)-Λ_B]
        +1/2∫ d^4x√(-g) {1/κ[ g^μν f(Φ)R_μν-f(Φ)∇_μ∇_ν  g^μν+g_μνf(Φ)□  g^μν]- g^μν_μΦ_νΦ}
        -1/2∫ d^4x T^(m)_μν√(-g)  g^μν


    0=   ∫ d^4x √(-g)  g^μν{g_μν[-f(Φ)/2κ R+1/2_Φ^Φ+V(Φ)+Λ_B]+f(Φ)/κR_μν-_μΦ_νΦ-T^(m)_μν}
        +1/κ∫ d^4x√(-g) (-f(Φ)∇_μ∇_ν  g^μν+g_μνf(Φ)□  g^μν)

The integral in the second line of Eq.(<ref>) can be modified to a more useful one, but first it is necessary to prove one general property that will help us doing the modification. 


∇_μ A^μ=_μ A^μ+^μ_μA^


    ^μ_μ   =1/2g^μρ(g_ρ,μ+g_μρ,-g_μ,ρ)=1/2g^μρg_ρ,μ+1/2g^μρg_μρ,-1/2g^μρg_μ,ρ
       =1/2g^μρg_ρ,μ+1/2g^μρg_μρ,-1/2g^ρμg_ρ,μ=1/2g^μρg_μρ,



    1/√(-g)√(-g)/ x^   =1/2g g/ x^=1/2g^-1 g/ x^=1/2ε_μ_0…μ_3 g^0μ_0⋯ g^3μ_3/ x^(ε^ν_0…ν_3 g_0ν_0⋯ g_3ν_3)
       =1/2ε_μ_0…μ_3ε^ν_0…ν_3 g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)
       =1/2|[ _μ_0^ν_0 _μ_0^ν_1 _μ_0^ν_2 _μ_0^ν_3; _μ_1^ν_0 _μ_1^ν_1 _μ_1^ν_2 _μ_1^ν_3; _μ_2^ν_0 _μ_2^ν_1 _μ_2^ν_2 _μ_2^ν_3; _μ_3^ν_0 _μ_3^ν_1 _μ_3^ν_2 _μ_3^ν_3 ]|g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)
       =1/2_μ_0^ν_0 _μ_1^ν_1 _μ_2^ν_2 _μ_3^ν_3 g^0μ_0⋯ g^3μ_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)
       =1/2g^0ν_0 g^1ν_1 g^2ν_2 g^3ν_3(g_0ν_0,⋯ g_3ν_3+⋯+g_0ν_0⋯ g_3ν_3,)
       =1/2(g^0ν_0g_0ν_0,+g^1ν_1g_1ν_1,+g^2ν_2g_2ν_2,+g^3ν_3g_3ν_3,)
       =1/2(g^0ρg_0ρ,+g^1ρg_1ρ,+g^2ρg_2ρ,+g^3ρg_3ρ,)=1/2g^μρg_μρ,


Combining equations (<ref>), (<ref>), (<ref>) we get

∇_μ A^μ=_μ A^μ+1/√(-g)√(-g)/ x^A^=1/√(-g)_μ(√(-g) A^μ)

Let us now modify the integral of the last term of Eq.(<ref>) by using Eq.(<ref>), Eq.(<ref>) and the fact that the variation of the metric g^μν vanishes at infinity.

    ∫ d^4x√(-g) (g_μνf(Φ)□  g^μν-f(Φ)∇_μ∇_ν  g^μν)=
    ∫ d^4x√(-g)(g_μν f g^ρ∇_ρ∇_ g^μν-f ∇_μ∇_ν g^μν)=
    ∫ d^4x√(-g)[f ∇_ρ∇_(g_μν g^ρ g^μν)-∇_μ(f ∇_ν  g^μν)+(∇_μ f)∇_ν g^μν]=
    ∫ d^4x√(-g)[f ∇_ρ∇_(g_μν g^ρ g^μν)+(∇_μ f)∇_ν g^μν]-∫ d^4x _μ(√(-g) f ∇_ν g^μν)_0=
    ∫ d^4x√(-g){∇_ρ[f ∇_(g_μν g^ρ g^μν)]-(∇_ρ f)∇_(g_μν g^ρ g^μν)+∇_ν( g^μν∇_μ f)- g^μν∇_μ∇_ν f }=
    ∫ d^4x√(-g)[-(∇_ρ f)∇_(g_μν g^ρ g^μν)- g^μν∇_μ∇_ν f ]+∫ d^4x _ρ[√(-g) f ∇_(g_μν g^ρ g^μν)]_0+∫ d^4x _ν(√(-g)  g^μν∇_μ f)_0=
    ∫ d^4x√(-g)[-∇_(g_μν g^ρ g^μν∇_ρ f)+g_μν g^ρ g^μν∇_∇_ f- g^μν∇_μ∇_ν f]=
    ∫ d^4x√(-g)  g^μν(g_μν□ f-∇_μ∇_ν f)-∫ d^4x _(√(-g) g_μν g^ρ g^μν∇_ρ f)_0

