title: Black Hole versus Naked Singularity via Axial Perturbation
authors: Parthapratim Pradhan
text: 
  
  Link Budget Analysis for Reconfigurable Smart Surfaces in Aerial Platforms
      Safwan Alfattani, Wael Jaafar, Yassine Hmamouche, Halim Yanikomeroglu, and Abbas Yongaçoglu 
  This work is supported in part by a
  scholarship from King AbdulAziz University, Saudi Arabia, in part by
  the National Sciences and Engineering Research Council of Canada
  (NSERC), and in part by Huawei Canada.
  S. Alfattani is with the Electrical Engineering Department of King Abdulaziz University, Saudi Arabia, and with the
  	School of Electrical Engineering and Computer Science,
  	University of Ottawa, Ottawa, ON, Canada, (e-mail: smalfattani@kau.edu.sa).
  W. Jaafar and H. Yanikomeroglu are with the Department of Systems and
  Computer Engineering, Carleton University, Ottawa, ON, Canada, e-mails:
  (waeljaafar, halim@sce.carleton.ca)
  Y. Hmamouche is with the Mathematical and Electrical Engineering Department, IMT-Atlantique, Brest, France,  (e-mail: yassine.hmamouche@imt-atlantique.fr).
  A. Yongacoglu is with the
   School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada 
   (e-mail: yongac@uottawa.ca).
      December 30, 2023
  ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================
  
  
  
  
  
  
  
  We differentiate non-extremal black hole, extremal black hole  and naked singularity via metric 
  perturbations for Reissner-Nordström spacetime. First we study the axial perturbations for extremal 
  Reissner-Nordström black hole and compute the effective potential due to these perturbations. 
  Then we study the axial perturbations for the naked singularity case and compute the effective potential. 
  We show that for the non-extremal black hole, the effective potential outside the event horizon (r_+) is real and positive. While in between Cauchy horizon (r_-) and event horizon (r_-<r<r_+) 
  the effective potential is negative.  For the extremal black hole, the effective potential is always positive.  Also for naked singularity, the effective potential  is positive.
  From the effective potential diagram, we show that the structure of effective potentials for extremal BH looks like a potential barrier outside the horizon. While for non-extremal BH, 
  the structure of the effective potential looks like a potential well rather than a 
  potential barrier. For NS, the structure of the effective potentials is 
  neither a potential barrier nor a potential well. Preferably it looks like an exponential decay function. 
  We observe that an effective potential barrier's geometric construction due to axial perturbations could distinguish between the non-extremal black hole,  extremal black hole, and naked singularity. The stability of the extremal BH has been discussed.
  
  
  
  
  PACS numbers: 04.70.Bw  04.25.Nx  04.40.Nr
  
  
  25 cm
  
  
  
  § INTRODUCTION
  
  The fundamental difference between a black hole (BH) and a naked singularity (NS) is that the former object has a horizon structure while the latter object doesn't have any horizon structure. Again a BH can be categorized into two types: non-extremal BH and extremal BH. A non-extremal BH is characterized by a Hawking temperature, while extremal BH is characterized by zero Hawking temperature. In the case of  NS, the Hawking temperature is completely undefined since it is horizonless. Now the question is how to differentiate a BH from an NS? In this work, we will try to understand the difference between BH ( non-extremal BH & extremal BH) and NS via perturbative formalism. 
  We have considered here only axial (sometimes called odd-parity) perturbations.
  
  The Reissner-Nordström (RN) solution is a spherically symmetric solution of the coupled 
  Einstein and Maxwell system. It represents a BH having mass M and charge Q_∗. The fact that the RN
  BH consist of two horizons. One is the event horizon or outer horizon, and the other one is the Cauchy horizon or inner
  horizon. When two horizons are coincident, we will get extremal BH. In general relativity, a spherically symmetric charged BH solution can be obtained when M^2>Q_∗^2. When M^2=Q_∗^2, one gets an extremal BH solution. 
  Finally, when M^2<Q_∗^2, we get NS, which is visible to the external observer.
  
  The perturbation formalism of non-extremal (M^2>Q_∗^2) RN BH was first introduced
  by Moncrief <cit.> and then  Zerilli <cit.>.  The author in Ref. <cit.> 
  used the Hamiltonian variational principle for computations of Perturbation equations, and the author in Ref. <cit.> decomposed the gravitational and electromagnetic field perturbation equations using vector and tensor harmonics. After that Chandrasekhar <cit.> studied the same problem 
  via metric perturbations. He derived one-dimensional wave equations that govern the RN BH's axial perturbations via the procedure of metric perturbations.
  There are several related work we should mention here that have studied perturbations of RN BH in 
  different formalism <cit.>.
  
  
  However, in the present work we shall focus on  particularly the gravitational axial 
  perturbations of M^2=Q_∗^2 BH and  Q_∗^2>M^2 spacetime. 
  The case M^2>Q_∗^2 is already been studied in Ref. <cit.>. First we will study the 
  gravitational axial perturbations separately for M^2=Q_∗^2 BH. Then we will 
  differentiate  M^2>Q_∗^2 BH, M^2=Q_∗^2 BH and Q_∗^2>M^2 spacetime 
  visually by plotting the effective potential diagram. 
  The effective potential that we have used should be derived via metric perturbations. 
  The formalism and procedure that we have used here could be found in Ref. <cit.>.  In what follows, in the next section, we have described the basic formalism of metric perturbations for extremal RN BH. In  Sec. (<ref>), 
  we have studied the axial perturbations for extremal case. In  Sec. (<ref>), 
   we have examined the same for NS case. 
  And in each case we have plotted the effective potential barrier diagram to differentiate 
  M^2>Q_∗^2 BH, M^2=Q_∗^2 BH and Q_∗^2>M^2. Finally, we have given the 
  conclusions in Sec <ref>.
  
