Conformal symmetries of the energy-momentum tensor of spherically symmetric static spacetimes

Symmetries and conformal symmetries play a very important role in mathematical physics. One of the fundamental symmetries on a Riemannian manifold is that of the metric tensor $\mathbf{g}$ written mathematically as katzin

Here $\sigma$ is the conformal factor and $\pounds_{\xi}$ represents the Lie derivative operator relative to the vector field ${\xi}$, which gives isometries or Killing vectors (KVs) if the $\sigma$ is zero, homothetic motions (HMs) if it is a constant, and conformal Killing vectors (CKVs) if it is a function of the coordinates $x^{a}$. In component form we can write the above equation as

Apart from the metric tensor the other quantities fundamental to the Einstein field equations (EFEs)

are the stress-energy tensor, ${\bf T}$, which describes the matter field in the manifold, the Ricci tensor, $\bf R$, which is a contraction of the curvature tensor, and the Ricci scalar $R$. In Eq. (3) $\kappa$ is the coupling constant defined by $\kappa=8\pi G/c^{4}$, $G$ and $c$ are Newton’s gravitational constant and the speed of light. Thus, the symmetries of both the stress-energy tensor and the Ricci tensor, play a significant role. These symmetries known as matter collineations (MCs) and Ricci collineations (RCs) hall-book , respectively, satisfy the equations

Similarly, one can define collineations for the curvature and Weyl tensors hall-book .

Solutions of EFEs can be classified by requiring these symmetries and thus a complete list of metrics having certain symmetry can be obtained ESEFEs . Spacetimes have been classified on the basis, for example, of KVs bq1987 ; qz1988 ; qz1995 , HMs dz1997 , CKVs maartens1995 ; keane2004 , RCs tm1990 ; bk1993 ; fqz1995 ; hall1996 ; qz1998 ; ziad2003 ; saif2003 ; ugur1 ; cb2002 ; ugur2 ; ugur3 ; tsamparlis1 and MCs ugur4 ; sharif2003 ; sharif2004 ; ugur5 ; ugur6 . This also provides a way to find new solutions of EFEs which are otherwise very difficult to solve. These collineations have been generalized to define what are called conformal collineations (or inheritance collineations duggal1 ; duggal11 ; duggal2 ). Thus we obtain conformal matter collineations (CMCs)

or conformal Ricci collineations (CRCs) defined by

Conformal symmetry is physically significant as CKVs, for example, generate constants of motion along the null geodesics for massless particles which are conserved quantities. On the other hand, it is of mathematical interest to obtain classification by conformal collineations and to investigate their relation with collineations. Though there has been a good amount of literature on the study of CKVs, the interest in conformal collineations is relatively recent. The complete classification of spherically symmetric static space-times by their CRCs when the conformal factor $\phi$ is a constant has been carried out in Ref. bkk2003 . The CRCs with a non-constant $\phi$ have been studied for the Friedmann-Robertson-Walker spacetimes cb2002 , the general static spherically symmetric spacetimes crcsph , the non-static spherically symmetric spacetimes bokhari2020 and Kantowski-Sachs spacetimes hkb2018 . For pp-waves relationship between CRCs and CKVs has been studied in Ref. keane2004 . Recently, Akhtar et al. bokhari2018 have classified static plane symmetric space-times according to CRCs. Further, the CRCs for the Einstein-Maxwell field equations in the case of non-null electromagnetic fields have been investigated as well faridi .

In this paper we classify spherically symmetric static spacetimes by their CMCs. The equation (5) for CMCs in component form can be written as

where $\psi$ is the conformal factor which is a function of all the spacetime coordinates $x^{a}=(x^{0},x^{1},x^{2},x^{3})$. In this paper we use the usual component notation in local charts and a partial derivative will be denoted by a comma. Note that the above equation gives MCs if $\psi=0$, thus the classification of spherically symmetric static spacetimes by MCs sharif2003 becomes a special case of the classification obtained in this paper. We call a CMC as proper if it is neither a KV nor an MC. The set of all CMCs on the manifold is a vector space, but it may be infinite dimensional. If $T_{ab}$ is non-degenerate, i.e. $det(T_{ab})\neq 0$, then the Lie algebra of CMCs is finite dimensional but if $T_{ab}$ is degenerate, it may be infinite. Thus, in the case of a non-degenerate energy-momentum tensor, i.e. $det(T_{ab})\neq 0$, we use the standard results on conformal symmetries to obtain the maximal dimensions of the algebra of CMCs as 15. Since $T_{ab}$ describes the distribution and motion of matter contents of a manifold, and mathematically it is very similar to the the Ricci tensor, the study of MCs and CMCs has a natural geometrical as well as physical significance.