∫ d^4x√(-g)(g_μνf(Φ)□  g^μν-f(Φ)∇_μ∇_ν g^μν)=∫ d^4x√(-g)  g^μν[g_μν□ f(Φ)-∇_μ∇_ν f(Φ)]

The combination of equations (<ref>) and (<ref>) leads to


    0   =∫ d^4x √(-g)  g^μν{g_μν[-f(Φ)/2κ R+1/2_Φ^Φ+V(Φ)+Λ_B].
       .+1/κ[f(Φ)R_μν-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦ-T^(m)_μν}
    
    0   =-g_μν[f(Φ)/2κ R-1/2_Φ^Φ-V(Φ)-Λ_B]+1/κ[f(Φ)R_μν-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]
       -_μΦ_νΦ-T^(m)_μν

(T^(m)_μν-g_μνΛ_B)=f(Φ)/κ[R_μν-1/2g_μνR]+g_μν[_Φ^Φ/2+V(Φ)]
+1/κ[-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦ

We can define now the following tensor

    -T^(Φ)_μν=g_μν[_Φ^Φ/2+V(Φ)]+1/κ[-∇_μ∇_ν f(Φ)+g_μν□ f(Φ)]-_μΦ_νΦ

T^(Φ)_μν=_μΦ_νΦ-g_μν[_Φ^Φ/2+V(Φ)]+1/κ[∇_μ∇_ν f(Φ)-g_μν□ f(Φ)]
It is known that the covariant derivative of a scalar ∇_μΦ equals to the simple derivative _μΦ. Thus, in the above equation □≡ g_β∇^∇^β=∇_∇^=∇^2 and _Φ^Φ=∇_Φ∇^Φ=(∇Φ)^2. Hence, we can write Eq.(<ref>) as
T^(Φ)_μν=∇_μΦ∇_νΦ-g_μν[(∇Φ)^2/2+V(Φ)]+1/κ[∇_μ∇_ν f(Φ)-g_μν∇^2 f(Φ)]
The combination of equations (<ref>) and (<ref>) yields to

    (T^(m)_μν-g_μνΛ_B)=f(Φ)/κ[R_μν-1/2g_μνR]-T^(Φ)_μν

κ(T^(m)_μν+T^(Φ)_μν-g_μνΛ_B)=f(Φ)(R_μν-1/2g_μνR)

Finally, we present the above equations in the case of (4+n)-dimensional spacetime. As we already mentioned the only difference is that Greek indices become capital Latin indices. Moreover, we should have in mind that in this case κκ_(4+n) and Λ_B is the higher dimensional cosmological constant. Therefore, it is
[box=]equation
T_MN^(Φ)=∇_M Φ∇_NΦ-g_MN[(∇Φ)^2/2+V(Φ)]+1/κ_(4+n)[∇_M ∇_N f(Φ)-g_MN∇^2 f(Φ)]

where now it is

[box=]equation
(∇Φ)^2=∇_KΦ∇^KΦ, □=∇^2=∇_K∇^K


The field equations are
[box=]equation
κ_(4+n)(T^(m)_MN+T^(Φ)_MN-g_MNΛ_B)=f(Φ)(R_MN-1/2g_MNR)




§ VARIATION WITH RESPECT TO THE SCALAR FIELD

By varying the action of equation (<ref>) with respect to the scalar field Φ and _μΦ, we obtain a new equation that relates functions f(Φ) and V(Φ) to the field Φ and its derivatives. The procedure that one should follow is depicted below. Firstly, we write the action (<ref>) in the following form in order to be more convenient for evaluation.
S=∫ d^4x√(-g) _tot
where
_tot=f(Φ)/2κR-1/2_μΦ^μΦ-V(Φ)-Λ_B+_m
The variation of the action (<ref>) with respect to Φ and _μΦ will provide us the Euler-Lagrange equation.