  
  Why is it important to study the perturbations of M^2=Q_∗^2 BH   separately?
  There are several good reasons to study them. First, M^2=Q_∗^2 BHs  are special class of BHs.
  They are playing a key role in classical gravity as well as in quantum gravity. They can be treated as a theoretical toy in quantum gravity to explore the characteristics of quantum gravity. 
  
  Second, the M^2=Q_∗^2 BHs are often called as supersymmetric. They are invariant under several supercharges. They saturate the BPS (Bogomolny-Prasad-Sommerfield) bound. They are stable. 
  Their BH entropy has been calculated in string theory. 
  They have no Hawking temperature; therefore, they do not Hawking radiate. 
  
  Third, the fact that M^2=Q_∗^2 BHs don't have any trapped surfaces. They don't have any bifurcation two
  sphere (S^2). The proper distance between the outer and inner horizons in the extremal RN
  BH is infinite even though the vanishing coordinate distance. Typically, a BH is defined to be extreme if it has zero surface gravity, which “measures the equilibrium temperature for the thermal distribution
  of the radiation” <cit.>. Another exciting feature of M^2=Q_∗^2 BH is that it has 
  Couch-Torrence symmetry <cit.> that means it is conformally invariant under a spatial inversion while its non-extremal counterparts don't have such a feature.
  
  
  
  Distinguishing the BH and NS have been started by a pioneer work of de Felice <cit.>
  (See also<cit.>) where the author showed the classical instability of an NS. More precisely, he observed for a Kerr BH  that there is “a sharp discontinuity from positive to negative values when a→ M”. In fact, in the regime m<a<1.008m, there exists stable circular orbits with negative energy with respect to ∞. 
  That indicates if accretion happens then, a Kerr naked singularity slows down to the BH state. 
  This is one of the motivations behind the present work. Before going to axisymmetric spacetime, it is crucial to see what happens in the spherically spacetime? That's why we have considered extremal RN BH. Do we see any interesting feature perturbatively in the effective
  potential when one has to be taken this condition M^2→ Q_∗^2? This forms the basis of our work. 
  There are several important work <cit.>; we should mention here that has to be considered the difference between  NS and BH for various types of BH. 
  Recently, in <cit.>, the authors differentiated BHs from NSs by using the images of accretion disks. In <cit.>, it was pointed out that NS is unstable under linearized perturbations. While in <cit.>, it was pointed out that NS is unstable in the context of the gravitational collapse of a scalar field.
  
  
  We find that for non-extremal BH, the effective potential outside the event horizon is real and positive.  While in between Cauchy horizon and event horizon (r_-<r<r_+) 
  the effective potential is negative.  For extremal BH, the effective potential is always positive.  
  For NS, the effective potential is also positive. 
  Alternatively, from the effective potential diagram, we can argue that the structure of effective potentials for extremal BH looks like a potential barrier outside the horizon and the potentials
   V_1^(-) and V_2^(-)  are real and positive everywhere outside the event horizon.      
  For non-extremal BH, the structure of the effective potentials look like a 
  potential well rather than a potential barrier and potentials are negative 
  in the region r_-<r<r_+.  While for NS, the structure of the effective potentials is 
  neither a potential barrier nor a potential well. Rather it looks like an exponential decay function. These are the prime differences between non-extremal 
  BH, extremal BH, and NS. Using these features, one could differentiate between non-extremal BH, extremal BH, and NS in RN geometry.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  §  THE METRIC PERTURBATIONS IN EXTREMAL REISSNER-NORDSTRÖM SPACETIME
  
  The metric perturbations of non-extremal Reissner-Nordström BH have been studied
  in detail by S. Chandrasekhar <cit.>. Here, we are interested to study the  perturbation of 
   extremal BH and NS following the 
  approach of S. Chandrasekhar <cit.>. We have used the notation and convention throughout the work
  follwing the Ref.  <cit.>. We begin by writing the unperturbed extremal Reissner-Nordström 
  Spacetime as 
  
      ds^2   =    e^2ν(  dx^0)^2-e^2ψ(  dx^1)
      ^2-e^2μ _2(  dx^2)^2-e^2μ _3(dx^3)^2
         =    e^2ν(  dx^0)^2-e^2μ _2(  dx^2)^2-r^2  dΩ^2,
   
  where the co-ordinates are denoted as 
  
      x^0   ⟺    t 
      
      x^1   ⟺   φ
      
      x^2   ⟺    r 
      
      x^3   ⟺   θ,
  
  dΩ^2=e^2ψ(dx^1)^2+e^2μ _3(dx^3)^2 
  is the line element on the unit two sphere of the metric and the metric coefficients are 
  
      e^2ν   =    e^-2μ_2=(1-M/r)^2 ≡Υ/r^2
      
       e^μ_3   =    r 
      
       e^ψ   =    r sinθ
      ω   =    q_2=q_3=0 
      Υ   =    r^2 e^2ν.
  
  Here r is Schwarzschild like radial coordinates and having circumference 2π r. To study the perturbations 
  of any spherically symmetric system one should restrict on axisymmetric modes of perturbations. The fact that 
  the equations which govern the perturbations of a spherically symmetric spacetime could be separable in all four 
  coordinates t, r, θ and φ. Since the extremal RN metric (<ref>) is static and 
  spherically symmetric. Once it is perturbed, the spacetime becomes time dependent and axially symmetric. 
  The most general time dependent and axially symmetric spacetime can be written as
  
      ds^2 = e^2ν( dx^0)^2-e^2ψ( dx^1-ω dx^0-q_2  dx^2-q_3  dx^3)^2
       -e^2μ_2( dx^2)^2-e^2μ_3( dx^3)^2
  
  This is a perturbed extremal RN spacetime. 
  The above spacetime  metric (<ref>) contains seven functions: 
  ν, ψ, μ_2, μ_3, ω, q_2, q_3. Again they are 
  functions of t, x^2 and  x^3 only. Whereas the Einstein's equation contains only six independent equations for 
  the metric components therefore it could not be possible to determine seven functions arbitrarily hence there is a 
  constraint which is a possible combinations of six independent function. This constraint  of the metric 
  coefficients becomes an indentity: 
  
      (ω_,2-q_2,0)_,3-(ω_,3-q_3,0)_,2+(q_2,3-q_3,2)_,0.
  
