Abstract
We treat a very crude model of an exploding star, in the weak field approximation of the Brans-Dicke theory, in a scenario that resembles some characteristics data of a Type Ia Supernova. The most noticeable feature, in the electromagnetic component, is the relationship between the absolute magnitude at maximum brightness of the star and the decline rate in one magnitude from that maximum. This characteristic has become one of the most accurate method to measure luminosity distances to objects at cosmological distances [1, 2] . An interesting result is that the active mass associated with the scalar field is totally radiated to infinity, representing a mass loss in the ratio of the "tensor" component to the scalar component of 1 to (2 ω + 3) (ω is the Brans-Dicke parameter), in agreement with a general result of Hawking [3] . Then, this model shows explicitly, in a dynamical case, the mechanism of radiation of scalar field, which is necessary to understand the Hawking result.
Introduction
Although the problem of constructing an exact solution in the Brans-Dicke (BD) theory, that may be considered a generalization of Vaidya solution [4] , has been addressed in the past [5] , this solution has not been found yet. That generalization must reduce to Vaidya solution corresponding to the external gravitational field of a radially radiating star, in the limit of GR, and also, in the limit that the radiation flux vanishes, we must be able to obtain one of the empty space solution of Brans [6] .
The interest of having a solution in the BD theory associated with electromagnetic radiation is to use it in the construction of a model of an exploding star that includes the ejection of a shell of mater, electromagnetic radiation in a similar way as in a model in GR [7, 8, 9] , and scalar field radiation. This may be in consonance with a star model in which a highly evolved core implodes to nuclear densities while at the same time the outer most layers of matter are blown at high speed. This idea can be realized in the weak field approximation of BD theory using the method of Barros and Romero [10] to construct a generalization of Vaidya solution that it is matched through a spherical singular hypersurface to a satisfactory interior solution.
The formalism of a singular hypersurface in BD theory has been developed and used in the description of bubbles dynamics in extended inflation models of the Universe [11, 12] , among others applications. While Barrabes et al [13] treat a singular hypersurface in BD theory in the Einstein representation, another authors [11, 12] develop the formalism of a singular hypersurface in the Jordan-Fierz (J-F) representation. We shall follow the last approach in which it is usually considered that the metric corresponds to the physical one.
In the weak field method of Barros and Romero if the metric field equation is considered in the Einstein representation, it does not depend on the scalar field. As a consequence, the dynamics of the singular hypersurface can be obtained in a similar manner as in the corresponding model of GR. Of course, the scalar field and the metric in the JF representation will depend on the matter model assumed for the ejected shell, while the radiation involved in the process will include two components, one corresponding to the scalar field and the other to the electromagnetic radiation, both tuned with the evolution of their sources. Thus, once we have determined the evolution of the sources, we know the interior and exterior solutions and the content and variation of the radiated energy at infinity in both components.
In section 2 we present a brief summary of the results of the model of Barros and Romero [10] . Section 3 is devoted to present the results of the theory of surface layers in BD theory, useful in the weak field limit. In section 4 we discuss a general model of an stellar explosion by constructing the metric and the scalar field both in the interior and exterior regions of the singular hypersurface. We end the section with a derivation of the frequency shift of spectral lines in the background metric and the luminosity of the exploding object as seen by a distant marshalljennifer@example.org. In section 5 we treat a particular model by matching two Vaidya solutions, with time dependent masses m − and m + , respectively, through a shell of dust [9] , to obtain the solution of Einstein equation, necessary to construct the BD solution in its linearized form, for the same matter distribution. The time evolution of the shell of dust is completely determined by the GR matching conditions for the metric alone. The resulting system of equations of motion consists of two ordinary differential equations for four unknowns. To complete the system, we obtain, first, an equation that shows how the total rest mass of the shell changes due to the balance between the total energy per unit time incident on the shell, J − , and the total energy per unit time emitted by the shell, J + , both at the shell surface. Then, according to the general characteristic of the scenario in which we are interested in, we may consider that J − ≪ J + , and therefore, J − it is neglected in the energy balance equation. This assumption is equivalent to consider m − as constant. Another assumption is to take, essentially, J + proportional to the radius of the shell through a constant, χ, that it is characteristic of the radiation production mechanism within the shell. This choice is similar to that one made by Hamity and Gleiser [7] , on different grounds, that leads to a satisfactory model of a stellar explosion, with characteristic parameters such as maximum luminosity and time decay from that maximum, corresponding to a Type Ia supernova [2] . This is precisely the scenario in which we propose to compute the output of scalar energy from the ejected shell. For this particular example we compute the effective active mass of the system as a function of an external observer time. This effective mass presents two components: the "tensor" component, related to the radiation of electromagnetic energy, and the scalar component, related to the radiation of scalar field. We end the section with a numerical integration of the system of differential equations from a given set of initial conditions. We show a comparative set of results for different values of the separation constant χ, the Brans-Dicke parameter ω, and the initial active mass of the shell. One interesting result is that the active mass, associated with the scalar field, is totally radiated to infinity; for the given system of initial conditions the process takes place in a short time, well before the star reaches its maximum luminosity if the initial active mass of the shell is of the order of or greater than 10 −2 M ⊙ . This represents a mass loss in the ratio of the "tensor" component to the scalar component of 1 to (2 ω + 3), in agreement with a general result of Hawking [3] . Then, this model shows explicitly, in a dynamical case, the mechanism of radiation of scalar field, which is necessary to understand the Hawking result. In the last section we summarize the main results of our work.