    S=0=∫ d^4x (√(-g) _tot)=∫ d^4x[(√(-g) _tot)/ΦΦ+(√(-g) _tot)/(_μΦ)(_μΦ)](_μΦ)=_μ(Φ)


    0   =∫ d^4x [(√(-g) _tot)/ΦΦ+(√(-g) _tot)/(_μΦ)_μ(Φ)]
    
       =∫ d^4x [(√(-g) _tot)/ΦΦ+_μ((√(-g) _tot)/(_μΦ)Φ)-_μ((√(-g) _tot)/(_μΦ))Φ]


However
∫ d^4x _μ((√(-g) _tot)/(_μΦ)Φ)=0

because at the limits of the integration Φ=0. Thus, combining equations (<ref>) and (<ref>) we get

    0=∫ d^4x [(√(-g) _tot)/Φ-_μ((√(-g) _tot)/(_μΦ))]Φ

(√(-g) _tot)/Φ=_μ[(√(-g) _tot)/(_μΦ)]

Eq.(<ref>) constitutes the Euler-Lagrange equation. Substituting now the quantity _tot from Eq.(<ref>) into Eq.(<ref>) we obtain

    √(-g) /Φ[f(Φ)/2κR-V(Φ)]=_μ[√(-g) /(_μΦ)(-1/2_νΦ^νΦ)]


    √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g) /(_μΦ)(g^ρν_νΦ_ρΦ)]


    √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g)( g^ρν^μ_ν_ρΦ+g^ρν_νΦ^μ_ρ)]


    √(-g)(1/2κdf/dΦR-dV/dΦ)=-1/2_μ[√(-g)( g^ρμ_ρΦ+g^μν_νΦ)_2g^μρ_ρΦ]

[box=]equation
√(-g)(1/2κdf/dΦR-dV/dΦ)=-_μ(√(-g) g^μρ_ρΦ)






CHAPTER: ENERGY-MOMENTUM TENSOR'S COMPONENTS



Firstly, the non-diagonal and non-zero components of energy-momentum tensor T^M_N are going to be calculated. Subsequently, the diagonal components will be calculated and finally the zero-valued  components will be presented.


𝐓^0_1:



    (<ref>)[N=1]M=0 T^0_1   =^0ϕ_1ϕ+^0χ_1χ+∇^0∇_1f=g^0K_Kϕ_1ϕ+g^0K_Kχ_1χ+g^0K∇_K∇_1f
       =g^01(_1ϕ)^2+g^01(_1χ)^2+g^01∇_1^2f=g^01_e^-2A[(_1ϕ)^2+(_1χ)^2+∇_1^2f]
       =e^-2A[(_1ϕ)^2+(_1χ)^2+∇_1^2f]


∇_1^2f=∇_1(∇_1f)=∇_1(_1f)=_1(_1f)-^L_11_0_Lf=_1^2f

    _1^2f   =_1(_ϕ f_1ϕ+_χ f_1χ)
       =_ϕ^2f(_1ϕ)^2+_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ^2f(_1χ)^2+_ϕ_χ f_1ϕ_1χ+_χ f_1^2χ
       =_ϕ^2f(_1ϕ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ^2f(_1χ)^2+_χ f_1^2χ

Combining equations (<ref>)-(<ref>), we obtain
T^0_1=e^-2A[(_1ϕ)^2+(_1χ)^2+_1^2f]
or
T^0_1=e^-2A[(1+_ϕ^2f)(_1ϕ)^2+(1+_χ^2f)(_1χ)^2+2_χ_ϕ f_1χ_1ϕ+_ϕ f_1^2ϕ+_χ f_1^2χ]



𝐓^1_0:



    (<ref>)[N=0]M=1 T^1_0   =^1ϕ_0ϕ+^1χ_0χ+∇^1∇_0f=g^1K_Kϕ_0ϕ+g^1K_Kχ_0χ+g^1K∇_K∇_0f
       =g^10(_0ϕ)^2+g^11_1ϕ_0ϕ+g^10(_0χ)^2+g^11_1χ_0χ+g^10∇_0^2f+g^11∇_1∇_0f
       =g^10_e^-2A[(_0ϕ)^2+(_0χ)^2+∇_0^2f]+g^11_e^-2A(1-2m/r)[_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f]