In the next section, we setup the CMC equations for static spherically symmetric spacetimes. In Section III these equations are solved when the energy-momentum tensor is degenerate, while in Section IV we obtain results when the tensor is non-degenerate. We find that the degenerate case always gives infinite dimensional Lie algebras of CMCs. We conclude with a brief summary and discussion in Section V. Throughout the paper, we will consider four-dimensional space-times, and space-time indices will be denoted by small Latin letters (e.g., $a$, $b$, $c$,…) and the metric has signature $(+,-,-,-)$.

The metric (8) has time-independent coefficients, and using the field equations (3) with $c=1$ and $G=\frac{1}{8\pi}$, i.e., $\kappa=1$, the components of the energy-momentum tensor $T_{ab}$ become

Similarly, the matter tensor form can be written as

Using the above energy-momentum tensor components, the Ricci curvature scalar given in (13) can be cast into the form

Further, when we state the energy-momentum tensor components given above in terms of the Ricci tensor components (9)-(11) we find that

In GR, physical fields are described by the symmetric tensor $T_{ab}$ which is the energy-momentum tensor of the field. We have the covariant decomposition/identity for $T_{ab}$ as follows tp2019 :

where $h_{ab}=g_{ab}-u_{a}u_{b}$ is the projection tensor, and the quantities $\rho,p,q_{a}$ and $\pi_{ab}$ are the physical variables representing the mass density, the isotropic pressure, the heat flux and the traceless stress tensor, respectively, as measured by the observers $u^{a}$. In the above decomposition, $T_{ab}$ is described by two scalar fields ($\rho,p$), one spacelike vector ($q^{a},\,q_{a}u^{a}=0$), and a traceless symmetric 2-tensor ($\pi_{ab},\,g_{ab}\pi^{ab}=0$). The irreducible parts of $T_{ab}$ are defined as

where $u^{a}$ is a timelike unit four vector field normalized by $u^{a}u_{a}=1$. The energy-momentum tensor $T_{ab}$ given in (24) represents the general anisotropic fluid, and reduces to an anisotropic fluid without heat flux if $q^{a}=0$, an isotropic non-perfect fluid if $\pi^{ab}=0$, a perfect fluid if $q^{a}=0$ and $\pi^{ab}=0$, and a dust if $p=0,q^{a}=0$ and $\pi^{ab}=0$. GR is a classical theory of relativity, however, in the field equations (3), the classical spacetime geometry is also related to the stress-energy tensor of quantum matter. To overcome this inconsistency, we need to embed GR (or its generalizations) within some quantum mechanical framework, i.e., quantum gravity. For the metric ansatz (8), it is customary to choose the fluid to be at rest because the spacetime is static, i.e., $u^{a}=u^{0}\delta^{a}_{0}$. Then, using the normalization condition of the four-velocity, that is, $u^{a}u_{a}=1$, one can find $u^{0}=e^{-\nu/2}$. Thus, we find from (27) that for this choice of observers the heat flux vanishes ($q^{a}=0$), which is expected from the symmetries of the metric. Also, under the latter choice of observers, the remaining physical variables $\rho,p$ and $\pi_{ab}$ that follow from (25), (26) and (28) are

If the choice of observers as a timelike four-vector field $u^{a}=u^{0}\delta^{a}_{0}$ is not appropriate for any reason, then we must apply the normalization condition of the four-velocity with $u^{a}u_{a}=-1$ by choosing the four-velocity as $u^{a}=u^{1}\delta^{a}_{1}$ which is a spacelike four-vector, since one can always normalize the four-velocity to $\pm 1$. Then we find for the metric (8) that $u^{1}=e^{-\lambda/2}$. For the spacelike four-vector, the projection tensor has the form $h_{ab}=g_{ab}+u_{a}u_{b}$. Then this choice effects the mass density, the isotropic pressure $p$ and the traceless stress tensor $\pi_{ab}$ due to Eqs. (25), (26) and (28) such that