  It should be noted that in unperturbed extremal RN metric (<ref>), the axial metric components 
  ω= q_2=q_3=0 while in perturbed extremal RN metric, these components are not equal to zero. 
  This is the basic difference between perturbed spacetime and unperturbed spacetime. 
  
  Note that a general perturbation of the extremal RN spacetime should result in ω, q_2 and 
  q_3 are small quantities of the first order. While the functions ν, ψ, μ_2, μ_3 
  are having the small increments like δν, δψ, δμ_2, δμ_3. There are two possible kinds of metric perturbations in extremal RN spacetime. One is called polar (even parity) 
  perturbation, and another one is called axial (or odd parity) perturbation. This strictly depends upon the transformation of the parameter φ→ -φ on the metric. Simply, the perturbation leading to non-vanishing values of  ω, q_2 and q_3 are called axial perturbations. 
  The perturbation leading to small increments of δν, δψ, δμ_2, δμ_3 
  are called polar perturbations. 
  
  
  Now we shall linearize the Maxwell equations. This can be grouped into two sets of four equations. In first 
  set, we write the four odd equations.  
  
      (r^2 sinθ/√(Υ)  F_12)_,3 + ( r^2 sinθ  F_31)_, 2   =    0
      (sinθ√(Υ)  F_01)_,2 + ( r^2 sinθ/√(Υ) F_12)_, 0   =    0
      (sinθ√(Υ) F_01)_,3 + ( r^2 sinθ  F_13)_, 0   =    0
  
  
      (r^2/√(Υ)  F_01)_,0 + (√(Υ)  F_12)_,2 
      + F_13_,3
   
  
      =     r^2 sinθ  F_02  Q_02 + r^2 sinθ/√(Υ)  F_03  Q_03 
      -sinθ√(Υ) F_23  Q_23
  
  
  In the second set, we write the four even equations. They are represented by 
  
      ( r^2 sinθ  F_02)_,2 + (r^2 sinθ/√(Υ)  F_03)_, 3   =    0 
      
      -(sinθ √(Υ)  F_23)_,2 + (r^2 sinθ/√(Υ)  F_03)_, 0   =    0
      (sinθ √(Υ)  F_23)_,3 + ( r^2 sinθ  F_02)_, 0   =    0
  
  
      F_02_,3 - ( √(Υ)  F_03)_, 2 + 
      (r^2/√(Υ)  F_23)_,0
  
  
      =   sinθ√(Υ)  F_01 Q_23 + r^2 sinθ/√(Υ) F_12 Q_03 - r^2 sinθ  F_13  Q_02
  
  where the partial derivative of a function f is given with the notation, 
  
      ( f )_, μ   =   ∂ f/∂ x^μ
  
  The function Q_AB are given by,
  
      Q_AB   =   ∂ q_A/∂ x^B - ∂ q_B/∂ x^A
      Q_A0 = ∂ q_A/∂ x^0 - ∂ω/∂ x^A ( A,B = 2,3)
  
  It should be noted that the first set of equations (<ref>) having the quantities which reverse 
  their signs when φ is replaced by -φ. Whereas the second set of equations (<ref>) 
  having the quantities which are invariant to the reversal in the sign of φ. That's why we call them 
  odd or axial quantities and even or polar quantities. 
  
  Further it should be noted that F_02=-M/r^2 is the only non-vanishing components of the Maxwell 
  tensor for the unperturbed extremal metric, therefore the linearized version of equations (<ref>) and 
  (<ref>) which governs the perturbations are 
  
      ( √(Υ)  F_01sinθ)_,r +   r^2/√(Υ)  F_12,0sinθ   =    0
      √(Υ)( F_01sinθ)_, θ + r^2  F_13,0sinθ   =    0
      r^2/√(Υ)  F_01,0 + ( √(Υ) F_12)_,r + F_13,θ   =    - M ( ω_,2 - q_2,0)sinθ
      r^2/√(Υ) F_03,0   =   ( √(Υ) F_23)_,r
      δ F_02,0 - M/r^2( δψ + δμ_3 )_,0 + e^ν/r sinθ( F_23sinθ)_, θ   =    0
      [ δ F_02 - M/r^2( δν + δμ_2 ) ]_,θ 
      +  ( √(Υ) F_ 30 )_,r +  r^2/√(Υ)  F_23, 0   =    0
  
  Since the perturbed components of Ricci tensor of extremal RN metric can not be zero like 
  Schwarzschild BH therefore these components can be computed from the following identity:
  
      δ R_(a)(b)   =    -2 [η^(n)(m){δ F_(a)(n)F_(b)(m) +
       F_(a)(n)δ_(b)(m)}-η_(a)(b)  M δ F_(0)(2)/r^2].
  
  From this identity, we will get 
  
      δ R_(0)(0)   =   δ R_(1)(1)= - δ R_(2)(2) =δ R_(3)(3)=- 2M/r^2δ F_(0)(2)
      δ R_(0)(1)   =    - 2M/r^2δ F_(1)(2),  δ R_(0)(3)= + 2M/r^2δ F_(2)(3)
      δ R_(1)(2)   =    + 2M/r^2δ F_(0)(1),  δ R_(2)(3)= + 2M/r^2δ F_(0)(3)
      δ R_(0)(2)   =   δ R_(1)(3)= 0
  
  Here,  Z_(α)(β) is the (α)(β) tetrad component of the tensor Z. 
  Also, R and F are denoted as the Ricci and Maxwell tensors, and a comma denotes 
  partial differentiation.
  