T^1_0=e^-2A{[(_0ϕ)^2+(_0χ)^2+∇_0^2f]+(1-2m/r)[_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f]}

    ∇_0∇_0f   =_0^2f-^L_00_Lf=_0^2f-^0_00_0f-^1_00_1f-^4_00_4f
       =_0^2f-(m/r^2-_1m/r)_0f-(_0m/r-_1m/r+m/r^2+2m_1m/r^2-2m^2/r^3)_1f-e^2A(A'-2mA'/r-_4m/r)_4f


    ∇_1∇_0f   =_1_0f-^L_10_Lf=_1_0f-^1_10_1f-^4_10_4f
       =_1_0f-(_1m/r-m/r^2)_1f+e^2AA'_4f


    _0^2f   =_0(_ϕ f_0ϕ+_χ f_0χ)
       =_ϕ^2f(_0ϕ)^2+_χ_ϕ f_0χ_0ϕ+_ϕ f_0^2ϕ+_χ^2f(_0χ)^2+_ϕ_χ f_0ϕ_0χ+_χ f_0^2χ
       =_ϕ^2f(_0ϕ)^2+2_χ_ϕ f_0χ_0ϕ+_χ^2f(_0χ)^2+_ϕ f_0^2ϕ+_χ f_0^2χ


    _1_0f   =_1(_ϕ f_0ϕ+_χ f_0χ)
       =_ϕ^2f_1ϕ_0ϕ+_χ_ϕ f_1χ_0ϕ+_ϕ f_1_0ϕ+_χ^2f_1χ_0χ+_ϕ_χ f_1ϕ_0χ+_χ f_1_0χ
       =_ϕ^2f_1ϕ_0ϕ+_χ^2f_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ+_χ f_1_0χ

From equations (<ref>)-(<ref>) we have

    T^1_0   =e^-2A[(_0ϕ)^2+(_0χ)^2+_0^2f+(1-2m/r)(_1_0ϕ+_1χ_0χ+_1_0f).
       . +_1(m/r)_0f-_0(m/r)_1f+e^2A_4(m/r)_4f]

or

    T^1_0   =e^-2A{(1+_ϕ^2f)(_0ϕ)^2+(1+_χ^2f)(_0χ)^2+2_χ_ϕ f_0ϕ_0χ+_ϕ f_0^2ϕ+_χ f_0^2χ.
       +(1-2m/r)[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_ϕ_χ f(_1ϕ_0χ+_1χ_0ϕ)+_ϕ f_1_0ϕ+_χ f_1_0χ]
       . +(_1m/r-m/r^2)(_ϕ f_0ϕ+_χ f_0χ)-_0m/r(_ϕ f_1ϕ+_χ f_1χ)+e^2A_4m/r(_ϕ f_4ϕ+_χ f_4χ)}

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𝐓^4_0:



    (<ref>)[N=0]M=4T^4_0   =^4ϕ_0ϕ+^4χ_0χ+∇^4∇_0f=g^4K(_Kϕ_0ϕ+_Kχ_0χ+∇_K∇_0f)
       =g^44_1(_4ϕ_0ϕ+_4χ_0χ+∇_4∇_0f)=_4ϕ_0ϕ+_4χ_0χ+∇_4∇_0f


    ∇_4∇_0f   =_4_0f-^L_40_Lf=_4_0f-^0_40_0f-^1_40_1f
       =_4_0f-A'_0f-_4m/r_1f


    _4_0f   =_4(_ϕ f_0ϕ+_χ f_0χ)
       =_ϕ^2f_4ϕ_0ϕ+_χ_ϕ f_4χ_0ϕ+_ϕ f_4_0ϕ+_χ^2f_4χ_0χ+_ϕ_χ f_4ϕ_0χ+_χ f_4_0χ
       =_ϕ^2f_4ϕ_0ϕ+_χ^2f_4χ_0χ+_ϕ_χ f(_4χ_0ϕ+_4ϕ_0χ)+_ϕ f_4_0ϕ+_χ f_4_0χ

Equations (<ref>)-(<ref>) yield to
T^4_0=_4ϕ_0ϕ+_4χ_0χ+_4_0f-A'_0f-_4m/r_1f
or

    T^4_0=(1+_ϕ^2f)_4ϕ_0ϕ   +(1+_χ^2f)_4χ_0χ+_χ_ϕ f(_4χ_0ϕ+_4ϕ_0χ)
       +_ϕ f_4_0ϕ+_χ f_4_0χ-A'_0f-_4m/r_1f

