Lie symmetry approach to the time-dependent Karmarkar condition

The gravitational collapse of stellar objects is of much interest in relativistic astrophysics, requiring the solution of time-dependent relativistic field equations. Gravitational collapse problems were pioneered by Oppenheimer and Synder opp , not long after General Relativity was formulated by Einstein. Initially, the Schwarzschild solution was used until the discovery of the Vaidya solution vaid which accommodates null-radiation in the exterior atmosphere due to energy being radiated away from the collapsing body of fluid. There have been numerous attempts at obtaining solutions of the Einstein field equations for describing a radiating body, simultaneously undergoing gravitational collapse, and these efforts typically employ boundary conditions, equations of state, initial static configurations and separation of variables bon ; pret ; bog . The boundary of a collapsing star divides the spacetime into two distinct regions, the interior, $\mathcal{M}^{-}$, and the exterior spacetime, $\mathcal{M}^{+}$. The interior spacetime must match smoothly to the exterior spacetime in order to generate a physically viable model of a radiating star. Early attempts were made by Glass glas in which the Darmois and Lincherowitz matching conditions were utilised. Santos then established the appropriate boundary conditions for a spherically symmetric, shear-free, time-dependent metric which matches smoothly to the exterior Vaidya metric san . The Santos matching condition requires a non-vanishing pressure at the boundary for a star dissipating energy in the form of a radial heat flux. This is a necessary condition that ensures continuity of momentum flux across the boundary. The Santos junction conditions have been generalised to include shear nai ; thir1 , the cosmological constant as well as the electromagnetic field mah1 ; thir2 . Herrera and co-workers her1 ; her2 ; her3 have established important, fundamental results concerning matter distribution, stability of the shear-free condition, energy conditions and thermodynamic properties of gravitational collapse processes.
In developing models describing gravitational collapse, assumptions concerning the gravitational potentials and matter content of the gravitating body are often made. These have included acceleration-free and expansion-free collapse, Weyl-free collapse, anisotropic pressure configurations, the inclusion of shear and bulk viscosity and stipulations of equation of state mah2 ; gov ; red . Differential equations which arise, typically with respect to invariance of the junction conditions, lend themselves to the application of Lie symmetry methods and this can help to determine novel, exact solutions mso ; abe1 ; abe2 . In addition to restricting the matter content, conditions on the spacetime geometry are just as important. It is of interest to consider gravitational fields, typically represented by a Riemannian metric of four dimensions, to be immersed in a flat spacetime of higher dimension. Randall-Sundram and Anchordoqui-Bergliaffa re-established the conjecture that 4-dimensional spacetime might be embedded in higher dimensional flat space and much effort is made to achieve class one embedding mau1 . In general, an n-dimensional Riemannian spacetime is said to be of class p if it can be embedded into a flat space of dimension n + p. The Karmarkar condition kar is a necessary (but not sufficient) condition for a spacetime to be of class one pan . The derivation of the Karmarkar condition is purely geometric in nature providing relations among the components of the Riemann tensor. This in turn relates the metric potentials to one another and has thus assisted in model development. Of interest is to note that the Karmarkar condition together with pressure isotropy gives the interior Schwarzschild solution as the only bounded matter configuration. Recent attempts at modelling compact objects such as 4U 1538-52,PSR J1614-2230, Vela X-1 and Cen X-3 using the Karmarkar condition have produced models with favourable physical characteristics, consistent with observations bha ; mau2 . It is expected that the Karmarkar condition should also be favourable for time-dependent systems, perhaps improving stability which is an issue for shear-free spacetimes her2 .
Lie symmetry analysis is a very powerful tool for the study of nonlinear differential equations and we make use of this in generating solutions of differential equations obtained via Karmarkar’s condition. The steps that we follow in the analysis are: (i) we determine the Lie symmetries for each master partial differential equation for each model, (ii) the one-dimensional optimal system is determined in each case for the admitted Lie symmetries, (iii) we define similarity transformations from the Lie symmetries by using the Lie invariants which are used to reduce the master equation into an ordinary differential equation and (iv) exact closed-form solutions are then obtained. These steps have been applied before for various gravitational models with interesting results. Exact solutions describing charged radiation have been derived by applying Lie symmetries abe4 . Moreover, in abe2 Lie symmetries have been used to derive expanding and shearing models of radiating relativistic stars, while shear-free radiating stars were considered in abe3 . In GG , a new solution which describes a Euclidean star is derived by Lie symmetries and has the physical property of satisfying all the energy conditions and admitting a barotropic equation of state. For the field equations of the Schwarzschild model, conservation laws, invariant functions and differential operators derived by using the Lie symmetry analysis in ch5 ; ch6 . The Emden–Fowler equation which can describe gravitational spherically symmetric solutions was investigated via symmetry analysis in ch7 . For other applications of symmetry analysis in gravitational physics, we refer the reader to ch8 ; ch9 ; ch10 ; ch11 and the references therein.
The plan of this paper is as follows: In Section 2 we present the gravitational field equations for our analysis. In Section 3, the basic properties and definitions of Lie symmetries are given. The definition of the one-dimensional optimal system is discussed. The latter is necessary in order to perform a complete classification of the admitted similarity transformations. In Sections 4, 5 and 6 we solve the Karmarkar condition with respect to conformally flat, Weyl-free collapse and shear-free metrics respectively. A physical application is then given in Section 7 which describes radiating, Weyl-free collapse. In Section 8 we discuss the merits of the radiating stellar model obtained and in Section 9 we conclude on the suitability and novelty of our methods. Appendices A, B and C complete the presentation of this study where we present exact solutions for the three cases given in Sections 4, 5 and 6. These may be used for future studies.