  
  
  §  AXIAL  PERTURBATIONS IN EXTREMAL REISSNER-NORDSTRÖM SPACETIME
  
  First we will treat the axial perturbations following the linerization of coupled Einstein-Maxwell equations 
  around the extremal RN BH. Now putting the unperturbed metric coefficients in the  φ r and φ θ 
  components of the Ricci tensor, we now get
  
      [Υ (q_2,3-q_3,2)sin^3θ]_,3+r^4(ω_,2-q_2,0)_,0 sin^3θ   =    2r^2√(Υ) sin^2θ δ R_(1)(2)
         =    4M √(Υ) F_(0)(1) sin^2θ,
  
  and 
  
      [Υ (q_2,3-q_3,2)sin^3θ]_,2-r^4/Υ (ω_,3-q_3,0)_,0sin^3θ   =    -2r^2sin^2θ δ R_(1)(3)
         =    0,
  
  Now the linearized components of the Maxwell tensors are
  
      [√(Υ)  F_(0)(1)]_,2+ r^2/√(Υ) F_(1)(2),0   =    0, 
      √(Υ) [F_(0)(1)sinθ]_,3+r^2 F_(1)(3),0 sinθ   =    0, 
      r^2/√(Υ) F_(0)(1),0 + [√(Υ)  F_(1)(2)]_,2+F_(1)(3),3   =    -M (ω_2-q_2,0)sinθ.
  
  Now introducing the functions 
  
      B(r, θ)    =    F_(0)(1) sinθ, 
       Q(r, θ)    =   Υ (q_2,3-q_3,2)sin^3θ.
  
  Putting these functions in Eq. (<ref>) and Eq. (<ref>) and using the linearized Maxwell 
  equations [<ref>,<ref>, <ref>], we get the differential equations 
  
      1/r^41/sin^3θ ∂ Q(r, θ)/ ∂θ   =    
      -(ω_,2-q_2,0)_,0+4M  √(Υ)/r^4 B(r, θ)/sin^2θ ,
   
  and 
  
      Υ/r^41/sin^3θ ∂ Q(r, θ)/ ∂ r   =    
      (ω_,3-q_3,0)_,0,
  
  Assuming the perturbations have the time dependence e^iσ t therefore we obtain the 
  following equations 
  
  
      1/r^41/sin^3θ ∂ Q(r, θ)/ ∂θ   =   
      -iσω_,2-σ^2q_2+4M  √(Υ)/r^4 B(r, θ)/sin^2θ,
  
  and
  
      Υ/r^41/sin^3θ ∂ Q(r, θ)/ ∂ r   =    
      i σω_,3+σ^2 q_3,
  
  Differentiating Eq. (<ref>) with respect to θ and Eq. (<ref>) with respect to 
  r, and summing the equations we get 
  
      r^4∂/ ∂ r(Υ/r^2 ∂ Q(r, θ)/ ∂ r)   +   sin^3θ ∂/ ∂θ(1/sin^3θ ∂ Q(r, θ)/ ∂θ)+σ^2r^4/Υ  Q(r, θ) 
         =    4M sin^3θ √(Υ) ∂/ ∂θ( B(r, θ)/sin^2θ).
   
  Again, differentiate Eq. (<ref>) with respect to r, Eq. (<ref>) with respect to θ, 
  and Eq. (<ref>) with respect to t. Putting these  in the latter  equation and using Eq. (<ref>),
  one obtains
  
      [Υ/r^2 {√(Υ)  B(r, θ) }_,2]_,2+√(Υ)/r^2( B(r, θ)_,3/sinθ)_,3sinθ-r^2/√(Υ)  B(r, θ)_,0,0   =    M (ω_,2,0-q_2,0,0) sin^2θ.
  
  Thus Eqs. (<ref>) and (<ref>) govern  the axial perturbations. 
  
  Next, the variables r and θ in Eqs. (<ref>) and (<ref>) may be separated 
  by the following substituations <cit.> 
  
      B(r, θ)    =    3  B(r) C^-1/2_l+1(cosθ),
  
  and 
  
      Q(r, θ)    =    Q(r) C^-3/2_l+2(cosθ),
   
  where C^j_i denotes the Gegenbauer function of order i and index 
  j <cit.>. It should be noted  that the relations given in Ref.  
  <cit.> between the Gegenbauer functions and the Legendre functions 
  are valid only for l≥ 2 <cit.>.  The valid expression for the 
  i=3 and j=-3/2 Gegenbauer function is 
  
      C^-3/2_3(cosθ)=1/2(cos^3θ-3cosθ)
  
  The Gegenbauer functions may be computed directly for all orders and indices from the Ref <cit.>
  
      C^j_i(cosθ)    =   ∑_k=0^iΓ(j+k)Γ
      (j+i-k)/k!(i-k)![Γ(j)]^2cos(i-2k)θ.
  
  Putting Eqs. (<ref>) and (<ref>) in Eqs. (<ref>) and 
  (<ref>) we get the radial equations
  
      Υ d/ dr(Υ/r^4 d Q(r)/ dr)
      -2n Υ/r^4  Q(r)+σ^2  Q(r)    =    -8nM Υ^3/2/r^3  B(r)
  
  and 
  
      d/ dr[Υ/r^2 d/ dr(√(Υ)  B(r))]
      -2(n+1)√(Υ)/r^2  B(r)+(σ^2r^2/√(Υ)
      -4M^2√(Υ)/r^4)  B(r)    =   
      -M  Q(r)/r^4 .
  
  where 
  
      n=(l-1)(l+2)/2
  
  Here, n is the principal quantum number, and l is an orbital quantum number. 
  