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𝐓^0_4:



    (<ref>)[N=4]M=0T^0_4   =^0ϕ_4ϕ+^0χ_4χ+∇^0∇_4f=g^0K(_Kϕ_4ϕ+_Kχ_4χ+∇_K∇_4f)
       =g^01_e^-2A(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)=e^-2A(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)


    ∇_1∇_4f   =_1_4f-^L_14_Lf=_1_4f-^1_14_1f=_1_4f-A'_1f


    _1_4f=   =_1(_ϕ f_4ϕ+_χ f_4χ)
       =_ϕ^2f_1ϕ_4ϕ+_χ_ϕ f_1χ_4ϕ+_ϕ f_1_4ϕ+_χ^2f_1χ_4χ+_ϕ_χ f_1ϕ_4χ+_χ f_1_4χ
       =_ϕ^2f_1ϕ_4ϕ+_χ^2f_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ)+_ϕ f_1_4ϕ+_χ f_1_4χ


Inserting the expressions of the quantities ∇_1∇_4f and _1_4f from equations (<ref>) and (<ref>) respectively, into equation (<ref>) we get
T^0_4=e^-2A(_1ϕ_4ϕ+_1χ_4χ+_1_4f-A'_1f)
or

    T^0_4=e^-2A[(1+_ϕ^2f)_1ϕ_4ϕ   +(1+_χ^2f)_1χ_4χ+_ϕ_χ f(_1χ_4ϕ+_1ϕ_4χ) 
       +_ϕ f_1_4ϕ+_χ f_1_4χ-A'(_ϕ f_1ϕ+_χ f_1χ)]



𝐓^4_1:



    (<ref>)[N=1]M=4T^4_1   =^4ϕ_1ϕ+^4χ_1χ+∇^4∇_1f=g^4K(_Kϕ_1ϕ+_Kχ_1χ+∇_K∇_1f)
       =g^44_1(_4ϕ_1ϕ+_4χ_1χ+∇_4∇_1f)=_4ϕ_1ϕ+_4χ_1χ+∇_4∇_1f

It is obvious from equations (<ref>) and (<ref>) that

T^4_1=e^2AT^0_4



𝐓^1_4:



    (<ref>)[N=4]M=1T^1_4   =^1ϕ_4ϕ+^1χ_4χ+∇^1∇_4f=g^1K(_Kϕ_4ϕ+_Kχ_4χ+∇_4∇_1f)
       =g^10_e^-2A(_0ϕ_4ϕ+_0χ_4χ+∇_0∇_4f)+g^11_e^-2A(1-2m/r)(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)
       =e^-2A(_0ϕ_4ϕ+_0χ_4χ+∇_0∇_4f)+e^-2A(1-2m/r)(_1ϕ_4ϕ+_1χ_4χ+∇_1∇_4f)

From equations (<ref>), (<ref>) and (<ref>) we deduce that
T^1_4=e^-2AT^4_0+(1-2m/r)T^0_4
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𝐓^0_0:



    (<ref>)[N=0]M=0T^0_0   =^0ϕ_0ϕ+^0χ_0χ+∇^0∇_0f+-□ f
       =g^01(_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f)+-□ f
       =e^-2A(_1ϕ_0ϕ+_1χ_0χ+∇_1∇_0f)+-□ f

where

≡ -(ϕ)^2+(χ)^2/2-V(ϕ,χ)=-^Lϕ_Lϕ+^Lχ_Lχ/2-V(ϕ,χ)

□ f≡∇^2f=∇^K∇_Kf

We are going to calculate firstly the quantities  and □ f.

    =-1/2(^Lϕ_Lϕ+^Lχ_Lχ)-V(ϕ,χ)
       =-1/2(^0ϕ_0ϕ+^1ϕ_1ϕ+^4ϕ_4ϕ+^0χ_0χ+^1χ_1χ+^4χ_4χ)-V(ϕ,χ)
       =-1/2(g^0K_Kϕ_0ϕ+g^1K_Kϕ_1ϕ+g^4K_Kϕ_4ϕ+g^0K_Kχ_0χ+g^1K_Kχ_1χ+g^4K_Kχ_4χ)-V(ϕ,χ)
       =-1/2{2g^01_e^-2A(_1ϕ_0ϕ+_1χ_0χ)+g^11_e^-2A(1-2m/r)[(_1ϕ)^2+(_1χ)^2]+g^44_1[(_4ϕ)^2+(_4χ)^2]}-V(ϕ,χ)
       =-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]}-1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ)


