Lemaitre-Tolman-Bondi Static Universe in Rastall-like Gravity

TL;DR

This paper introduces a new Rastall-like gravity theory that allows for stable Lemaitre-Tolman-Bondi static universes, overcoming limitations of general relativity and other modified gravity theories.

Contribution

It proposes a novel Rastall-like gravity model that enables stable LTB static solutions, which are not possible in GR or original Rastall gravity.

Findings

Stable LTB static universe solutions found in Rastall-like gravity

Demonstrated stability against scalar perturbations

Shows potential for alternative cosmological models

Abstract

In this work, we try to obtain a stable Lemaitre-Tolman-Bondi (LTB) static universe, which is spherically symmetric and radially inhomogeneous. However, this is not an easy task, and fails in general relativity (GR) and various modified gravity theories, because the corresponding LTB static universes must reduce to the Friedmann-Robertson-Walker (FRW) static universes. We find a way out in a new type of modified gravity theory, in which the conservation of energy and momentum is broken. In this work, we have proposed a novel modification to the original Rastall gravity. In some sense, our Rastall-like gravity is essentially different from GR and the original Rastall gravity. In this Rastall-like gravity, LTB static solutions have been found. The stability of LTB static universe against both the homogeneous and the inhomogeneous scalar perturbations is also discussed in details. We show that a LTB static universe can be stable in this Rastall-like gravity.

Equations

$$d{s}^2=-d{t}^2+\frac{A^{\prime 2}(r,t)}{1-K(r)}d{r}^2+A^2(r,t)d{\Omega }^2,$$

$$G^{\nu }{}_{\mu }=R^{\nu }{}_{\mu }-\frac{1}{2}R{\delta }^{\nu }{}_{\mu }=\mathrm{diag}\left(-\frac{K_0}{A_0^2}-\frac{K_0^{\prime }}{A_0{A}_0^{\prime }},-\frac{K_0}{A_0^2},-\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }},-\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }}\right),$$

$$T^{\mu \nu }=(\rho +p)U^{\mu }{U}^{\nu }+p{g}^{\mu \nu },$$

$$T^{\nu }{}_{\mu }=\mathrm{diag}\left(-\rho ,p,p,p\right),$$

$$\frac{K_0}{A_0^2}=\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }}=\frac{d{K}_0}{d{A}_0^2},$$

$$d{s}^2=-d{t}^2+a_0^2\left(\frac{d{\tilde{r}}^2}{1-\tilde{\mathcal{K}}{\tilde{r}}^2}+{\tilde{r}}^{2}d{\Omega }^2\right),$$

$$G^{\nu }{}_{\mu }+M^{\nu }{}_{\mu }=8\pi G_N{T}^{\nu }{}_{\mu },$$

$$T^{\nu }{}_{\mu ;\nu }=X_{\mu }.$$

$$T^{\nu }{}_{\mu ;\nu }=X_{\mu }=\lambda R_{,\mu },$$

$$G^{\nu }{}_{\mu }=\kappa \left(T^{\nu }{}_{\mu }-\lambda R{\delta }^{\nu }{}_{\mu }\right),\mathrm{or}\mathrm{equivalently}{R}_{\mu \nu }+\left(\kappa \lambda -\frac{1}{2}\right)g_{\mu \nu }R=\kappa T_{\mu \nu },$$

$$\frac{\kappa }{4\kappa \lambda -1}\left(3\kappa \lambda -\frac{1}{2}\right)=4\pi G_N.$$

$$T^{\nu }{}_{\mu ;\nu }=X_{\mu }=Y^{\nu }{}_{\mu ;\nu },$$

$$G^{\nu }{}_{\mu }+\kappa Y^{\nu }{}_{\mu }=\kappa T^{\nu }{}_{\mu },$$

$$\kappa Y=R+\kappa T,$$

$$\kappa Y^{\nu }{}_{\mu }=\left(R+\kappa T\right){\mathcal{J}}^{\nu }{}_{\mu },\mathrm{where}{\mathcal{J}}^{\nu }{}_{\mu }\equiv \mathrm{diag}\left(0,1,0,0\right),$$

$$-\frac{K_0}{A_0^2}-\frac{K_0^{\prime }}{A_0{A}_0^{\prime }}$$

$$-\frac{K_0}{A_0^2}+R_0+\kappa T_0$$

$$-\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }}$$

$$\frac{w{K}_0}{A_0^2}+\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }}\left(1+2w\right)=0.$$