  With the further substituations 
  
      H_1^(-)   =    -2 √(2n) √(Υ) B(r),
  
  and
  
      H_2^(-)   =   1/r Q(r),
  
  and changing  to the Regge-Wheeler “tortoise” coordinate r_* 
  defined by 
  
      d/dr_* = Υ/r^2d/dr
  
  with these definitions, we get a pair of coupled  differential equations 
  of second order
  
      Λ^2H_1^(-)   =   Υ/r^5[{2(n+1)r-3M+4M^2/r}H_1^(-)+
      3M H_1^(-)+2M √(2n)H_2^(-)],
   
  and
  
      Λ^2H_2^(-)   =   Υ/r^5[{2(n+1)r
      -3M+4M^2/r}H_2^(-)-3M H_2^(-)+2M√(2n)H̅_1^(-)].
  
  where, Λ^2=d^2/dr_*^2+σ^2. 
  
  The above pair of coupled second order differential equations can be decoupled by further substituations 
  
      Z_1^(-)   =    +q_1 H_1^(-)+√(-q_1q_2)H_2^(-)
      
      Z_2^(-)   =    -√(-q_1q_2)H_1^(-) +q_1 H_2^(-),
  
  where 
  
      q_1   =    M (3+√(9+8n)) 
       
      q_2   =    M (3-√(9+8n)).
  
  It is proved that Z_1^(-) and Z_2^(-) satisfy one-dimensional Schroödinger-type 
  wave-equations, 
  
      Λ^2 Z_1^(-)   =    V_1^(-) Z_1^(-),
  
  and
  
      Λ^2 Z_2^(-)   =    V_2^(-) Z_2^(-),
  
  where the effective potential for axial perturbations of extremal RN BH after substituting the value of 
  Υ are 
  
      V_1^(-)   ≡    V_1^(-)(r)= (r-M)^2/r^6[2(n+1)r^2-(3-√(9+8n))Mr+4 M^2],
  
  and 
  
      V_2^(-)   ≡    V_2^(-)(r)= (r-M)^2/r^6[2(n+1)r^2-(3+√(9+8n))Mr+4 M^2].
  
  These equations governing the axial perturbations to the pair of one dimensional wave equations for the 
  extremal Reissner Nordström BH. Now we will see the behavior  of these potentials graphically 
  [See Fig. (<ref>)) and Fig. (<ref>))]. It is evident from the plot that the height of 
  the potential barrier increases when n increases.  
  
  
  
  
  
  
   §.§ Stability Analysis of Extremal RN BH via the Effective potentials V_1^(-) and V_2^(-)
  
  The shape of the effective potentials depends on n. The barrier-height is increasing when the value of n is increasing. First, we will analyze the effective potential 
  V_1^(-). 
  
  
  
    §.§.§ Properties of the Effective potential V_1^(-)
  
  The extrema of the effective potential is determined by 
  solving the following equation
  
      d V_1^(-)/dr   =   (r-M )(a r^3+b r^2+c r+d)/r^7 = 0,
  
  where 
  
      a    =    -4(n+1 ) 
      
      b    =   (17+8n-3√(9+8n))M
      
      c    =    - (31-5√(9+8n))M^2
      
      d    =    24 M^3.
  
  It follows from the above equation that the only solution of Eq. (<ref>) is   
  
      r=M
  
  or
  
      a r^3+b r^2+c r+d = 0
  
  The first solution i.e. r=r_x=M is the horizon of extremal RN BH. Let us check the stability of this 
  extremal horizon. To do so, we have to calculate the second derivative of effective potential. Thus 
  one find 
  
      d^2 V_1^(-)/dr^2   =   2/r^8[a_1 r^4+b_1 r^3+c_1 r^2+d_1 r+e_1].
   
  where 
  
      a_1    =    6(n+1 ) 
      
      b_1    =    -6(7+4n-√(9+8n))M
      
      c_1    =    20(6+n-√(9+8n))M^2 
      
      d_1    =    -15(11-√(9+8n))M^3
      
      e_1    =    84M^4.
  
  Now 
  
      (d^2 V_1^(-)/dr^2)_r_x=M   =   2/M^4(2n+3+√(9+8n))> 0     ∀ n
  
  Thus the radius of the orbit r=r_x=M is a stable orbit for ∀ n[It should be emphasized that r_x=M is not only the horizon of extremal RN BH. It is also the radius of the null photon sphere of extremal RN BH. 
  As it was already proved in <cit.> that there exists stable photon orbit at radius r_x=M. We here strongly suggest that indeed there exists a stable photon orbit at r_x=M which is reverified via axial gravitational perturbations.]. Hence the extremal horizon of extremal RN BH under 
  gravitational axial perturbation is stable. 
  
  Now we analyze the second solution i. e. a r^3+b r^2+c r+d = 0. To determine the nature of the roots 
  we have to compute the discriminate 
  
      Δ = 1728 (n+1 ) (17+8n-3√(9+8n)) (31-5√(9+8n))M^6
      -96(17+8n-3√(9+8n))M^6 
      
      -(17+8n-3√(9+8n))^2 (31-5√(9+8n))^2 M^6
      -16(n+1 ) (31-5√(9+8n))^3 M^6 
      
      -248832 (n+1 )M^6
  
  It implies that Δ<0 for ∀ n. So the cubic equation has only one real root 
  and two complex conjugate roots. 
  For simplicity, we consider the value of n=2. In this case we find a=-12, b=18M, c=-6M^2 and 
  d=24M^3 therefore the value of Δ=-1703376<0. Then the radial equation becomes 
  
      2r^3-3Mr^2+M^2r-4M^3    =   (r-r_1) (r-r_2) (r-r_3)= 0
  
  The only real root of this equation is 
  
      r_1   =   1/6[3+(216-3√(5181))^1/3
      +3^1/3(72+√(5181))^1/3]M
  
  Taking the value of √(5181)=71.979≈ 71.98, we get the value of real root 
  
      r_1   ≈    1.82 M
  
  and other roots r_2 and r_3 are imaginary. Note that r_1>r_x. We can 
  check now whether this radius r_1 which is located outside the horizon is stable 
  or unstable.  
  