=-e^-2A/2{2(_1ϕ_0ϕ+_1χ_0χ)+(1-2m/r)[(_1ϕ)^2+(_1χ)^2]}-1/2[(_4ϕ)^2+(_4χ)^2]-V(ϕ,χ)

    □ f   =∇^K∇_Kf=g^KL∇_L∇_Kf
       =2g^01∇_0∇_1f+g^11∇_1∇_1f+g^22∇_2∇_2f+g^33∇_3∇_3f+g^44∇_4∇_4f


∇_1∇_1f and ∇_0∇_1f are known from equations (<ref>) and (<ref>) respectively.


    ∇_2∇_2f   =_2^2f_0-^L_22_Lf=-^0_22_0f-^1_22_1f-^4_22_4f
       =r_0f-(2m-r)_1f+r^2e^2AA'_4f


    ∇_3∇_3f   =_3^2f_0-^L_33_Lf=-^0_33_0f-^1_33_1f-^4_33_4f
       =[r_0f-(2m-r)_1f+r^2e^2AA'_4f]sin^2θ=(∇_2∇_2f)sin^2θ


    ∇_4∇_4f   =_4_4f-^L_44_0_Lf=_4^2f

Combining equations (<ref>), (<ref>), (<ref>)-(<ref>) we obtain

    □ f   =2g^01∇_0∇_1f+g^11∇_1∇_1f+g^22∇_2∇_2f+g^33∇_3∇_3f+g^44∇_4∇_4f
       =2e^-2A[_0_1f+(m/r^2-_1m/r)_1f+e^2AA'_4f]+e^-2A(1-2m/r)_1^2f
       +g^22∇_2∇_2f+g^22/sin^2θ(∇_2∇_2f)sin^2θ_2g^22∇_2∇_2f+_4^2f
       =2e^-2A_0_1f+e^-2A2(m/r^2-_1m/r)__1(1-2m/r)_1f+2A'_4f+e^-2A(1-2m/r)_1^2f
       +2e^-2A/r^2[r_0f-(2m-r)_1f+r^2e^2AA'_4f]+_4^2f
       =e^-2A_0_1f+e^-2A_1(_0f)+e^-2A_1[(1-2m/r)_1f]+2A'_4f
       +e^-2A/r^22r__1(r^2)_0f+2e^-2A/r_e^-2A/r^2_1(r^2)(1-2m/r)_1f+2A'_4f+_4^2f
       =e^-2A_0_1f+e^-2A/r^2r^2_1(_0f)+e^-2A/r^2r^2_1[(1-2m/r)_1f]+4A'_4f
       +e^-2A/r^2_1(r^2)_0f+e^-2A/r^2_1(r^2)[(1-2m/r)_1f]+_4^2f
       =e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f)


□ f=e^-2A_0_1f+e^-2A/r^2_1[r^2_0f+r^2(1-2m/r)_1f]+e^-4A_4(e^4A_4f)

Substituting ∇_1∇_0f from equation (<ref>) into (<ref>) we get
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    T^0_0=e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f+_1f/r(m/r-_1m)]+A'_4f+-□ f

T^0_0=e^-2A[_1ϕ_0ϕ+_1χ_0χ+_1_0f-_1(m/r)_1f]+A'_4f+-□ f

Using equation (<ref>) as well, we obtain

    T^0_0   =e^-2A[(1+_ϕ^2f)_1ϕ_0ϕ+(1+_χ^2f)_1χ_0χ+_χ_ϕ f(_1χ_0ϕ+_1ϕ_0χ)+_ϕ f_1_0ϕ. 
       +._χ f_1_0χ+_ϕ f_1ϕ+_χ f_1χ/r(m/r-_1m)]+A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f

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𝐓^1_1:



    (<ref>)[N=1]M=1T^1_1   =^1ϕ_1ϕ+^1χ_1χ+∇^1∇_1f+-□ f
       =g^1K(_Kϕ_1ϕ+_Kχ_1χ+∇_K∇_1f)+-□ f
       =g^10_e^-2A(_0ϕ_1ϕ+_0χ_1χ+∇_0∇_1f)+g^11_e^-2A(1-2m/r)[(_1ϕ)^2+(_1χ)^2+∇_1^2f]+-□ f

Equations (<ref>) and (<ref>) combined with (<ref>) give
T^1_1=T^0_0+(1-2m/r)T^0_1


𝐓^2_2:



    (<ref>)[N=2]M=2T^2_2   =^2ϕ_2ϕ+^2χ_2χ+∇^2∇_2f+-□ f
       =g^2K(_Kϕ_2ϕ+_Kχ_2χ+∇_K∇_2f)+-□ f
       =g^22_e^-2A/r^2(_2ϕ_2ϕ_0+_2χ_2χ_0+∇_2∇_2f)+-□ f
       =e^-2A/r^2∇_2∇_2f+-□ f


    (<ref>)(<ref>)T^2_2   =e^-2A/r^2[r_0f-(2m-r)_1f+r^2e^2AA'_4f]+-□ f
       =e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ f

Thus, we have
T^2_2=e^-2A/r[_0f+(1-2m/r)_1f]+A'_4f+-□ f
or

    T^2_2   =e^-2A/r[(_ϕ f_0ϕ+_χ f_0χ)+(1-2m/r)(_ϕ f_1ϕ+_χ f_1χ)]
       +A'(_ϕ f_4ϕ+_χ f_4χ)+-□ f



𝐓^3_3:



    (<ref>)[N=3]M=3T^3_3   =^3ϕ_3ϕ+^3χ_3χ+∇^3∇_3f--□ f
       =g^3K(_Kϕ_3ϕ+_Kχ_3χ+∇_K∇_3f)--□ f
       =g^33_e^-2A/r^2sin^2θ(_3ϕ_3ϕ_0+_3χ_3χ_0+∇_3∇_3f)--□ f
       =e^-2A/r^2sin^2θ∇_3∇_3f+-□ f


    (<ref>)(<ref>)T^3_3   =e^-2A/r^2sin^2θ(∇_2∇_2f)sin^2θ--□ f=e^-2A/r^2∇_2∇_2f+-□ f

T^3_3=T^2_2
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𝐓^4_4:



    (<ref>)[N=4]M=4T^4_4   =^4ϕ_4ϕ+^4χ_4χ+∇^4∇_4f+-□ f
       =g^4K(_Kϕ_4ϕ+_Kχ_4χ+∇_K∇_4f)+-□ f
       =g^44_1[(_4ϕ)^2+(_4χ)^2+∇_4∇_4f]+-□ f
       =(_4ϕ)^2+(_4χ)^2+∇_4∇_4f+-□ f


The combination of equations (<ref>) and (<ref>) gives
T^4_4=(_4ϕ)^2+(_4χ)^2+^2_4f+-□ f
Moreover, it is

    _4^2f   =_4(_ϕ f_4ϕ+_χ f_4χ)
       =_ϕ^2f(_4ϕ)^2+_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ^2f(_4χ)^2+_ϕ_χ f_4ϕ_4χ+_χ f_4^2χ
       =_ϕ^2f(_4ϕ)^2+_χ^2f(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χ

Therefore, equations (<ref>) and (<ref>) are combined to give
T^4_4=(1+_ϕ^2f)(_4ϕ)^2+(1+_χ^2f)(_4χ)^2+2_χ_ϕ f_4χ_4ϕ+_ϕ f_4^2ϕ+_χ f_4^2χ+-□ f



𝐓^0_2:



    (<ref>)[N=2]M=0 T^0_2   =^0ϕ_2ϕ+^0χ_2χ+∇^0∇_2f=g^0K_Kϕ_2ϕ_0+g^0K_Kχ_2χ_0+g^0K∇_K∇_2f
       =g^01_e^-2A∇_1∇_2f=e^-2A∇_1∇_2f


∇_1∇_2f=(_1_2f_0-^L_12_Lf)=-^L_12_Lf=-^2_12_2f_0=0

Thus, equations (<ref>) and (<ref>) give

T^0_2=0


𝐓^0_3:



    (<ref>)[N=3]M=0 T^0_3   =^0ϕ_3ϕ+^0χ_3χ+∇^0∇_3f=g^0K_Kϕ_3ϕ_0+g^0K_Kχ_3χ_0+g^0K∇_K∇_3f
       =g^01_e^-2A∇_1∇_3f=e^-2A∇_1∇_3f