The spherically symmetric line element is given by

where$~{}d\Omega^{2}$ is the line element of the two-sphere, that is,

An energy-momentum tensor incorporating heat flux, $q^{a}=(0,q^{1},0,0)$, is used,

where, $\rho$ is the energy density, $p_{r}$ the radial pressure, $p_{t}$ the tangential pressure and $q_{a}$ the heat flux vector. The timelike four-velocity of the fluid is $u_{a}$ and $\chi_{a}$ is a spacelike unit four-velocity along the radial direction. These quantities must satisfy $u_{a}u^{a}=-1$, $u_{a}q^{a}=0$, $\chi_{a}\chi^{a}=1$ and $\chi_{a}u^{a}=0$. In co-moving coordinates we have

The four-acceleration and expansion scalar are given by

Einstein’s time-dependent field equations are then given by

In the above, $\rho$, $p_{r}$, $p_{t}$ and $q$ are the energy density, radial pressure, tangential pressure and radial heat flux respectively.

In the context of geometry a differential equation (DE) may be considered as a function $H=H(y^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},...)$ in the jet-space $B=B\left(y^{i},u^{A},u_{,i}^{A},u_{,ij}^{A},...\right)$, where $\left\{y^{i}\right\}$ are the independent variables and $u^{A}$ are the dependent variables, while comma means derivative with respect to the variable $y^{i}$, that is $u_{,i}^{A}=\frac{\partial u^{A}}{\partial y^{i}}$.

Consider the infinitesimal one-parameter point transformation

with generator

The vector field $\mathbf{X}$ which defines the infinitesimal transformation (13)-(14) is called a Lie point symmetry of the DE $H$ if there exists a function $\kappa$ such that the following condition holds Stephani ; Bluman ; ibra

where

is the $n$th extension vector. Coefficient $\eta_{i}^{A}$ of the first extension vector is defined as

coefficient $\eta_{ij}^{A}$ of the second extension vector is given by the expression

while coefficient $\eta_{ij...j_{n}}^{A}$ of the $n$th extension vector is defined as

The main application of Lie point symmetries of a DE is focused on the construction of invariant functions which can be used for the determination of invariant solutions also known as similarity solutions.

For the Lie point symmetry $\mathbf{X}$ of the differential equation $H$ we define the Lagrange system Stephani ; Bluman ; ibra as

whose solution provides the characteristic functions
$W^{\left[0\right]}\left(y^{k},u\right),~{}W^{\left[1\right]}\left(y^{k},u,u_{i}\right)$, $W^{\left[2\right]}\left(y^{k},u,u_{,i},u_{ij}\right),...~{}$.
The characteristic functions, can be applied to reduce the order of the DE (in the case of ordinary differential equations) or the number of the dependent variables (in the case of partial differential equations).