In GR, it is conventional to restrict the possible energy-momentum tensors by imposing energy conditions. The energy conditions for the energy-momentum tensor $T_{ab}$ to represent some known matter fields are the conditions that are coordinate-independent restrictions on $T_{ab}$. In literature, there are five categories of the energy conditions. These are the trace energy conditions (TEC), null energy conditions (NEC), weak energy conditions (WEC), strong energy conditions (SEC) and dominant energy conditions (DEC). The TEC means that the trace of the energy-momentum tensor $T=g^{ab}T_{ab}$ should always be positive (or negative depending on the signature of metric). The NEC mathematically states that $T_{ab}k^{a}k^{b}\geq 0$ for any null vector $k^{a}$, i.e., $k^{a}k_{a}=$. On the other hand, the WEC requires that $T_{ab}t^{a}t^{b}\geq 0$ for all timelike vectors $t^{a}$. The SEC states that $T_{ab}t^{a}t^{b}\geq\frac{1}{2}Tt^{c}t_{c}$ for all timelike vectors $t^{a}$. The DEC includes the WEC, as well as the additional requirement that $T^{ab}t_{b}$ is a non-spacelike vector, i.e., $T_{ab}T^{b}_{\,\,c}t^{a}t^{c}\leq 0$ carrol ; capo2015 . For a perfect fluid, the energy conditions are described as

Thus it is seen that the energy conditions are simple constraints on various linear combinations of the energy density $\rho$ and the pressure $p$. The matter including both positive energy density and positive pressure, which is called ”normal” matter, satisfies all the standard energy conditions. On the contrary, ”exotic” matter violates any one of the energy conditions. For example, the SEC is satisfied by ”all known forms of energy”, but not by the dark energy where $p=-\rho$. We note that for the static and spherically symmetric metric (8) one can find the null vector as $k^{a}=e^{-\nu/2}\delta^{a}_{0}+e^{-\lambda/2}\delta^{a}_{1}$ by using the condition $k^{a}k_{a}=0$. Then the NEC for the general anisotropic fluid (24) becomes

which yields the perfect fluid energy condition taking $T_{2}=pr^{2}$.

For the perfect fluid, it is easily seen that $T_{0}=\rho e^{\nu},\,\,T_{1}=pe^{\lambda},\,\,T_{2}=pr^{2}$, i.e., $T_{2}=r^{2}e^{-\lambda}T_{1}$, and $T_{3}=sin^{2}\theta T_{2}$, which yields $\pi_{ab}=0$ as it should be, and these give rise to the following relations

where $\rho$ and $p$ are, respectively, density and pressure of the fluid, which are

Further, the condition of isotropy of the pressure for the perfect fluid matter yields that

which is equivalent with

In this case the energy conditions for a barotropic equation of state $p=p(\rho)$ are given by

The linear form of a barotropic equation of state is given by $p=w\rho$ where $w$ is the equation of state parameter. For $w=0,1/3$ and $1$, we get dust, incoherent radiation and stiff matter, respectively.

For the spherically symmetric static spacetimes (8), Eq. (7) takes the form:

In the above equations the summation convention is not assumed. For the non-degenerate energy-momentum tensor $T_{ab}$, after some tedious calculations similar to those done in Ref. crcsph we see that the general solution of equations (41)-(44) can be written as

with the conformal function given by

where $A_{\ell}(t,r),\,(\ell=1,2,3,4,5)$, are integration functions and $a_{j}\,(j=1,2,3)$ are constant parameters, which give the three KVs of spherically symmetric spacetimes

Further, the functions $A_{\ell}(t,r)$ in the above equations (45)-(49) are subject to the following constraint equations

where the dot represents the derivative with respect to time $t$, and $j=1,2,3$. When we solve the above constraint equations for possible cases of non-degenerate energy-momentum tensor $T_{ab}$, we obtain the corresponding CMCs for the spherically symmetric static spacetimes (see Section IV). In the following section we find CMCs for the degenerate energy-momentum tensor of the spherically symmetric static spacetimes.

Subcase (D-A2-i). In this subcase we find

where $T^{\prime}_{0}\neq 0$ and $\alpha=2,3$. If $T^{\prime}_{0}=0$, i.e., $T_{0}=c$ (a constant), then the CMCs takes the following form

where $a$ is an integration constant. The corresponding Lie algebra of the vector fields in this subcase is infinite dimensional because the vector fields given in (57) and (58) have arbitrary components. For this subcase, using (21)-(23) we have

which shows that all Ricci tensor components are nonzero, e.g. the Ricci tensor in this subcase is non-degenerate. Furthermore, in this subcase we have the equation of state $p=0$ (dust) for the perfect fluid.

Subcase (D-A2-ii). Considering the constraints of this subcase, we obtain that

where $a_{1}$ is a constant of integration. Here, we again have infinite dimensional Lie algebra of vector fields. Using (21)-(23), we find

which are independent relations for any form of the energy-momentum tensor. It is obvious that for this case the choice of observers as a timelike four-vector field $u^{a}=e^{-\nu/2}\delta^{a}_{0}$ is not appropriate, since it gives $\rho=0$ due to the condition $T_{0}=0$. Thus we use the spacelike four-velocity $u^{a}=e^{-\lambda/2}\delta^{a}_{1}$ of the observers for the metric (8), which implies that Eqs. (31) and (32) give

Thus, the dust fluid is also allowed in this subcase. Furthermore, all Ricci tensor components for this subcase are non-zero. This means that the Ricci tensor is non-degenerate even though the matter tensor is degenerate.

Subcase (D-A3-i). Here, by choosing a timelike four-velocity, the conditions $T_{0}\neq 0\neq T_{1}$ and $T_{2}=0=T_{3}$ mean that the fluid represents an anisotropic fluid without heat flux, and then the physical quantities become

Using the transformations $d\bar{r}=\psi\sqrt{\mid T_{1}\mid}dr$, where $\bar{r}=\bar{r}(t,r)$, one finds

where $f(t)$ and $g(t)$ are functions of integration, and the conformal factor $\psi$ has the form

For this case, it follows from the condition $T_{q}=0$ that

We find from Eqs. (63) and (68) that

These equations show that it should be satisfied the conditions $R_{0}\geq e^{\nu}/2$ and $R_{2}\geq 0$ in order to be valid the SEC and TEC, respectively. There is only one constraint equation $2\nu^{\prime\prime}+{\nu^{\prime}}^{2}-\nu^{\prime}\lambda^{\prime}+\frac{2}{r}(\nu^{\prime}-\lambda^{\prime})=0$ following from the condition $T_{2}=0$ for this subcase. Also, all the Ricci tensor components are again non-zero, i.e. $\det(R_{ab})\neq 0$, even if the matter tensor is degenerate.

Subcase (D-A3-ii). For this subcase, considering the constraint $T_{0}=0=T_{1}$, we find

which yields the following solution

where $\lambda_{0}$ is an integration constant. The above solution is just the Schwarzschild metric which gives $R_{ab}=0$, i.e., all $R_{i}$’s and $T_{i}$’s vanish identically. Therefore there is a contradiction with the condition $T_{2}\neq 0\neq T_{3}$ as an assumption that is not possible in this subcase.

Subcase (D-A4-i). In this subcase, the constraint $T_{0}=0$ gives

and

where $T_{1}=\frac{1}{r}\left(\nu+\lambda\right)^{\prime}$. For this case the choice of a timelike four velocity of the observers is not allowed since $T_{0}=0$ gives $\rho=0$. So we need to choose a spacelike four-velocity such as $u^{a}=e^{-\lambda/2}\delta^{a}_{r}$. Using Eqs. (31) and (32), this choice gives rise to an anisotropic fluid without heat flux as follows:

Here Eq. (72) has the following solution

where $\lambda_{1}$ is a constant of integration. The equations given in (73) are second order ordinary differential equations in terms of $\nu$. Then, using the $\lambda(r)$ given by (75) in any of the three equations of (73), one can solve the obtained second order differential equation to find $\nu(r)$ as

where $\nu_{0}$ and $\nu_{1}$ are constants of integration. The Ricci scalar of the obtained metric given by (75) and (76) is

and the $R_{i}$’s and $T_{i}$’s for this solution are

Further, the matter density and pressure that come from Eqs. (74), (75) and (80) are

which implies that the equation of state parameter $w$ yields an incohorent radiation, i.e., $w=1/3$. Finally, we conclude that the Ricci tensor is also degenerate in this case.

For this subcase, we obtain that $X^{0}=X^{0}(x^{a})$ and $X^{j}=X^{j}(r,\theta,\phi)$ where the form of $X^{j}$ is the same as in equations (46) - (48), and the constraint equations (51) - (56) yield the following solution

where $b_{j},d_{j}$ and $\ell$ are constants. In this case, the conformal factor is given by

where we have used the transformation $dr=(T_{2}/T_{1})^{1/2}d\bar{r}$. By considering (79) and (80), the latter transformation yields $\bar{r}=-\sqrt{2}\ln\left(r+\sqrt{r(r-\lambda_{1})}-\frac{\lambda_{1}}{2}\right)$. Thus, in addition to the three KVs given in (50), it follows that the remaining CMCs and corresponding conformal factors are