∇_1∇_3f=(_1_3f_0-^L_13_Lf)=-^L_13_Lf=-^3_13_3f_0=0

Equations (<ref>) and (<ref>) yield to

T^0_3=0
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𝐓^1_2:



    (<ref>)[N=2]M=1T^1_2   =^1ϕ_2ϕ_0+^1χ_2χ_0+∇^1∇_2f=g^1K∇_K∇_2f=g^10∇_0∇_2f+g^11∇_1∇_2f
       =e^-2A∇_0∇_2f+e^-2A(1-2m/r)∇_1∇_2f


∇_0∇_2f=_0_2f_0-^L_02_0_Lf=0

Combining equations (<ref>), (<ref>), (<ref>) we get

T^1_2=0


𝐓^1_3:



    (<ref>)[N=3]M=1T^1_3   =^1ϕ_3ϕ_0+^1χ_3χ_0+∇^1∇_3f=g^1K∇_K∇_3f=g^10∇_0∇_3f+g^11∇_1∇_3f
       =e^-2A∇_0∇_3f+e^-2A(1-2m/r)∇_1∇_3f


∇_0∇_3f=_0_3f_0-^L_03_0_Lf=0

From equations (<ref>) and (<ref>) into (<ref>) we obtain

T^1_3=0


𝐓^2_0:



(<ref>)[N=0]M=2T^2_0=^2ϕ_0_0ϕ+^2χ_0_0χ+∇^2∇_0f=g^2K∇_K∇_0f=g^22∇_2∇_0f_0 (<ref>)=0
T^2_0=0


𝐓^2_1:



(<ref>)[N=1]M=2T^2_1=^2ϕ_0_1ϕ+^2χ_0_1χ+∇^2∇_1f=g^2K∇_K∇_1f=g^22∇_2∇_1f_0 (<ref>)=0
T^2_1=0
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𝐓^2_3:



(<ref>)[N=3]M=2T^2_3=^2ϕ_0_3ϕ+^2χ_0_3χ+∇^2∇_3f=g^2K∇_K∇_3f=g^22∇_2∇_3f

∇_2∇_3f=_2_3f_0-^L_23_Lf=-^3_23_3f_0=0

From equations (<ref>) and (<ref>) it is obvious that

T^2_3=0


𝐓^2_4:



(<ref>)[N=4]M=2T^2_4=^2ϕ_0_4ϕ+^2χ_0_4χ+∇^2∇_4f=g^2K∇_K∇_4f=g^22∇_2∇_4f

∇_2∇_4f=_2_4f_0-^L_24_Lf=-^2_24_2f_0=0

Equations (<ref>) and (<ref>) are combined to give

T^2_4=0


𝐓^3_0:



(<ref>)[N=0]M=3T^3_0=^3ϕ_0_0ϕ+^3χ_0_0χ+∇^3∇_0f=g^3K∇_K∇_0f=g^33∇_3∇_0f_0 (<ref>)
T^3_0=0


𝐓^3_1:



(<ref>)[N=1]M=3T^3_1=^3ϕ_0_1ϕ+^3χ_0_1χ+∇^3∇_1f=g^3K∇_K∇_1f=g^33∇_3∇_1f_0 (<ref>)
T^3_1=0



𝐓^3_2:



(<ref>)[N=2]M=3T^3_2=^3ϕ_0_2ϕ+^3χ_0_2χ+∇^3∇_2f=g^3K∇_K∇_2f=g^33∇_3∇_2f_0 (<ref>)
T^3_2=0


𝐓^3_4:



(<ref>)[N=4]M=3T^3_4=^3ϕ_0_4ϕ+^3χ_0_4χ+∇^3∇_4f=g^3K∇_K∇_4f=g^33∇_3∇_4f

∇_3∇_4f=_3_4f_0-^L_34_Lf=-^3_34_3f_0=0

The combination of equations (<ref>) and (<ref>) yield to
T^3_4=0
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𝐓^4_2:



(<ref>)[N=2]M=4T^4_2=^4ϕ_0_2ϕ+^4χ_0_2χ+∇^4∇_2f=g^4K∇_K∇_2f=g^44∇_4∇_2f_0 (<ref>)
T^4_2=0


𝐓^4_3:



(<ref>)[N=3]M=4T^4_2=^4ϕ_0_3ϕ+^4χ_0_3χ+∇^4∇_3f=g^4K∇_K∇_3f=g^44∇_4∇_3f_0 (<ref>)

T^4_3=0



*
unsrt

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