Consider $A\left(t,r\right)=B\left(t,r\right)$ and $Y\left(t,r\right)=rB\left(t,r\right)$. In this case, the line element (1) is,

A spacetime with line element (26) is conformally flat and admits fifteen Conformal Killing vector fields (CKVs). For the conformally flat metric (26) where $B\left(t,r\right)$ is the unique unknown function, the Karmarkar condition becomes,

We apply the Lie theory to equation (27) from which we obtain the Lie point symmetry vectors

The admitted Lie symmetries form a seven-dimensional Lie algebra $G_{A}$, i.e. $\dim G_{A}=7$, and the associated commutators are shown in Table 1. Moreover, in Table 2 we present the Adjoint representation for the elements of the Lie algebra $G_{A}$.

We continue by using the results in Tables 1 and 2 to derive the one-dimensional optimal system for the partial differential equation (27). Consider the generic symmetry vector

From Table 2 we see that by applying the following adjoint representations

where for specific values of $\varepsilon_{1,}~{}\varepsilon_{2},~{}\varepsilon_{5}$ and $\varepsilon_{6}$ it follows

Thus, the two vector fields $Z^{\prime}$ and $Z$ are equivalent and lead to the same similarity solution. Coefficient constants $\alpha_{2},~{}\alpha_{4}$ and $\alpha_{7}$ are called relative invariants of the full adjoint action. Thus, in order to derive the relative invariants we solve the following system of partial differential equation olver

where $C_{ij}^{k}$ are the structure constants of the Lie algebra $G_{A}$ as presented in 1. Hence, system (31) becomes

from where it follows $\phi\left(\alpha_{i}\right)=\phi\left(\alpha_{2},\alpha_{4},\alpha_{7}\right)$, that is, the relative invariants are $\left\{\alpha_{2},\alpha_{4},\alpha_{7}\right\}$.

For $\alpha_{7}=0$, we find the equivalent symmetry vector

On the other hand for $\alpha_{2}=0$, the equivalent symmetry vector is

while when $\alpha_{4}=0$, the resulting equivalent symmetry vector is derived

Similarly, the following equivalent symmetry vectors are obtained:

Thus, a one-dimensional optimal system is constructed from the one-dimensional Lie algebras:

We proceed by using the Lie symmetries to determine similarity transformations, in the following we present the application of Lie point symmetries which lead to exact solutions expressed in closed-form functions.

We proceed with our study on the Karmarkar condition by assuming Weyl-free collapse spacetimes, where

that is, the line element (1) becomes

with $\omega\neq 0$. Recall that in the limit where $\omega=0,$ the latter line element describes the conformally flat spacetime (26). The Karmarkar condition for the above line element is

This admits a six-dimensional Lie algebra, $G_{B}$, consisted by the Lie symmetry vectors

In Table 3 we present the commutators of the Lie algebra $G_{B}$ while in Table 4 the Adjoint representation is given, necessary for the derivation of the one-dimensional optimal system.

Hence, the system of the partial differential equations (31) which provides the relative invariants is simplified as

which means that $\phi\left(\alpha_{I}\right)=\phi\left(\alpha_{1},\alpha_{2},\alpha_{6}\right)$.

Assume the generic symmetry vector $W=\alpha_{1}Y_{1}+\alpha_{2}Y_{2}+\alpha_{3}Y_{3}+\alpha_{4}Y_{4}+\alpha_{5}Y_{5}+\alpha_{6}Y_{6}$, then when $\alpha_{1}\alpha_{2}\alpha_{6}\neq 0$, the equivalent symmetry vector is

For $\alpha_{6}=0$, the equivalent vector field is $W^{\prime\prime}=\alpha_{1}^{\prime\prime}Y_{1}+\alpha_{2}^{\prime\prime}Y_{2},~{}$for $\alpha_{4}=0$, it follows $W^{\prime\prime\prime}=\alpha_{1}^{\prime\prime\prime}Y_{1}+\alpha_{3}^{\prime\prime\prime}Y_{3}+\alpha_{6}^{\prime\prime\prime}Y_{6},~{}$while when $\alpha_{1}=0$ we have the equivalent symmetry vector $~{}W^{\prime\prime\prime\prime}=\alpha_{2}^{\prime\prime\prime\prime}Y_{2}+\alpha_{6}^{\prime\prime\prime\prime}Y_{6}.~{}$

Similarly, the following equivalent symmetry vectors are obtained:

We conclude that the one-dimensional optimal system consisted by the one-dimensional Lie algebras: