Coordinate effect: Vaidya solutions without integrating the field equations

TL;DR

This paper extends Vaidya's algorithm to arbitrary radial radiation and shows how to algebraically derive Vaidya solutions without solving field equations, highlighting the role of curvature coordinates.

Contribution

It introduces an algebraic method for deriving Vaidya solutions and explores the implications for variable mass functions in spacetime.

Findings

Algebraic derivation of Vaidya solutions without integrating field equations

Extension of Vaidya's algorithm to arbitrary radial radiation

Discussion on the problem of variable mass in spacetime

Abstract

We extend Vaidya's algorithm for the description of a central mass losing or gaining energy due to electromagnetic-type radiation (`null dust') to the case of arbitrary radial corpuscular radiation. We also demonstrate the remarkable possibility of purely algebraic deduction of the Vaidya solution without integrating the field equations, and interpret this possibility as an artifact of curvature coordinates. Since Vaidya's approach by itself cannot lead to certain dependence of mass on spacetime coordinates, the search for a corresponding mass-function represents an independent issue. In this regard, as a perspective, we discuss an outlook on the problem of variable masses as a whole.

Equations

$$d{s}^2=e^{\nu (r)}d{t}^2-e^{\lambda (r)}d{r}^2-r^{2}d{\Omega }^2,$$

$$d{s}^2=\left(1-\frac{2m}{r}\right)d{t}^2-{\left(1-\frac{2m}{r}\right)}^{-1}d{r}^2-r^{2}d{\Omega }^2,$$

$${G_{\mu }}^{\nu }\equiv {R_{\mu }}^{\nu }-\frac{1}{2}R{\delta }_{\mu }^{\nu }=\varkappa {T_{\mu }}^{\nu }=\varkappa \epsilon k_{\mu }{k}^{\nu },k_{\alpha }{k}^{\alpha }=0,k^2=k^3=0,$$

$${T^{\mu }}_{\alpha }{k}^{\alpha }=0,\mu =0⟹{G_0}^0+{G_1}^0{e}^{\frac{\nu -\lambda }{2}}=0,$$

$${T_{\alpha }}^{\alpha }=0⟹{G_0}^0+{G_1}^1=0,$$

$$d{s}^2=e^{\nu (r,t)}d{t}^2-e^{\lambda (r,t)}d{r}^2-r^{2}d{\Omega }^2$$

$$e^{-\lambda }\left({\lambda }^{\prime }-\frac{1}{r}\right)+\frac{1}{r}+\dot{\lambda }{e}^{-(\lambda +\nu )/2}=0,$$

$$e^{-\lambda }\left({\lambda }^{\prime }-{\nu }^{\prime }-\frac{2}{r}\right)+\frac{2}{r}=0,$$

$${T_2}^2={T_3}^3=0⟹{G_2}^2={G_3}^3=0,$$

$$e^{-\lambda }\left(\frac{{\lambda }^{\prime }-{\nu }^{\prime }}{r}-{\nu }^{\prime \prime }-\frac{{{\nu }^{\prime }}^2}{2}+\frac{{\lambda }^{\prime }{\nu }^{\prime }}{2}\right)+e^{-\nu }\left(\ddot{\lambda }+\frac{{\dot{\lambda }}^2}{2}-\frac{\dot{\lambda }\dot{\nu }}{2}\right)=0,$$

$${R_2}^2={R_3}^3=0,$$

$$-g_{11}=e^{\lambda (r,t)}={\left(1-\frac{2m(r,t)}{r}\right)}^{-1}\equiv D^{-1}.$$

$$g_{00}=e^{\nu (r,t)}=\frac{{\dot{\lambda }}^2{e}^{\lambda }{r}^2}{{\left(e^{\lambda }+{\lambda }^{\prime }r-1\right)}^2}.$$