$$K_0=\mathcal{C}{A}_0^{-2w/(1+2w)},$$

$$\kappa \left(1+2w\right){\rho }_0=\frac{K_0}{A_0^2}.$$

$${\rho }_0=\frac{\mathcal{C}}{\kappa \left(1+2w\right)}{A}_0^{-2(1+3w)/(1+2w)}.$$

$$g_{\mu \nu }={\bar{g}}_{\mu \nu }+h_{\mu \nu },$$

$$h_{\mu }{}^{\nu }=\mathrm{diag}\left(-2\Phi ,2\Psi ,2\Psi ,2\Psi \right).$$

$$\delta T_{\mu }{}^{\nu }=\left({\rho }_0+p_0\right)\left(h_{\mu \sigma }{U}^{\sigma }{U}^{\nu }+{\bar{g}}_{\mu \sigma }{U}^{\sigma }\delta U^{\nu }+{\bar{g}}_{\mu \sigma }\delta U^{\sigma }{U}^{\nu }\right)+\left(\delta \rho +\delta p\right){\bar{g}}_{\mu \sigma }{U}^{\sigma }{U}^{\nu }+\delta p{\delta }_{\mu }{}^{\nu }.$$

$$\delta R_{\mu \sigma }=\frac{1}{2}\left({\nabla }_{\lambda }{\nabla }_{\mu }{h}_{\sigma }{}^{\lambda }+{\nabla }_{\lambda }{\nabla }_{\sigma }{h}_{\mu }{}^{\lambda }-{\nabla }_{\mu }{\nabla }_{\sigma }h\right)-\frac{1}{2}{g}^{\alpha \beta }{\nabla }_{\alpha }{\nabla }_{\beta }{h}_{\mu \sigma }.$$

$$\delta R$$

$$\delta R_{\mu }{}^{\nu }-\frac{1}{2}\delta R{\delta }_{\mu }{}^{\nu }+\kappa \delta Y_{\mu }{}^{\nu }=\kappa \delta T_{\mu }{}^{\nu },$$

$$\Phi =\Psi .$$

$$4\ddot{\Psi }+\square \Psi -\frac{1}{2}\delta R$$

$$-\square \Psi -\frac{2K_0^{\prime }}{A_0{A}_0^{\prime }}\Psi +\frac{1}{2}\delta R+\kappa \left(3\delta p-\delta \rho \right)$$

$$-\square \Psi -2\left(\frac{K_0}{A_0^2}+\frac{K_0^{\prime }}{2A_0{A}_0^{\prime }}\right)\Psi -\frac{1}{2}\delta R$$

Full text

I Introduction

As is well known, modern cosmology began with the application of general relativity (GR) to the universe, soon after the birth of GR. The first cosmological model developed by Einstein himself in 1917 Einstein:1917ce is the well-known Einstein static universe, which is homogeneous, isotropic, and spatially closed. It can be static by the help of a positive cosmological constant counteracting the attractive effects of gravity on ordinary matter. However, in 1929, Hubble found that the universe is expanding (rather than static), by examining the relation between distance and redshift of galaxies Hubble:1929ig . On the other hand, in 1930, Eddington Eddington:1930zz argued that the Einstein static universe is unstable with respect to homogeneous and isotropic scalar perturbations in GR, and hence the universe cannot be static in the presence of perturbations. Therefore, Einstein abandoned the idea of static universe (and the cosmological constant as the “ biggest blunder ” in his life Gamov:1970 ).

Recently, the Einstein static universe has been revived to avoid the big bang singularity in the emergent universe scenario Ellis:2002we ; Ellis:2003qz . In such kind of scenario, the Einstein static universe is the initial state for a past-eternal inflationary cosmological model and then evolves to an inflationary era. So, there is no big bang singularity, and no exotic physics is involved. The quantum gravity regime can even be avoided, if the size of Einstein static universe is big enough. In fact, it is argued that the Einstein static state is favored by entropy considerations as the initial state for the universe Gibbons:1987jt ; Gibbons:1988bm ; Mulryne:2005ef .