The weak field limit
In the weak field approximation the solutions of BD equations are simply related to the solutions of GR for the same matter distribution. Following Barros and Romero [10] the corresponding field equations are:
The "bar" on top of a symbol means that it is considered in the Einstein representation. For instance, g ab (G 0 , x) is the solution of eq.(4). The same symbol without the bar is in the J-F representation which it is usually considered as the physical one. Thus, the physical metric is obtained from eq.
(1). The box operator 2 corresponds to the wave equation for the flat metric; T ab is the (matter) energy momentum tensor. In first orderT ab = T ab ; T is the trace of T ab ; ǫ(x) is a first order term in the energy density; φ 0 is an arbitrary constant which satisfies |φ 0 | ≫ |ǫ|;
0 . Actually, in order that BD theory posses a Newtonian limit this constant must be related to the Newtonian gravitational constant G by [14] G 0 = 2ω + 3 2ω + 4 G .
Essentially, in the weak field approximation the metric calculated from BD equations is quasi-conformally related to the metric calculated from Einstein equations for the same matter configuration. The term quasi-conform means that in going from an Einstein solutionḡ ab (G, x) to the corresponding BD solution g ab (x), apart from the conformal factor [1 − G 0 ǫ(x)] we must replace G by the new ω-dependent "effective" gravitational constant G 0 given by (6).
Singular hypersurface in BD theory
The procedure sketched in the previous section requires to find first the metricḡ ab . To this end we have to specify the tensor T ab corresponding to the matter. We choose to have a singular hypersurface of matter, Σ, that separates the spacetime, M , into the external region M + , with metric g + ab , and the internal region M − , with metric g
The regions M + and M − are non-empties; this a necessary requirement to show that the weak field condition (2) guarantees the condition (3) in M .
Before we propose the solutionsḡ ± ab and ǫ ± in the regions M ± we have to consider the matching conditions at Σ that such solutions have to satisfy. Of course, the matching conditions of the metricsḡ + andḡ − are the same as in GR [15] :
whereK + andK − are the corresponding extrinsic curvatures of Σ as "seen" from M + or M − in terms of the metricsḡ ± ab , respectively;K + andK − are tensors in Σ;g is the induced metric on Σ by the metricḡ of the spacetime M . Finally, S is the surface energy momentum tensor in Σ.
The extrinsic curvatureK + can be expressed in terms of its components in a coordinate basis {e i } in Σ:
whereN Σ + is the unit normal vector to Σ, pointing from M − to M + , with its components expressed in M + ; In eq. (8) ";" indicates the covariant derivative using the Riemannian connection of M + associated withḡ
On the other hand, eq. (5) is in the J-F representation. The corresponding matching conditions on the scalar field φ through Σ are:
while its normal derivative has a discontinuity given by [12] :
where trS is the trace of S,
and a similar expression for Σ − .
4 The model
General characteristics of the model
To construct a simple model of an stellar explosion we choose Σ as the history of an spherical surface with metric ds
Also, we consider M + filled with a coherent unpolarized radial flow of electromagnetic radiation represented by T
where k + a is a null vector. Then, the exterior metric solution of eq. (4) is represented by Vaidya's metric [4] :
with
The scalar field in the exterior region satisfies eq. (5) with T = 0. An outgoing wave solution is:
where f (µ) is an arbitrary function at the moment.
We consider a general model corresponding to:
• A time dependent spherical shell of matter with an energy momentum tensor, compatible with the symmetry of the problem, given by
where v is the velocity of a fluid element in Σ, η the surface energy density and p the isotropic pressure within the shell.