  We see that
  
      (d^2 V_1^(-)/dr^2)_r=r_1   ≈    -261/M^4 <0
  
  Thus the radius r=r_1 which is located at r>M  has a local maximum.  Now we will see  
  the more generalized case. 
  To do so we consider the horizon location away from extremality i.e. r=(1+χ)M, 
  where χ<<1. In this limit 
  
      (d^2 V_1^(-)/dr^2)_r=(1+χ)M   =   2/M^4[2n+3+√(9+8n)-(24n+51+15√(9+8n))χ+ O(χ^2) ]
  
  This immediately implies that 
  
      (d^2 V_1^(-)/dr^2)_r=(1+χ)M< 0     ∀ n
  
  This means that outside the horizon the effective potential barrier has a local maximum. 
  
  
  
    §.§.§ Properties of the Effective potential V_2^(-)
  
  Proceeding analogously the extrema of the effective potential is determined by  
  
      d V_2^(-)/dr   =   (r-M )(a r^3+b r^2+c r+d)/r^7 = 0,
  
  where 
  
      a    =    -4(n+1 ) 
      
      b    =   (17+8n-3√(9+8n))M
      
      c    =    - (31-5√(9+8n))M^2
      
      d    =    24 M^3.
  
  There are two possible solutions. Either r=M or  a r^3+b r^2+c r+d = 0. The first one is 
  extremal horizon. To determine the stability of it we have to compute 
  
      d^2 V_2^(-)/dr^2   =   2/r^8[a_1 r^4+b_1 r^3+c_1 r^2+d_1 r+e_1].
   
  where 
  
      a_1    =    6(n+1 ) 
      
      b_1    =    -6(7+4n+√(9+8n))M
      
      c_1    =    20(6+n+√(9+8n))M^2 
      
      d_1    =   -15(11+√(9+8n))M^3
      
      e_1    =    84 M^4.
  
  Now
  
      (d^2 V_2^(-)/dr^2)_r=M   =   2/M^4(2n+3-√(9+8n)) > 0     ∀ n
  
  Again it proves that the extremal horizon of extremal RN BH under axial gravitational perturbation is stable. 
  
  Now we would like to study the near-extremal situations i.e.  at  r=(1+χ)M. In this limit 
  
      (d^2 V_2^(-)/dr^2)_r=(1+χ)M   =   2/M^4[2n+3-√(9+8n)-(24n+51-15√(9+8n))χ+ O(χ^2) ]
  
  This also indicates that 
  
      (d^2  V_2^(-)/dr^2)_r=(1+χ)M< 0     ∀ n
  
  Thus outside the horizon the effective potential barrier has a local maximum. 
  It is evident from the graphical plots of V_1^(-) and V_2^(-) that 
  for extremal RN BHs the structure of effective potentials look like a potential 
  barrier  outside the horizon i.e. r>M. Also the potentials V_1^(-) and V_2^(-)  
  are real and positive everywhere outside the horizon. Hence following the statement of 
  Chandrasekhar <cit.> we could say that extremal RN BH is stable under 
  gravitational axial perturbations.
  
  
  
  § AXIAL PERTURBATION IN NAKED SINGULARITY CASE OF REISSNER-NORDSTRÖM SPACETIME
  
  Since in this work we are interested to study both M^2=Q_∗^2 and M^2<Q_∗^2 situations. In 
  previous section, we have studied the axial perturbations in case of M^2=Q_∗^2. In the 
  present section, we will study the NS case of RN spacetime. We will also observe how the potential 
  barrier vary for non-extremal case, extremal case and NS. In previous section we have already been
  calculated the effective potential due to axial perturbations for the extremal situations, here we 
  will calculate the effective potential for NS case. Therefore the metric function for Eq. (<ref>) 
  can be written as 
  
      e^2ν   =    e^-2μ_2= (1-M/r)^2+(Q_∗^2-M^2/r^2)
       =(Υ+Ξ/r^2).
  
  where Ξ=Q_∗^2-M^2. Clearly for Q_∗^2>M^2, Ξ>0 there  
  is no horizon. This is called NS situations. For Q_∗^2=M^2, Ξ=0 that means we get 
  extremal BH solution.  For Q_∗^2<M^2, Ξ<0 that means we get non-extremal BH 
  solution having two horizons namely the event horizon, r_+=M+(M^2-Q_∗^2)^1/2 and the 
  Cauchy horizon, r_+=M-(M^2-Q_∗^2)^1/2. 
  
  The fundamental difference between non-extremal BH, extremal BH, and NS is as follows. Since the 
  spacetime metric admits Killing vector field ξ^μ_(t)=(1,0,0,0). Thus the norm of the 
  Killing vector field is defined by the quantity ξ^μξ_μ and for 
  
  (i) non-extremal BH it is given by 
  
      ξ^μξ_μ   =   (r-r_+)(r-r_-)r^-2
  
  and
  
      ξ^μξ_μ   >    0 ()   r_-<r<r_+
         =    0 ()   r=r_±
         <    0 ()   0<r<r_-
  
  (ii) for extremal BH it is defined by 
  
      ξ^μξ_μ   =   (1-M/r)^2 > 0 ()  ∀ r
         =    0 ()   r=M
  
  
  (iii) for NS it is
  
      ξ^μξ_μ   =   (1-M/r)^2+(Q_∗^2-M^2/r^2) 
      > 0 () ∀ r
  
  Analogously, for Q_∗^2>M^2, the pair of  coupled second order differential equations 
  become
  
      Λ^2H_1^(-)   =   (Υ+Ξ/r^5)
      [{2(n+1)r-3M+4Q_∗^2/r}H_1^(-)+3M H_1^(-)
      +2Q_∗^2 √(2n)H_2^(-)],
   
  and
  
      Λ^2H_2^(-)   =   (Υ+Ξ/r^5)[{2(n+1)r
      -3M+4Q_∗^2/r}H_2^(-)-
       3M H_2^(-)+2Q_∗^2√(2n)H̅_1^(-)].
  
  where, Λ^2=d^2/dr_*^2+σ^2. 
  