Motivated by the emergent universe scenario, in the past decade, the Einstein static universe was extensively studied in many gravity theories. It has been reconsidered in GR, and the Einstein static universe can be stable against perturbations, if the universe contains a perfect fluid with $c_{s}^{2}>1/5$ Gibbons:1987jt ; Gibbons:1988bm ; Barrow:2003ni . It is also very interesting to consider the Einstein static universe in various modified gravity theories, for example, loop quantum cosmology Parisi:2007kv , $f(R)$ theory Boehmer:2007tr ; Seahra:2009ft ; Goheer:2008tn , $f(T)$ theory Wu:2011xa ; Li:2013xea , modified Gauss-Bonnet gravity ($f(G)$ theory) Bohmer:2009fc ; Huang:2015kca , Brans-Dicke theory delCampo:2007mp ; delCampo:2009kp ; Huang:2014fia ; Miao:2016obc , Horava-Lifshitz theory Wu:2009ah ; Boehmer:2009yz ; Heydarzade:2015hra , massive gravity Parisi:2012cg ; Mousavi:2016eof , braneworld scenario Gergely:2001tn ; Zhang:2010qwa ; Atazadeh:2014xsa , Einstein-Cartan theory Atazadeh:2014ysa , $f(R,T)$ gravity Shabani:2016dhj , hybrid metric-Palatini gravity Boehmer:2013oxa , Eddington-inspired Born-Infeld theory Li:2017ttl , degenerate massive gravity Li:2019laq , and so on Boehmer:2003iv ; Goswami:2008fs ; Canonico:2010fd ; Zhang:2016obw ; Carneiro:2009et ; Boehmer:2015ina ; Atazadeh:2015zma ; Atazadeh:2016yeh .

We note that (almost) all relevant works in the literature by now only considered the Einstein static universe, which is homogeneous and isotropic, and hence is described by a Friedmann-Robertson-Walker (FRW) metric. In other words, they assumed the cosmological principle.

However, as a tenet, the cosmological principle is not born to be true. Actually, this assumption has not yet been well proven on cosmic scales $\gtrsim 1\,{\rm Gpc}$ Caldwell:2007yu . Therefore, it is still of interest to test both the homogeneity and the isotropy of the universe carefully. In fact, they could be violated in some theoretical models, such as Lemaître-Tolman-Bondi (LTB) model LTB (see also e.g. Yan:2014eca ; Deng:2018yhb ; Deng:2018jrp ; Qiang:2019zrs and references therein) violating the cosmic homogeneity, and the exotic Gödel universe Godel:1949ga (see also e.g. Li:2016nnn and references therein), most of the Bianchi type I$\,\sim\,$IX universes Bianchi , Finsler universe Li:2015uda , violating the cosmic isotropy. On the other hand, many observational hints of the cosmic inhomogeneity and/or anisotropy have been claimed in the literature (see e.g. Deng:2018yhb ; Deng:2018jrp ; Qiang:2019zrs for brief reviews), including type Ia supernovae (SNIa), cosmic microwave background (CMB), baryon acoustic oscillations (BAO), gamma-ray bursts (GRBs), integrated Sachs-Wolfe effect, Sunyaev-Zel’dovich effect, quasars, radio galaxies, and so on. Therefore, on both the theoretical and the observational sides, it is reasonable to consider the cosmological models violating the cosmological principle.

It is natural to ask “ why must the initial state for the universe be homogeneous and isotropic? ” If the initial state is random, it has great probability to be inhomogeneous and/or anisotropic. Therefore, it is interesting to consider a static universe violating the cosmological principle.

In the present work, we are interested in the well-known LTB model LTB (see also e.g. Yan:2014eca ; Deng:2018yhb ; Deng:2018jrp ; Qiang:2019zrs and references therein). In this model, the universe is spherically symmetric and radially inhomogeneous, and we are living in a locally underdense void centered nearby our location. As is well known, without invoking dark energy or modified gravity, it is possible to explain the apparent cosmic acceleration discovered in 1998 by using the LTB model Alnes:2005rw ; GarciaBellido:2008nz ; Enqvist:2006cg ; Celerier:2012xr ; Ishak:2013vha ; Clifton:2008hv ; Zhang:2012qr ; Yan:2014eca ; Yousaf:2017xec . This fact further justifies the motivation to consider a LTB static universe.

However, it is not an easy task to obtain a stable LTB static universe in GR and various modified gravity theories. We will briefly discuss this issue in Sec. II. Knowing the cause of failure, we find a way out in a new type of modified gravity theory, namely Rastall-like gravity, which will be briefly introduced in Sec. III. Then, in Secs. IV and V, we obtain the LTB static solutions in Rastall-like gravity without and with a cosmological constant, respectively. The stability of LTB static universe against both the homogeneous and the inhomogeneous scalar perturbations is also discussed in details. In Sec. VI, some brief concluding remarks are given.