$$e^{\nu }=\frac{{\dot{m}}^2}{{m^{\prime }}^2}{\left(1-\frac{2m(r,t)}{r}\right)}^{-1}=\frac{{\dot{m}}^2}{{m^{\prime }}^2}{D}^{-1}.$$

$$m^{\prime }{e}^{\nu /2}=-\dot{m}{e}^{\lambda /2}.$$

$$d{s}^2=\frac{{\dot{m}}^2}{{m^{\prime }}^2}{D}^{-1}d{t}^2-D^{-1}d{r}^2-r^{2}d{\Omega }^2.$$

$$\epsilon =\frac{m^{\prime }}{4\pi r^2},$$

$$m(r,t)\equiv {\int }_0^{r}4\pi \epsilon (t,r)r^{2}dr.$$

$${G_{\mu }}^{\nu }\equiv {R_{\mu }}^{\nu }-\frac{1}{2}R{\delta }_{\mu }^{\nu }=\varkappa {T_{\mu }}^{\nu }=\varkappa \epsilon k_{\mu }{k}^{\nu },$$

$$k_{\nu }{k}^{\nu }=\{\begin{array}{l}1\\ 0\\ -1\end{array},k^2=k^3=0,$$

$$k^{\mu }=\alpha u^{\mu }+\beta n^{\mu },$$

$$u_{\nu }{u}^{\nu }=1,n_{\nu }{n}^{\nu }=-1,u_{\nu }{n}^{\nu }=0,$$

$$k_{\nu }{k}^{\nu }={\alpha }^2-{\beta }^2.$$

$$k^{\mu }=\alpha u^{\mu }+\beta n^{\mu }=\frac{u^{\mu }}{\sqrt{1-v^2}}+\frac{v{n}^{\mu }}{\sqrt{1-v^2}},$$

$$u^{\mu }={\delta }_0^{\mu }/\sqrt{g_{00}},n^{\mu }={\delta }_1^{\mu }/\sqrt{-g_{11}},$$

$$k^0=\frac{\alpha }{\sqrt{g_{00}}},k^1=\frac{\beta }{\sqrt{-g_{11}}},k_0=\alpha \sqrt{g_{00}},k_1=-\beta \sqrt{-g_{11}}.$$

$${T_{\alpha }}^{\mu }{k}^{\alpha }=\epsilon k^{\mu },\mu =0⟹{G_0}^0{k}^0+{G_1}^0{k}^1=\varkappa \epsilon k^0$$

$${T_{\alpha }}^{\alpha }=\epsilon ⟹{G_0}^0+{G_1}^1=\varkappa \epsilon ,$$

$${T_2}^2={T_3}^3=0⟹{G_2}^2={G_3}^3=0⟹{R_2}^2={R_3}^3=-\varkappa \epsilon /2.$$

$${G_1}^0=\varkappa \epsilon k_1{k}^0⟹\varkappa \epsilon k^0=\frac{{G_1}^0}{k_1},$$

Full text

1 Introduction

In general relativity (GR), the curvature coordinates play a special role. Indeed, for example, in the case of spherical symmetry and given the interval

[TABLE]

the angular part of the metric (the unit sphere) remains flat, greatly simplifying the calculations. Moreover, in case of mutual reciprocity of the first two metric coefficients, $g_{00}g_{11}=-1$, as is the case for the Schwarzschild solution,

[TABLE]

the Einstein equations become linear Padmanabhan2010 .

As opposed to the Schwarzschild solution (2), in Vaidya’s approach a mass, initially in vacuum, begins to emit (or absorb) the radiation, which, in turn, gravitates, i.e. is described by its energy-momentum tensor at the right side of Einstein’s equations. In a sense, we are dealing here with the extension of the Schwarzschild problem to the case of a variable radiating mass, therefore, Vaidya’s choice to search for the solution in curvature coordinates is reasonable. A nontrivial moment in this technique is the automatic inclusion of mass variability itself (the “mass function”) into the equations as a function of all 4-coordinates.