  These equations  can be decoupled by further substituations 
  
      Z_1^(-)   =    +q_1 H_1^(-)+√(-q_1q_2)H_2^(-)
      
      Z_2^(-)   =    -√(-q_1q_2)H_1^(-) +q_1 H_2^(-),
  
  where 
  
      q_1   =   (3M+√(9M^2+8n Q_∗^2)) 
       
      q_2   =   (3M-√(9M^2+8n Q_∗^2)).
  
  We have already proved that  Z_1^(-) and Z_2^(-) satisfy one-dimensional 
  Schroödinger-type wave-equations, 
  
      Λ^2 Z_1^(-)   =    V_1^(-) Z_1^(-),
  
  and
  
      Λ^2 Z_2^(-)   =    V_2^(-) Z_2^(-),
  
  where the effective potential for axial perturbations of NS case
  
      V_1^(-)   ≡    V_1^(-)(r)= (Υ+Ξ/r^6)[2(n+1)r^2-
      (3M-√(9M^2+8n Q_∗^2))r+4 Q_∗^2],
  
  and 
  
      V_2^(-)   ≡    V_2^(-)(r)= (Υ+Ξ/r^6)[2(n+1)r^2-
      (3M+√(9M^2+8n Q_∗^2))r+4 Q_∗^2 ].
  
  These equations governing the axial perturbations for Q_∗^2>M^2 cases
  of Reissner Nordström BH and for  M^2>Q_∗^2, it was given in S. Chandrasekhar's 
  book <cit.>.
  The simple way we can distinguish two geometric structures, namely M^2>Q_∗^2 and M^2<Q_∗^2 via graphical plot. It is evident from the graphical 
  plots of V_1^(-) and V_2^(-) that for non-extremal BH (M^2>Q_∗^2) the structure of the
  effective potentials look like a potential well rather than a potential barrier and potentials are negative in the 
  region r_-<r<r_+. But for NS (Q_∗^2>M^2), the structure of the effective potentials is neither a potential barrier nor a potential well. Rather it looks like an exponential decay function. 
  This is the prime difference between the non-extremal BH and the NS. 
  
  Therefore we have found three different types of effective potential structures for non-extremal BH, extremal BH, and NS. 
  Hence these three different types of potentials might allow us to distinguish between non-extremal BH, 
  extremal BH, and NS. Hence,  we can conclude that BH and NS are quite distinguished objects. 
  
  
  
  . 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § CONCLUSIONS
  
  We have studied the axial gravitational perturbations of extremal RN BH and NS to differentiate between these compact objects. We also compared our results with non-extremal RN BH. To do this, we derived one-dimensional Schröndinger type wave equations for the extremal RN BH and NS.  We showed that for non-extremal BH, the effective potential outside the event horizon is real and positive. While in between Cauchy horizon and event horizon, the effective potential is negative. For extremal BH and NS, the effective potential is always positive. Moreover, we found that for non-extremal BH,  the shape of effective potential looks like a  potential well, and it is negative in the regime r_-<r<r_+. 
  For extremal BH, the shape of the potential looks like a potential barrier rather than a potential-well, and it is positive outside the horizon. While for NS, the shape of effective potential is neither a potential barrier nor a potential well. Rather it looks like an exponential decay function. These are the main differences between these three compact objects. To reinforce our result, we plotted several effective potential diagrams. So that one can easily distinguish these three compact objects visually. Furthermore, we found that the extremal RN BH is stable under axial gravitational perturbations.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  99
  mon74 V. Moncrief, 
  “Odd-parity stability of a Reissner-Nordström black hole”, 
  Phys. Rev. D 9, 2707 (1974).
  
  mon74a V. Moncrief, “Stability of Reissner-Nordström black holes”,  
  Phys. Rev. D 10, 1057 (1974).
  
  mon75 V. Moncrief, “Gauge-invariant Perturbations of  Reissner-Nordström black holes”, 
  Phys. Rev. D 12, 1526 (1975).
  
  zer74 F. J. Zerilli, 
  “Perturbation analysis for gravitational and electromagnetic radiation in a 
  Reissner-Nordström geometry”, 
  Phys. Rev. D 9, 860 (1974).
  
  sc79 S. Chandrasekhar, 
  “On the equations governing the perturbations of the Reissner-Nordström black hole”,  
  Proc. R. Soc. London A365, 453 (1979).
  
  sc79a S. Chandrasekhar and B. C. Xanthopoulos, 
  “On the metric perturbations of the Reissner-Nordström black hole”, 
  Proc. R. Soc. London A367, 1 (1979).
  
  sc S. Chandrasekhar, 
  The Mathematical Theory of Black Holes, Oxford University Press, Oxford, (1983).
  
  fro V. P. Frolov and  I. D. Novikov, 
   Black Hole Physics: basic concepts and new developments (Kluwer, Dordrecht, 1998).
  
  nam78 J. McNamara, 
  “Instability of black hole inner horizons”, 
  Proc. R. Soc. London A358, 499 (1978).
  
  nam78a J. McNamara, 
  “Behaviour of scalar perturbations of a Reissner-Nordström black hole inside the event horizon”, 
  Proc. R. Soc. London A364, 121 (1978).
  
  mat79 R. Matzner, N. Zamorano and V. D. Sandberg, 
  “Instability of the Cauchy horizon of Reissner-Nordström black holes”, 
  Phys. Rev. D 19, 2821 (1979).
  
  sand79 Y. Gürsel, V. D. Sandberg, I. D. Novikov, and A. A. Starobinsky, 
  “Evolution of scalar perturbations near the Cauchy horizon of a charged black hole”, 
  Phys. Rev. D 19, 413 (1979).
  
  sand79a Y. Gürsel, I. D. Novikov, V. D. Sandberg, and A. A. Starobinsky, 
  “Final state of the evolution of the interior of a charged black hole”, 
  Phys. Rev. D 20, 1260 (1979).
  
  hart82 S. Chandrasekhar and J. B. Hartle, 
  “On crossing the Cauchy horizon of a Reissner-Nordström black-hole”, 
  Proc. R. Soc. London A384, 301 (1982).
  
  price72 R. H. Price, 
  “Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations”,  
  Phys. Rev. D 5, 2419 (1972).
  
  bicak72 J. Bičák, 
  “Gravitational collapse with charge and small asymmetries I. Scalar perturbations”,  
  Gen. Relativ. Grav. 3, 331 (1972).
  
  bur95 L. M. Burko, 
  “Dipole perturbations of the Reissner-Nordström solution: The polar case”, 
  Phys. Rev. D 52, 4518 (1995).
  
  bur99 L. M. Burko, 
  “Dipole perturbations of the Reissner-Nordström solution. II The axial case ”, 
  Phys. Rev. D 59, 084003 (1999).
  
  ori95 L. M. Burko and A. Ori, 
  “Are Physical Objects Necessarily Burnt Up by the Blue Sheet inside a Black Hole?”, 
  Phys. Rev. Lett. 74, 1064 (1995).
  
  ori97 L. M. Burko and A. Ori, 
  “Late-time evolution of nonlinear gravitational collapse”, 
  Phys. Rev. D 56, 7820 (1997). 
  
  and98 A. Anderson, A. M. Abrahams and C. Lea, 
  “Curvature-based gauge-invariant perturbation theory for gravity: A new paradigm",  
  Phys. Rev. D 58, 064015 (1998).
  
  his81 W. A. Hiscock, 
  “Evolution of the interior of a charged black hole”,  
  Phys. Lett. A 83, 110 (1981).
  
  fer05 S. Fernando, “Gravitational perturbation and quasi-normal modes of charged black holes 
  in Einstein-Born-Infeld gravity”,  
  Gen. Relativ. Grav. 37, 585 (2005).
  
  hod12 S. Hod,  
  “Stability of the extremal Reissner-Nordström  black hole to charged scalar perturbations”, 
  Phys. Lett. B 713, 505 (2012).
  
  
  friedman J.L. Friedman, “On the Born approximation for perturbations of a 
  spherical star and the Newman-Penrose  constants ”, 
  Proc. Roy. Soc. Lond. Ser. A. 335, 163 (1973).
  
  stegun  M. Abramowitz and I. A. Stegun (eds.), 
   Handbook of Mathematical Functions (Dover Publications, New York, 1965).
  
  pla11 P. Pradhan, P. Majumdar, 
  “Circular geodesics in Extremal Reissner Nordström Spacetimes ”
   Physics Letters  A 375 (2011) 474-479.
  
  couch84 W. E. Cough and R. J. Torrence, 
  “Conformal Invariance Under Spatial Inversion of Extreme Reissner-Nordström Black Holes”,  
  Gen. Relativ. Grav. 16, 789 (1984).
  
  
  defelice78 F. de Felice, 
  “Classical instability of a naked singularity”, 
  Nature 273, 8 (1978).
  
  calvini78 M. Calvini, F. de Felice, L. Nobili  
  “Are Naked Singularities Really Visible?”,
  Lett. Nuovo Cimento, 23 15 (1978).
  
  zhang16 H. Zhang, 
  “Naked Singularity, firewall, and Hawking radiation”, 
  Scientific Reports, 7 4000 (2017).
  
  pug11a D. Pugliese, H. Quevedo, R. Ruffini,  
  “Circular motion of neutral test particles in Reissner-Nordström spacetime”,
  Phys. Rev. D 83, 024021 (2011).
  
  pug11b D. Pugliese, H. Quevedo, R. Ruffini,   
  “Motion of charged test particles in Reissner-Nordström spacetime”,
  Phys. Rev. D 83, 104052 (2011).
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  cha17 C. Chakraborty,  P. Kocherlakota, M. Patil, S. Bhattacharyya, P. S. Joshi, A. Królak 
  “Distinguishing Kerr naked singularities and black holes using the spin precession of a test gyro in strong gravitational fields',
  Phys. Rev. D 95, 084024 (2017).
  
  cha17a C. Chakraborty, P. Kocherlakota, P. S. Joshi,  
  “Spin precession in a black hole and naked singularity spacetime”,
  Phys. Rev. D 95, 044006 (2017).
  
  
  raj19 R. Shaikha and P. S. Joshi, 
  “Can we distinguish black holes from naked singularities by the images of their accretion disks”, 
  JCAP 10, 064 (2019).
  
  pul07 G. Dotti, R. J. Gleiser, J. Pullin, 
  “Instability of charged and rotating naked singularities”, 
  Phys. Lett. B  644, 289 (2007).
  
  dem99 D. Christodoulou, 
  “The instability of naked singularities in the gravitational collapse of a scalar field”, 
  Ann. of Math., 149, 183 (1999).
  
   
  
  
  
  