• The metric g − ab has the general form
where we have chosen the area variable r as one of the coordinates, for simplicity.
• The metric g + ab in first order takes the form
• The interior of the shell is occupied by a central spherical body and, perhaps, electromagnetic radiation and traceless matter in a neighborhood of Σ − , such that in that region, close to Σ, the scalar field solution of eq. (5) is
i.e., it is the sum of a spherical wave moving towards decreasing r plus a spherical wave travelling towards increasing values of r. The functions g 1 and g 2 may be determined by imposing initial and boundary conditions in M − and on Σ.
From eqs. (7), (9) and (10) we know that the scalar field and the dynamics of the spherical shell are closely related; that dynamics is contained in the solution of eq.(4). From eq. (9) we obtain
To apply the matching condition (10) we need to know the components N ± a in the J-F representation of the normal vector N , with the requirement N ± r > 0. Those components may be obtained from the normalization conditions:
where the components v a = (X,Ṙ, 0, 0) has to be computed on both sides of Σ, separately. In particular on Σ − the time component X − = dξ(τ )/dτ ; similarly, on Σ + is X + = dµ(τ )/dτ . From (22) we obtain
The definitions of the symbols used in (23) are:
The expressions for X − , H(ξ, r), and F (ξ, r) can only be known symbolically and up to the first order in our approximation. However, this knowledge is enough for the purpose of the present work. In particular we have
where ϕ 00 (ξ, r) is the first order term in the expansion of the (00) component of the metric. A straightforward calculation in first order gives
the (') indicates the derivative of the function. Computing the derivative of expression (21) with respect to τ we obtain g
Therefore, from eqs.(27), (28) and (29) we have
Finally, taking into account that in first order (B − X − ) ≃Ṙ 2 , g ′ 1 (X − +Ṙ) = dg 1 /dτ , and eq.(10) we have
Equation (31) is valid in the first order of approximation, including the terms inṘ 2 . This last equation, along with eq. (7), complete the description of a general model.
Frequency shift and luminosity

Consider
1 in M + an observer at rest at spatial infinity and let t measure proper time along its world-line; the observer 4-velocity u = (∂/∂t) has components u µ = 1; u j = 0 with j = r, θ, φ. Electromagnetic radiation emitted at the surface of the shell with characteristic frequency ω e = k · v(= X + ) will be received at spatial infinity with frequency ω r = k · u(= 1). Thus,
We assume that (2
The total electromagnetic energy radiated by the shell in all direction per unit time (the luminosity of the system) measured by an observer at infinity is given by
whereÑ is the unit normal vector to the surface of constant r andũ the 4-velocity of an observer at rest on that surface. The total electromagnetic energy E em radiated in a time interval ∆t is given by
In the case of a collapsing shell (Ṙ < 0), eq.(33) shows that L → 0 faster thanṁ + as R → G 0 (2 m + − f ) due to an extra redshift factor.
A particular solution
To present a particular simple solution of the general model we assume that:
• The metricḡ − is also represented, in part, by a Vaidya solution, corresponding to the exterior gravitational field of a radially radiating spherically symmetric central body (the core of the exploding star), given by
i.e, we also consider that in the interior of M − , close to Σ, we have a coherent unpolarized radial flow of electromagnetic radiation represented by
where k − a is a null vector. with
• The hypersurface Σ is the history of a spherical shell of dust:
The equation of motion for the surface layer, as given by the matching conditions eqs. (7) and (8) are
We have introduced the total mass m 0 of the constituents if infinitely dispersed and at rest [15] . We assume that
The momentum densities (or energy current densities) along the unit normal vector N to Σ ± [Σ + (or Σ − ) refers to Σ when it is considered as part of M + (or M − )] measured by a local observer at rest on Σ (4-velocity v = ∂/∂τ ) are given by (
To write (46)we have used the orthogonality conditions N · v = 0, v · v = 1, N · N = −1, where all the scalar products are computed with the metricḡ. The total energy per time unit incident on (J − ), or emitted by (J + ), the shell is
Taking the τ -derivative of (39) to use the resulting equation in (40), consideringṘ = 0, and the definitions (41-44), we obtain [9] ṁ
which is an energy-balance equation. It is easy to show that equation (39) and the definitions (43) and (44) giveĀ
If the functions m + (τ ) and m − (τ ) are given, (39) and (40) represent a system of second-order ordinary differential equations for the unknowns R(τ ) and η(τ ).
The equation of motion as a first order system
We are interested in a supernova explosion as the scenario to study, in a specific example, the main characteristics of scalar radiation in the BD theory. To this end we shall make some further assumptions to simplify the equation system. Let us consider first eq. (48). Once the gigantic explosion of the star takes place its core may collapse so rapidly that it forms a sort of extremely compact (degenerate) matter. This compact object, which may be a neutron star 2 or a black hole, is referred to as a compact supernova remnant. It may also be present a diffuse supernova remnant as a consequence of the shock wave and ejected material expanding from this explosion, and the interstellar material it sweeps up along the way. In our very crude model of the explosion, part of the interior of the shell is represented by a radiating compact object of mass m − (τ ) whose action on the rest mass of the expanding shell is represented by J − in (48). It seems then reasonable to assume 3 that J − ≪ J + and neglect J − in (48). This assumption is equivalent to consider m − as a constant. Thus, eq. (48) becomesṁ
In consonant with the last assumption we assume that the rate of scalar radiation, that originates at Σ, to the interior and exterior of the shell of dust, are equal; i.e.:
According to (21) what this assumption actually means is that g 2 is constant; i.e., we have in part of M − , in first order, an incoming scalar wave that originates at Σ, plus a static scalar field associated with the central body.
Let us introduce now the function a(τ ) by
2 At the moment of a neutron star's birth, the nucleons that compose it have energies characteristic of free fall, which is to say about 100M eV per nucleon. That translates to 10 12 K or so. The star cools off very quickly, though, by neutrino emission, so that within a couple of seconds the temperature is below 10 11 K and falling fast. In this early stage of a neutron star's life neutrinos are produced copiously, and since if the neutrinos have energies less than about 10M eV they sail right through the neutron star and the surrounding matter without interacting, they act as a wonderful heat sink [16] . 3 Since J − and J + are both functions of time, the least favorable instant is at τ = 0. In the next subsection we shall see from the initial values of all the relevant variables that J
In terms of the parameter a and using eqs. (43), (44), and (49) we obtain a first order equation for R(τ ) in the form [7 
Similarly, from (50) and (52) we havemȧ
Replacing firstB given in (49) into (41) and then the resulting expression forX + into (54), we obtain
Eqs. (53) and (55) are two relations for the three unknowns,m, a and R. Condition (45) now reads a ≥ 0, 2a
To find an explicit solution it is necessary to fix one of the unknown functions and then solve for the resulting differential equations for the other two. To guide our intuition let us replace the last obtained expression forX + into (54) and then use (55) to replaceṁ in the result. Thus, we obtaiṅ
We assume now
The value of the constant χ is correlated to the time scale of the process and it may be conjecture that it is a characteristic of the radiation production mechanism within the shell. For χ = 0 we have a non-radiating system (a,m = constants). The dimension of χ equals the inverse of a square length; the adopted unit in this paper is M −2
⊙ . This choice is equivalent to that one made by Hamity and Gleiser [7] , on different grounds, that leads to a satisfactory model of a stellar explosion, with characteristic parameters such as maximum luminosity and time decay from that maximum, corresponding to a Type Ia supernova. This is precisely the scenario in which we propose to compute the output of scalar energy from the ejected shell 4 . Finally, from (56) and (57) we obtainȧ
Hence, we can write (53) and (55) in the forṁ
The system of first order differential equations, (58, 59, 60), that describes the motion of the shell, has to be considered in conjunction with the first order differential equation for the amplitude of the outgoing scalar wave, which according to eq.(31), and assumptions (38) and (51), becomeṡ
The initial condition for f (τ ) may be obtained from the requirement that at the time of the explosion, τ = 0, the exterior scalar field matches continuously to a BD static solution in the weak field approximation generated by a central body of active mass m + i [10] ; i.e.,
where f i and m + i are the initial values of f (τ ) and m + (τ ) respectively. In order to compare the results of our model, in the electromagnetic mode, with observational data for a stellar explosion, such as a Type Ia Supernova [17] , for instance, we need to compute the light curve, L(τ ), from eq. (33), for values of 0 ≤ τ ≤ τ 1 , where τ 1 corresponds to the time when the maximum luminosity has decline in one magnitude.
5 From eqs. (24), (25), (33) and (59) the expression for the light curve becomes
It is apparent from eq.(19) that the Keplerian mass M (the active mass), measured by an orbiting object around the exploding star, in the present approximation is
i.e., the total active mass is decomposed into the sum of a "tensor" component m + and a scalar component f /2 [18] . Therefore, from eq.(19) the total scalar energy radiated in the time interval ∆τ is given by
Thus, from eqs. (61) and (65) 
Numerical results
We have performed a numerical integration of the system (32,34,58,59,60,61,65), corresponding to the functions t(τ ), E em (τ ), a(τ ), R(τ ),m(τ ), f (τ ), and E φ (τ ), with initial conditions, at τ = 0:
The constant m − = 1M ⊙ , and χ was chosen to have three different values, corresponding to different cases. The initial value of the total energy per unit time emitted by the shell verifies 6 J + i > 3.5 × 10 37 ergs/s. Similarly, for these initial values and ω = 500, the initial velocity of the shell becomesṘ i ≈ 10000 km/s. The value of ω = 500 it is generally accepted [19] as the lowest bound implies from Solar system tests, although some more recent estimates [20] have raised this limit to ω > 40000, obtained using signal timing from the Cassini spacecraft [21] . The same authors [20] point out that this bound may be weaker on cosmological scales than in the solar system [22] . On the other hand, in a recent publication [23] it is shown that the parameter ω "runs" with the scale factor in a Friedman-Robertson-Walker metric in order that a BD theory serves as a successful model for dark matter -dark energy. In this model the value of the parameter ω is less than (−3/2) in the matter to scalar transition period; equal to (−3/2) in the BD scalar field dominated era; and greater than (3/2) in the scalar to acceleration transition period. In any case, in our BD model of an stellar explosion in the weak field approximation, the main characteristics of the model are quite insensitive to any value of ω greater than 2 since it results G 0 ≈ 1, except for the radiation rate of scalar field and the total energy radiated in the scalar mode as it can be seen from eqs. (62), (65), and (66).
For the given set of initial conditions we have computed the light curve L(τ ), for 0 ≤ τ ≤ τ 1 , and the total radiated active masses E em1 and E φ1 in the same time interval. In Table 1 we show a comparative set of results for different values of the constants χ and ω, and the initial values ofm i ; △t 1 = t m − t 1 , where t m 5 Actually, the maximum luminosity should correspond to the B-band [1] but in our very crude model of a stellar explosion we do not have the possibility to specify the B-band from the model of radial electromagnetic radiation corresponding to the Vaidya solution. 6 The corresponding value for a star like the Sun is J + ∼ 10 33 ergs/s.
[M⊙] (1)
1.04 × 10 52
(1)
(1) The active mass associated with the scalar field, f i /2, is totally radiated to infinity after a time t f < t 1 , well before the shell reaches its maximum brightness. For the values of t f , see the text. and t 1 are the observer times at the maximum luminosity, L m , and when it has decline in one magnitude, L 1 , respectively 7 . By f 1 , we indicate the final value of f at observer time t 1 . Whenm i = 10 −2 M ⊙ , f (τ ) becomes equal to zero at a time t f ≈ 716s; while ifm i = 10 −1 M ⊙ , it results t f ≈ 1.5s; in both cases t f ≪ t 0 , where t 0 is the observer time at whichṘ = 0; in all the cases t m < t 0 . The reason for this fast approaching to zero of f (τ ) is thatḟ is essentially proportional tom, through the first term on the right hand side of eq. (61). Finally, it may be important to mention that although we have worked with the exact Vaidya solution, for simplicity, the same numerical results are obtained if we drop in the equations all the terms quadratic in m.
Final comments
We have treated a very crude model of an exploding star in the weak field limit of BD theory. However, the model presents aspects in the electromagnetic energy radiated to infinity that resembles some characteristics data of a Type Ia Supernova. The most noticeable feature is shown in the first three lines of Table 1 ; i.e., the value of the constant χ relates the absolute magnitude at maximum brightness and the decline rate in one magnitude from that maximum. This characteristic has become one of the most accurate method to measure luminosity distances to objects at cosmological distances [1, 2] . We also notice from Table 1 , that we may scale the value of L m by changing just the initial valuem i , without affecting the decline rate △t 1 . These values seem to be independent of the value of ω within a wide range. Finally, the total electromagnetic energy, radiated in the time interval [0, t 1 ], is independent of the value of χ and ω; the total energy E φ1 , in those cases in which it is not limited by the initial value f i , it is larger the smaller the value of ω.
On the other hand, the total active mass associated with the scalar field, as given by f i /2, is totally radiated to infinity. This process takes place in a time lapse considerably smaller than the time in which the star reaches its maximum brightness after explosion, if the mass of the shell is of the order of or greater than 10 −2 M ⊙ . This represents a mass loss in the ratio of the "tensor" component to the scalar component of 1 to (2 ω + 3), in agreement with a general result of Hawking [3] .

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