Here, we will show the remarkable possibility of a purely algebraic derivation of the Vaidya metric solely on the basis of the Vaidya ansatz connecting the Schwarzschild seed solution with the general form of the solution in curvature coordinates.

Next, we demonstrate the universal nature of the Vaidya approach based on the application of the curvature coordinates using the mechanism of mass loss due to corpuscular radiation (e.g., stellar wind, massive neutrinos, etc.).

After discussing the peculiarities of using the curvature coordinates (primarily as opposed to isotropic coordinates), we proceed to consider technical as well as fundamental issues connected with the so-called first integral and specific conditions for the mass-function necessary for closing the system and compatibility check of each particular solution.

2 Symmetries in the Vaidya problem

The general relativistic problem of radiating mass was presented in the pioneering works of Vaidya Vaidya1943 ; Vaidya1951 ; Vaidya1953 . The corresponding Einstein equations with radial emission (‘null dust’),

[TABLE]

imply the following two types of symmetry conditions

[TABLE]

[TABLE]

which are in non-static curvature coordinates

[TABLE]

take the form:

[TABLE]

[TABLE]

where the prime denotes differentiation with respect to $r$, and the dot with respect to $t$. The third condition of symmetry, taken into account by Vaidya,

[TABLE]

or, explicitly,

[TABLE]

is in fact not independent. Indeed, given the zero trace of the energy-momentum tensor, we have from (9):

[TABLE]

so that (10) reduces to (11), which, in its turn, appears to be equivalent (up to a constant factor) to the equation (8).

The key point is the main Vaidya’s ansatz according to which, for reasons of correct asymptotics at infinity, it is assumed that the $g_{11}$-component of the metric (6) is represented in the Schwarzschild form, but with the replacement of the constant mass by a certain mass-function $m(r,t)$ Vaidya1951 :

[TABLE]

Then the problem reduces to determining the component of the metric $g_{00}$, which, as it turned out, in the curvature coordinates can be found, up to an unknown mass-function $m(r,t)$, algebraically from the condition (4).

3 Vaidya’s solution as an algebraic consequence of the main ansatz

Indeed, the condition (4) in the form (7) is an algebraic relation that can be easily solved with respect to $g_{00}$:

[TABLE]

Substituting the ansatz (12) instead of $e^{\lambda}$, we finally get:

[TABLE]

Alternatively, following Vaidya1951 , we substitute (12) directly into (7), which gives:

[TABLE]

This relation is equivalent to (14). In both cases, we get the Vaidya solution:

[TABLE]

So, due to radiation symmetries expressed in curvature coordinates as (7, 8, 10), two unknown functions $\lambda(r,t)$ and $\nu(r,t)$ are reduced with the described method to a single one, $m=m(r,t)$, without necessity of integration of the field equations (for the traditional approach see, for example, Mychelkin1990a ) and without using the explicit form of the energy-momentum tensor on the right side. The energy density being necessary for self-consistency of the task is established after substituting the metric (16) in the field equations, which gives

[TABLE]

or, in integral representation,

[TABLE]

The latter expression was used by Raychaudhuri Raychaudhuri1953 as the initial ansatz for the subsequent alternative deduction of the Vaidya metric (16). Importantly, from (18) one may see that the radiated mass-energy is represented by a function of both time and radial coordinate simultaneously.

4 Corpuscular radial emission

Let us consider a direct extension of the Vaidya algorithm, which does not require the integration of the Einstein equations as well, to the case of arbitrary radial corpuscular radiation (including bradyon and tachyon variants); in part, we follow here the unpublished preprint Mychelkin1990a , while correcting its inaccuracies and also modifying the interpretation of the results. Then, Einstein’s equations (3) can be generalized as follows:

[TABLE]

[TABLE]

which includes all situations: $k_{\nu}k^{\nu}=0$ (null-dust), $k_{\nu}k^{\nu}=1$ (bradyons) and $k_{\nu}k^{\nu}=-1$ (tachyons). Each of these conditions implies its own separate parameterization. In general, we adopt

[TABLE]

[TABLE]

i.e.

[TABLE]

Thus, the ordinary case (null-dust) is parameterized as $\alpha=\beta$, with equal contributions from time- and space-like parts.

First, we study the time-like version $k_{\nu}k^{\nu}=1$ (“bradyon dust emission”) and accept the following gauge conditions for the radial vector $k^{\mu}$ of the collinear initial particle velocity $0\leq v<1$ (in units of the speed of light):

[TABLE]

with basis unit vectors

[TABLE]

for which

[TABLE]

Then, the previous symmetry conditions (4) and (5) turn into

[TABLE]

and

[TABLE]

and the third condition (9) becomes:

[TABLE]

Now, unlike the null-dust case, from the equations (26)-(28) one needs to exclude the intrinsic energy density $\varepsilon$ using Einstein’s equations, for example:

[TABLE]

from which it also follows that

[TABLE]

Then, substituting (29) into (26) taking into account (25) and (6) we finally get in curvature coordinates:

[TABLE]

where

[TABLE]

Applying the Vaidya ansatz (12) to (31) after transformations, we find an analogue of the relation (15):

[TABLE]

or, in the final form,

[TABLE]

where again $m$ stands for $m(r,t)$.

Note that for $\beta/\alpha=v\rightarrow 0$ from (32) it follows $\dot{m}\rightarrow 0$ (as it should be from physical considerations), so $v\rightarrow 0$ in the expression (33) does not necessarily imply divergence.

Thus, the Vaidya metric for generalized (corpuscular) radial emission,

[TABLE]

is obtained again without integrating the field equations, due to the same radial symmetries in curvature coordinates. The considered non-null emission could also be represented as a source for the Vaidya-type metrics with non-zero radial pressure Culetu2016 .

As for the space-like sector (tachyon dust emission, compare with Foster1972 ) defined by the requirement $k_{\nu}k^{\nu}=-1$, analytically it differs from the one considered above only in the initial parameterization in the gauge condition of the type (23):

[TABLE]

where $1<v<\infty$ in this gauge. Such tachyon emission might be of interest, as modern experiment is not capable of excluding the possibility of the existence of background neutrinos in tachyonic state (see, e.g., Caban2006 ; Ehrlich2015 ; Mychelkin2015 ; Makukov2016 ).

5 Cosmological term in curvature and isotropic coordinates

The described Vaidya algorithm can be extended to the case of the simplest cosmological background Mallett1985 if we include the regular $\Lambda$-term into the ansatz (12), similar to the case of the Kottler (Schwarzschild–de Sitter) metric

[TABLE]

After substituting into the extended condition of the type (4),

[TABLE]

algebraic transformations lead to Vaidya-type metric in the curvature coordinates embedded into the de Sitter background, similar to (13), (14) and (16) with the only replacement of $D$ by $\tilde{D}$:

[TABLE]

and again without integration of the Einstein equations (in curvature coordinates only).

To see how much the situation changes by considering isotropic coordinates instead,

[TABLE]

we first consider the case of constant mass in the definition of $\tilde{D}$ (36), i.e. the standard Kottler metric in curvature coordinates (for which, as in the Schwarzschild case, $g_{00}g_{11}=-1$). Even in this simplest case deducing the transformation from curvature to isotropic coordinates is nontrivial, and was first obtained in Kozhanov1986b . Since this result is interesting in itself and is widely unknown, we briefly outline it here. Using the standard Tolman algorithm Tolman1949 , we get from the Kottler metric:

[TABLE]

Via the substitution found by the polynomial method from Zhuravsky1941a

[TABLE]

the elliptic integral (40) is reduced to the Weierstrass canonical form

[TABLE]

whose inversion is the elliptic Weierstrass function with invariants $g_{2}$ and $g_{3}$,

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Therefore, according to (41) and (43), the transformation from curvature to isotropic coordinates is:

[TABLE]

Hence, the interval for the Kottler metric in isotropic coordinates, after some algebra Kozhanov1986b , assumes the final form:

[TABLE]

Now, $\Lambda$-term enters the functions $\wp$ and $\wp^{\prime}=d\wp(x,g_{2},g_{3})/dx$ through the parameter $g_{3}$, in accord with (43).

However, moving in (40) to general Vaidya case, when $\lambda(r)\rightarrow\lambda(r,t)$, we get, instead of the constant $m$, an unknown mass function $m(r,t)$. In such case, integration and, accordingly, conversion to isotropic coordinates becomes impossible.

6 Relation of Vaidya algorithm to isotropic and null coordinates

Although there is no explicit transformation of the Vaidya metric with an unknown mass-function to other frames, we can still try to apply the Vaidya algorithm directly in isotropic coordinates

[TABLE]

In such a case, since the null-dust symmetries (4), (5) and (9) are preserved, the field equations corresponding to (7), (8) and (10) will be as follows:

[TABLE]

[TABLE]

and

[TABLE]

where the derivatives with respect to $R$ are now denoted as primes. Further, following Vaidya’s algorithm, we again consider the Schwarzschild metric as an initial reference, but in isotropic, rather than curvature, coordinates,

[TABLE]

Based on this, we adopt the ‘isotropic” ansatz as direct analog of (12):

[TABLE]

where the constant mass is replaced by the mass-function $m=m(R,t)$.

However, unlike in curvature coordinates, the remaining component of the desired isotropic metric $g_{00}(R,t)$ does not follow algebraically from the corresponding equation (47). Instead, substituting (51) in (47), we arrive at the following differential equation with respect to $g_{00}=e^{\nu}$:

[TABLE]

where $y=e^{3\nu/2}$. The solution of this equation (and of the whole system (47)-(49)) cannot be obtained if the explicit dependence $m=m(R,t)$ is unknown. Thus, the Vaidya algorithm does not work in isotropic coordinates, and the transformation from curvature to isotropic coordinates does not exist either.

By direct analogy with the Schwarzschild case, there is another commonly used representation of the Vaidya metric in the Eddington-Finkelstein null coordinates Vaidya1953 , namely:

[TABLE]

Expression (52) differs from its Schwarzschild counterpart only by the replacement of $m=\text{const}$ with the variable mass function $m(u)$. But now the transfer from curvature to null coordinates deserves a deeper consideration. Traditionally, the metric (52) can be obtained from the original Vaidya metric (16) using the differential transformation

[TABLE]

where $r^{*}$ corresponds to the so-called ingoing ($+$) or outgoing ($-$) tortoise coordinates. For the Schwarzschild problem in standard curvature coordinates this transformation represents the usual way to go to the metric (52) with $m=\text{const}$. Before applying this transformation to the variable $m\neq\text{const}$, one must first represent the factor $\dot{m}^{2}/m^{\prime 2}$ in (16) as

[TABLE]

which corresponds to fixing the value of (53) on the lines of constant $u$. In this particular case, (16) takes the “desired form”,

[TABLE]

and then the metric (52) is obtained formally using the traditional algorithm (53). However, this transition to $m(u)$ is dubious, since in case of an unspecified mass function $m=m(r,t)$, the expression (53) cannot be integrated. In other words, the final representation of the trailing (leading) coordinate $u=t\pm r^{*}$ is defined only up to the general quadrature

[TABLE]

which, in this case, cannot be explicitly integrated (see also in Culetu2016 ). Thus, the Vaidya metric in null coordinates (52) appears, in a sense, symbolic rather than operational.

Summing up the last two sections, one might say that when searching for explicit transformations of Vaidya’s solution from the curvature coordinates, one always has to know the explicit form of the mass function.

7 First integral and mass-function

The Einstein equations in the Vaidya metric can only be integrated up to the so-called “first integral”. This brings us to the discussion of the remaining equation (8). Substituting the metric components (12) and (14) into (8), we arrive at an important identity:

[TABLE]

From here, in particular, we can get the known first integral Vaidya1951 :

[TABLE]

Substitution of (57) into (16) leads to another commonly used representation of the Vaidya metric:

[TABLE]

where for brevity we write $m$ instead of $m(r,t)$.

Vaidya interpreted (57) as a differential equation that should be resolved with respect to $m$ Vaidya1951 . Formally, the formula (57) is a relation between two unknown functions $m(r,t)$ and $f(m)$ to be used for the definition of $m$.

From the standpoint of searching for possible restrictions on the form of $f(m)$, on the contrary, one can substitute a general metric (58) directly into the Einstein equations (3). Then, in particular, using the $22$- and $01$-components, we get

[TABLE]

Further, given that $f=f(m)$ Vaidya1951 , we can write $\dot{f}(m)=f_{,m}\dot{m}$ and, similarly, $f^{\prime}(m)=f_{,m}m^{\prime}$ (thus $\frac{f^{\prime}}{f}=\frac{\dot{f}}{f}\frac{m^{\prime}}{\dot{m}}$) and multiplying now (59) by $\dot{m}/m^{\prime}$, one gets

[TABLE]

But $2\dot{m}/r=-\dot{D}$, so by integrating (60), we get instead of (57) some more general valid form of the first integral, where the role of the integration constant $C$ is played, generally speaking, by an arbitrary function of the radial coordinate $r$:

[TABLE]

In Gupta1988 , an expression similar to (61) is obtained, and arguments are made in favor of choosing $C(r)=1$, which, as expected, coincides with the original functional (57), and, in turn, actually turns the Vaidya metric (58) back to its original form (16) with the only unknown mass function $m=m(r,t)$.

8 Discussion

We have seen that for the presentation of Vaidya’s solution in closed form in curvature or null coordinates it is necessary to know the explicit dependence of the mass-function on spacetime coordinates, and this is a separate issue. There were attempts to bring the metric in the form (58) asymptotically to the standard Schwarzschild form (55) by imposing a physically acceptable constraint of the type $f(m)=\pm\dot{m}$ Mallett1985 ; Vaidya1951 ; Gupta1988 . However, direct substitution of the metric (55) into the field equations shows that although the condition (8) is satisfied, the other equation (7) is incompatible with the requirement $f(m)=\pm\dot{m}\neq 0$.

In fact, this could be expected, because it follows from (15) and (16) that the mass function $m(r,t)$ cannot be a function of only $r$ or only $t$. In particular, this means that the known phenomenological Eddington-Jeans power-law, $\dot{m}=-km^{n}$, frequently used to describe stellar evolution, cannot be incorporated into the Vaidya approach.

We emphasize that, in any case, the freedom to choose the mass function $m(r,t)$ is limited by the functional relation (56), or its consequence (59), which, however, is difficult to satisfy in practice. For example, from (56) we can again obtain that the condition $m=m(t)$ is not compatible with the Vaidya metric, and the same is true for the simplest dependencies of both multiplicative $m(r,t)=f(r)\phi(t)$ and additive $m(r,t)=\alpha f(r)+\beta\phi(t)$ types.

In general, the specific nature of solutions involving the use of curvature coordinates in GR is not an exception. Thus, the Fisher solution Fisher1948 of the Einstein equations with minimal scalar field in curvature coordinates is also found only up to a non-trivial (not analytically solvable) identity, although in isotropic and other coordinates this problem is solved in exact form Janis1968 ; Xanthopoulos1989 (for more details on this point, see Makukov2020 ). Besides, it is worth to note that the extension of the Kottler metric onto variable cosmological term is uniquely related to curvature coordinates as well Dymnikova . The curvature and related null coordinates also play crucial role in deducing the Kerr rotational solution Kerr1963 ; Newman1965 and in performing the Choptuik dynamical chaos algorithm for collapsing scalar field Choptuik1993 .

There exist also more mathematical studies (e.g., Shaikh2019 ) on geometric symmetries generated by the structure of the Riemann-Christoffel and Weyl conformal curvature tensors, without examining the mass-function properties111Acting in this way, the authors in Shaikh2019 analyze the pure radiation case in Vaidya’s (null) coordinates (52), and also compare it with the analogous Ludwig-Edgar solution. They found, among other things, that both metrics are similar in a sense that both are Ricci-simple but dissimilar with respect to belonging to pseudosymmetric or semisymmetric manifolds. Note also that examination of the structure of the energy-momentum tensor for background radiation in Shaikh2019 is in concordance with our approach (cf. (17) as an application of the Einstein tensor)..

9 Outlook on the problem of variable mass in general relativity

The Vaidya problem describes the behavior of radial emission background radiated from/onto the central non-rotating mass. For this reason, strictly speaking, the right side of the Einstein equations must contain, apart from the emission, a massive source, which is absent in explicit form in the Vaidya approach. The situation resembles the transfer from the Laplace equation to the Poisson equation with a point source in Newtonian gravity. In general relativistic case, this might lead to the following extension of the Vaidya problem:

[TABLE]

where $\rho(t,\vec{r})=\delta(\vec{r})M(t)$. In this case the rate of loss of the central mass $\dot{M}$ is determined by the radiation intensity, i.e.

[TABLE]

where $m$ characterizes the background radiation. With such modification there is the possibility from the start to adopt a definite law for the central mass time-dependence. Following general Vaidya’s algorithm, we may choose some static metric as a fiducial one. For it, we believe that the most natural choice (dictated by the conformance to observations) is the generalized Papapetrou-like exponential metric,

[TABLE]

with $\phi(t,R)=M(t)/R$, as opposed to Vaidya’s preference of the Schwarzschild-like curvature coordinates (6). Naturally, using (64) implies inclusion into consideration of the scalar background, i.e. adding the scalar field energy-momentum tensor into the right side of the Einstein equations:

[TABLE]

as well as the related Klein-Gordon equation. Negative sign in (65) means that scalar field is taken in antiscalar regime Makukov2018 .

However, it is worth to note that the direct application of such algorithm may encounter certain difficulties due to restrictions concerning the application of distributions in the Einstein equations Geroch1987 . This problem might be surmounted by applying a smoothing algorithm to the central mass (i.e. to $\delta$-distribution) Makukov2016 . In this case, the singular Newtonian-type potential in the exponential metric (64) may be transformed as

[TABLE]

where $\sigma$ is the smoothing scale. In principle, such approach could allow to relate a definite mass-loss law to observed radial emission.

10 Conclusion

We have analyzed a peculiar feature of the Vaidya metric, which is that, unlike in traditional methods, its derivation does not require direct integration of the Einstein equations. The resulting algebraic algorithm can also be successfully used, in particular, to extend the Vaidya approach to corpuscular emission and, furthermore, to cosmological background taken into account.

The price for this unique possibility is that the solution of Einstein’s equations under consideration can de facto be obtained only up to an unknown mass-function. At the same time, both the mass-function $m(r,t)$ and the functional (57) must identically satisfy the non-trivial nonlinear partial differential equation (56), or, on equal footing, its corollary (59). As for the search for the explicit forms of $m(r,t)$, in any case each concrete expression (if it exists) should be carefully tested with the full system of the Einstein equations.

In conclusion, the issue of the closed representation of variable masses in arbitrary coordinates in general relativity, strictly speaking, remains open. The Vaidya solution was the first significant step in this direction, but implemented for a particular case of curvature coordinates. As an alternative, we propose an approach related to explicitly time-dependent mass as described in Outlook.

Acknowledgements.

The work is partially supported within the grant No. AP08052312 of the Ministry of Education and Science of the Republic of Kazakhstan.