II The failure of LTB static universe

in GR and various modified gravity theories

The LTB metric, in comoving coordinates ($r$, $\theta$, $\phi$) and synchronous time $t$, is given by LTB ; Alnes:2005rw ; GarciaBellido:2008nz ; Enqvist:2006cg ; Celerier:2012xr ; Ishak:2013vha ; Clifton:2008hv ; Zhang:2012qr ; Yan:2014eca

[TABLE]

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,d\phi^{2}$, and a prime denotes a derivative with respect to $r$. $K(r)$ is an arbitrary function of $r$, playing the role of spatial curvature. In general, $A(r,t)$ is an arbitrary function of $r$ and $t$, playing the role related to the scale factor. Obviously, the LTB metric reduces to the well-known FRW metric if $A(r,t)=a(t)\,r$ and $K(r)=Kr^{2}$.

Let us consider the LTB static universe, in which $A=A_{0}(r)$ is independent of the time $t$, and the subscript “ 0 ” indicates the quantities related to the static solutions. In this case, $\dot{A}=0$ and $\ddot{A}=0$, where a dot denotes a derivative with respect to $t$. Now, by definition, the Einstein tensor is given by

[TABLE]

where $\delta^{\nu}{}_{\mu}={\rm diag}\left(1,\,1,\,1,\,1\right)$, $R_{\mu\nu}$ is the Ricci tensor, $R=R^{\mu}{}_{\mu}$, and the Greek indices $\mu$, $\nu$ run over 0, 1, 2, 3. We note that the last three diagonal components $G^{1}{}_{1}\not=G^{2}{}_{2}=G^{3}{}_{3}$ in general. On the other hand, if the universe contains a perfect fluid, the corresponding energy-momentum tensor reads

[TABLE]

or equivalently

[TABLE]

where 4-velocity $U^{\mu}=dx^{\mu}/d\tau$ satisfies $g_{\mu\nu}U^{\mu}U^{\nu}=-1$, while $\tau$ is the proper time, $\rho$ and $p$ are energy density and pressure, respectively. In a proper (comoving) frame, $U^{\mu}=(1,\,0,\,0,\,0)$. In general, $\rho$ and $p$ are functions of $r$ and $t$. In the case of LTB static universe, $\rho=\rho_{0}(r)$ and $p=p_{0}(r)$ are independent of $t$.

In GR, the field equations are $G^{\nu}{}_{\mu}=8\pi G_{N}T^{\nu}{}_{\mu}$, where $G_{N}$ is the Newtonian gravitational constant. Since the last three diagonal components of $T^{\nu}{}_{\mu}$ are equal, we must have $G^{1}{}_{1}=G^{2}{}_{2}=G^{3}{}_{3}=8\pi G_{N}p\,$. In the case of LTB static universe, from Eq. (2), it requires

[TABLE]

which means $K_{0}/A_{0}^{2}={\cal K}={\rm const.}$ Introducing $a_{0}=\left|{\cal K}\right|^{-1/2}$, $\tilde{r}=A_{0}(r)/a_{0}$, $\tilde{\cal K}={\cal K}a_{0}^{2}=\pm 1$ if ${\cal K}\not=0$, or introducing $\tilde{r}=A_{0}(r)/a_{0}$, $\tilde{\cal K}=0$, $a_{0}={\rm const.}\not=0$ if ${\cal K}=0$, the metric of LTB static universe becomes

[TABLE]

which is nothing but the one of FRW static universe. So, the LTB static universe in GR fails. In addition, we have $8\pi G_{N}p_{0}=G^{1}{}_{1}=-K_{0}/A_{0}^{2}=-{\cal K}={\rm const.}$, which is also independent of $r$. Actually, one can find $\partial_{r}p_{0}=0$ from the conservation equations $T^{\nu}{}_{\mu;\nu}=0$. In fact, this gives an instructive hint for the failure of LTB static universe in GR.

Furthermore, we have also considered the LTB static universes in various modified gravity theories, such as $f(R)$ theory, $f(T)$ theory, Brans-Dicke theory, modified Gauss-Bonnet gravity ($f(G)$ theory), and they all failed because the corresponding LTB static universes all reduced to the FRW static universes. In fact, one can always recast the field equations of modified gravity theory as the form of

[TABLE]

where $M^{\nu}{}_{\mu}$ is the modification term with respect to GR. In the case of LTB static universe, if we require that the conservation equations $T^{\nu}{}_{\mu;\nu}=0$ hold (which are actually equivalent to $M^{\nu}{}_{\mu;\nu}=0$ because $G^{\nu}{}_{\mu;\nu}=0$ always), the last three diagonal components of $M^{\nu}{}_{\mu}$ should be equal in various modified gravity theories such as $f(R)$ theory, $f(T)$ theory, Brans-Dicke theory, and modified Gauss-Bonnet gravity ($f(G)$ theory). Since the last three diagonal components of $T^{\nu}{}_{\mu}$ are also equal, it is necessary to require $G^{1}{}_{1}=G^{2}{}_{2}=G^{3}{}_{3}$. Following the similar derivations in GR, the LTB static universes in these modified gravity theories also fail, because they must reduce to the FRW static universes.

III The Rastall-like gravity theory

III.1 The original Rastall gravity theory

From the discussions in Sec. II, it is easy to see that the conservation equations $T^{\nu}{}_{\mu;\nu}=0$ might be responsible for the failure of LTB static universe in GR and various modified gravity theories. In order to obtain a successful LTB static universe, a possible way out might be breaking the conservation equations $T^{\nu}{}_{\mu;\nu}=0$. In the literature, there exists some models breaking the conservation of energy and momentum. For instance, the famous steady state model of the expanding universe proposed by Hoyle, Bondi and Gold in 1948 Hoyle:1948zz ; Bondi:1948qk requires continuous creation of matter (particle creation) from nothing. In fact, it is not completely unacceptable to consider such kind of models.

In this work, we are interested in the so-called Rastall gravity theory proposed in 1972 Rastall:1973nw . Rastall argued that the fundamental assumption $T^{\nu}{}_{\mu;\nu}=0$ of GR is questionable in fact. All one can assert with fair confidence is Rastall:1973nw

[TABLE]

In Rastall:1973nw , Rastall proposed to consider the assumption

[TABLE]

where $\lambda$ is a constant, and $R=R^{\mu}{}_{\mu}$. Since $G^{\nu}{}_{\mu;\nu}=0=\kappa\left(T^{\nu}{}_{\mu}-\lambda\,\delta^{\nu}{}_{\mu}R\right)_{;\nu}$ always, the assumption in Eq. (9) is consistent with the modified field equations Rastall:1973nw

[TABLE]

where $\kappa$ is a non-zero constant. Contracting Eq. (10) gives $\left(4\kappa\lambda-1\right)R=\kappa T$, where $T=T^{\mu}{}_{\mu}$, while the model parameters $\kappa\lambda=1/4$ should be excluded. Rastall Rastall:1973nw argued that

[TABLE]

When $\lambda=0$, we have $\kappa=8\pi G_{N}$, and GR is recovered. Obviously, Rastall gravity theory is a new kind of modified gravity theory violating the conservation of energy and momentum.

In the past decade, Rastall gravity was extensively studied in the literature (the original Ref. Rastall:1973nw has been cited more than 160 times to date), and we refer to Capone:2009xm ; Batista:2011nu ; Fabris:2012hw ; Silva:2012gn ; Santos:2014ewa ; Bronnikov:2016odv ; Yuan:2016pkz ; Heydarzade:2016zof ; Heydarzade:2017wxu ; Moradpour:2017shy ; Spallucci:2017mto ; Lobo:2017dib ; Darabi:2017tay ; Xu:2017bix ; Kumar:2017qws ; Visser:2017gpz ; Moradpour:2017tbp ; Salako:2018uzo ; Lin:2018coh ; Das:2018dzp ; Abbas:2018ffk ; Abbas:2018mxs ; Bamba:2018zil ; Sadeghi:2018vrf ; Ziaie:2019jfl ; Sakti:2019krw ; Abbas:2019evn ; Halder:2019akt ; Khyllep:2019odd ; Li:2019jkv for example. In particular, very recently, it is claimed in Li:2019jkv that Rastall gravity is strongly favored by 118 galaxy-galaxy strong gravitational lensing systems, with $\kappa\lambda=0.163\pm 0.001$. This new observational evidence further justified the serious studies of Rastall gravity. In fact, the corresponding cosmology, black hole, compact star, wormhole, thermodynamics in Rastall gravity have been extensively considered in the literature. It attracted much attention in the recent years, and grows rapidly now.

III.2 A novel modification to the Rastall gravity theory

However, the original Rastall gravity theory is not suitable for our purpose. This can be clearly seen from Eq. (10). Since the last three diagonal components of the term $\lambda\,\delta^{\nu}{}_{\mu}R$ are equal, and the last three diagonal components of $T^{\nu}{}_{\mu}$ are also equal for a perfect fluid, we must have $G^{1}{}_{1}=G^{2}{}_{2}=G^{3}{}_{3}$ again. Then, following the similar discussions in Sec. II, the LTB static universe in the original Rastall gravity theory also fails. So, we should introduce a novel modification to the Rastall gravity theory.

In Rastall gravity, Eq. (8) is firm, but the choice of $X_{\mu}$ might be changed. For simplicity, we propose to consider a fairly general form

[TABLE]

where $Y^{\nu}{}_{\mu}\not=T^{\nu}{}_{\mu}$ (note that if $Y^{\nu}{}_{\mu}=\lambda\,R\,\delta^{\nu}{}_{\mu}$, the original Rastall gravity theory Rastall:1973nw can be recovered). In this new form, since $G^{\nu}{}_{\mu;\nu}=0=\kappa\left(T^{\nu}{}_{\mu}-Y^{\nu}{}_{\mu}\right)_{;\nu}$ always, the assumption in Eq. (12) is consistent with the modified field equations

[TABLE]

where $\kappa$ is a non-zero constant. Contracting Eq. (13), we find that the trace of $Y^{\nu}{}_{\mu}$ is given by

[TABLE]

where $Y=Y^{\mu}{}_{\mu}$, $R=R^{\mu}{}_{\mu}$ and $T=T^{\mu}{}_{\mu}$. So far, the choice of $Y^{\nu}{}_{\mu}$ is still pending. Before we make a particular choice of $Y^{\nu}{}_{\mu}$, here are some general remarks:

1. (R1)

Regardless of the matter distribution in the universe, if $Y^{\nu}{}_{\mu}=0$, this Rastall-like gravity theory reduces to GR. 2. (R2)

If the matter distribution in the universe is isotropic and homogeneous, and if $Y^{\nu}{}_{\mu}$ is also isotropic and homogeneous (namely its last three diagonal components are equal and independent of spatial coordinates), the universe should be described by a FRW metric, and the Rastall-like gravity theory equivalently reduces to GR with an effective $T^{\nu}{}_{\mu,\,{\rm eff}}=T^{\nu}{}_{\mu}-Y^{\nu}{}_{\mu}$. 3. (R3)

If the matter distribution in the universe is isotropic and homogeneous, but $Y^{\nu}{}_{\mu}$ is anisotropic and/or inhomogeneous (namely its last three diagonal components are not equal and/or depend on spatial coordinates), because the anisotropic and/or inhomogeneous energy-momentum-exchange between matter and geometry will change the matter distribution in the universe, the universe becomes anisotropic and/or inhomogeneous (and hence is not described by a FRW metric). 4. (R4)

If the matter distribution in the universe is anisotropic and/or inhomogeneous, regardless of $Y^{\nu}{}_{\mu}$, the universe is not described by a FRW metric.

Next, let us step forward, and try to specify the choice of $Y^{\nu}{}_{\mu}$. We assume that the universe contains a perfect fluid whose energy-momentum tensor is given by Eqs. (3) or (4). To obtain a successful LTB static universe, $Y^{\nu}{}_{\mu}$ should satisfy three conditions:

1. (C1)

$Y^{\nu}{}_{\mu}$ is diagonal, because $G^{\nu}{}_{\mu}$ and $T^{\nu}{}_{\mu}$ are both diagonal (n.b. Eqs. (2) and (4)). 2. (C2)

Its trace should be related to geometry and matter according to Eq. (14). 3. (C3)

Due to the discussions in Sec. II, $Y^{1}{}_{1}\not=Y^{2}{}_{2}=Y^{3}{}_{3}$ is required to obtain a successful LTB static universe, while $G^{1}{}_{1}\not=G^{2}{}_{2}=G^{3}{}_{3}$ and $T^{1}{}_{1}=T^{2}{}_{2}=T^{3}{}_{3}$ (n.b. Eqs. (2) and (4)).

Obviously, there is a large room for the choice of $Y^{\nu}{}_{\mu}$. The simplest one is given by

[TABLE]

and we will briefly mention other reasonable choices in Sec. VI. When $R+\kappa T=0$, this Rastall-like gravity theory reduces to GR (if $\kappa\not=8\pi G_{N}$, one can simply rescale $T^{\nu}{}_{\mu,\,{\rm new}}=\kappa T^{\nu}{}_{\mu}/(8\pi G_{N})$, but $\kappa>0$ is required to ensure the energy density $\rho_{\rm new}\propto\kappa\rho\geq 0$). In some sense, our Rastall-like gravity is essentially different from GR and the original Rastall gravity (see the discussions in Sec. VI).

IV LTB static universe in Rastall-like gravity

In this section, we consider the LTB static universe in Rastall-like gravity. The field equations are given by Eq. (13), and $Y^{\nu}{}_{\mu}$ is given by Eq. (15). We assume that the universe contains a perfect fluid whose energy-momentum tensor is given by Eqs. (3) or (4), and $p=w\rho$, where the equation-of-state parameter $w$ is a constant.

IV.1 LTB static solutions

In the case of LTB static universe, $A$, $\rho$ and $p$ are all independent of the time $t$, and hence they are functions only depending on the spatial coordinate $r$, namely $A_{0}(r)$, $\rho_{0}(r)$ and $p_{0}(r)$, while we also denote $K=K_{0}(r)$. Due to the non-minimal coupling between geometry and matter, $T^{\nu}{}_{\mu;\nu}=Y^{\nu}{}_{\mu;\nu}\not=0$, one can find that $\rho_{0}^{\,\prime}\not=0$ and $p_{0}^{\,\prime}\not=0$, namely they are not homogeneous. The static solutions are determined by the field equations in Eq. (13), namely

[TABLE]

Noting $R_{0}=2K_{0}/A_{0}^{2}+2K_{0}^{\prime}/(A_{0}A_{0}^{\prime})$ and $T_{0}=3p_{0}-\rho_{0}$, Eq. (17) is not independent of Eqs. (16) and (18). Multiplying Eq. (16) by $w$, and then adding Eq. (18), we have

[TABLE]

Noting $K_{0}^{\prime}/A_{0}^{\prime}=dK_{0}/dA_{0}$, Eq. (19) can be regarded as an ordinary differential equation of $K_{0}$ with respect to $A_{0}$, and its solution is given by

[TABLE]

where $\cal C$ is an integral constant, and we require $w\not=-1/2$. Obviously, $K_{0}/A_{0}^{2}$ is not a constant if $w\not=-1/3$ and ${\cal C}\not=0$ (n.b. $A_{0}=A_{0}(r)$ varies with the spatial coordinate $r$), and hence the LTB static universe does not reduce to the FRW static universe. Multiplying Eq. (18) by $2$, and then subtracting Eq. (16), we have

[TABLE]

Substituting Eq. (20) into Eq. (21), it is easy to get

[TABLE]

Eqs. (20) and (22) are the explicit expressions of the LTB static solutions.

IV.2 Stability analysis

To become a successful LTB static universe, it should be stable against perturbations. Fortunately, the comprehensive perturbation theory in LTB cosmology has been developed in Clarkson:2009sc . Because of the spherical symmetry of the LTB spacetime, perturbations can be decoupled into two independent modes, namely the polar and the axial modes Clarkson:2009sc ; Nishikawa:2014sga . Since we are interested in the evolution of the density perturbations, we focus on the polar mode Nishikawa:2014sga ; February:2012fp . Following e.g. February:2012fp and Sec. III of Clarkson:2009sc , a first approximation is to neglect the mode-mixing, and focus only on the scalar perturbations February:2012fp . In the Regge-Wheeler (RW) gauge Clarkson:2009sc ; Nishikawa:2014sga ; February:2012fp , the perturbed metric is given by

[TABLE]

where $\bar{g}_{\mu\nu}$ is the background (static) metric, and February:2012fp

[TABLE]

The perturbation of energy-momentum tensor is given by

[TABLE]

Note that the perturbations $\Phi$, $\Psi$, $\delta\rho$, $\delta p=w\delta\rho$, $\delta U^{\mu}$ are all functions of $t$, $r$, $\theta$, $\phi$. The perturbation of Ricci tensor induced by the perturbation of the metric is given by

[TABLE]

The perturbation of Ricci scalar reads

[TABLE]

where $\Box$ is the d’Alembertian. Substituting them into the perturbation of the field equation (13), i.e.

[TABLE]

its space-space “$\,i\not=j\,$” components tell us

[TABLE]

Substituting Eq. (29) into the diagonal components and the time-space “$\,0\hskip 1.19995pti\,$” components of Eq. (28), they become

[TABLE]

One can check that Eq. (31) is not independent of Eqs. (30) and (32). Noting $g_{\mu\nu}U^{\mu}U^{\nu}=-1$, we have $\delta U_{0}=-h_{00}/2=-\Phi$. Substituting Eq. (29) into Eq. (27), it is easy to get

[TABLE]

Noting $\Box\Psi=-\ddot{\Psi}+\nabla^{2}\Psi$ in the case of LTB static universe ($\nabla^{2}$ is the Laplacian), and then substituting Eq. (34) into Eq. (30), we obtain

[TABLE]

Once $\Psi$ is available, $\delta\rho$, $\delta U_{i}$ and $\delta U_{0}$ are ready by using Eqs. (35), (33) and $\delta U_{0}=-\Psi$, respectively. If $\Psi$ is stable, they are also stable. Multiplying Eq. (30) by $w$, and then adding Eq. (32), we have

[TABLE]

Using Eqs. (16) and (18), it becomes

[TABLE]

Considering a harmonic decomposition Clarkson:2009sc (see also e.g. Li:2017ttl ; Huang:2015kca ; Harrison:1967zza ),

[TABLE]

Eq. (37) can be separated into two differential equations, namely

[TABLE]

where the wave number $k$ is related to the degree $n$ Clarkson:2009sc ; Harrison:1967zza ; notek2 . One can obtain the spatial part of $\Psi$ by solving Eq. (39) following e.g. Clarkson:2009sc ; Harrison:1967zza , but it does not determine the stability of the perturbation $\Psi$. In fact, the stability of the perturbation $\Psi$ depends on the temporal part, $\psi_{n}(t)$, which can be stable on the condition $wk^{2}>0$ (n.b. Eq. (40)). The LTB static universe is stable against the inhomogeneous ($k^{2}>0$) scalar perturbation if $w>0$. From Eq. (21), it is easy to see that $w>0$ requires a closed ($K_{0}>0$) universe. Unfortunately, the LTB static universe is unstable against the homogeneous ($k^{2}=0$) scalar perturbation, since $\psi_{n}(t)\propto t$ diverges when $t\to\infty$. So, the LTB static universe fails in this case.

V LTB static universe in

Rastall-like gravity with a cosmological constant

Let us come back to the discussions in Sec. III.2. In fact, a cosmological constant $\Lambda$ can be allowed in the Rastall-like gravity theory. Since $\Lambda_{;\nu}=0$, it is easy to see that $\left(G^{\nu}{}_{\mu}+\Lambda\delta^{\nu}{}_{\mu}\right)_{;\nu}=0=\kappa\left(T^{\nu}{}_{\mu}-Y^{\nu}{}_{\mu}\right)_{;\nu}$ always holds. So, Eq. (12) is also consistent with the modified field equations

[TABLE]

Contracting Eq. (41), we find that the trace of $Y^{\nu}{}_{\mu}$ is given by

[TABLE]

The general remarks (R1)$\,\sim\,$(R4) in Sec. III.2 are still valid. To obtain a successful LTB static universe, the conditions (C1) and (C3) in Sec. III.2 are still valid, while the condition (C2) should be changed to Eq. (42). Again, there is a large room for the choice of $Y^{\nu}{}_{\mu}$. The simplest one is given by

[TABLE]

where ${\cal J}^{\nu}{}_{\mu}$ is defined in Eq. (15), and we will briefly mention other reasonable choices in Sec. VI. When $R+\kappa T-4\Lambda=0$, this Rastall-like gravity theory reduces to GR (if $\kappa\not=8\pi G_{N}$, one can simply rescale $T^{\nu}{}_{\mu,\,{\rm new}}=\kappa T^{\nu}{}_{\mu}/(8\pi G_{N})$, but $\kappa>0$ is required to ensure the energy density $\rho_{\rm new}\propto\kappa\rho\geq 0$).

V.1 LTB static solutions

We assume that the universe contains a perfect fluid whose energy-momentum tensor is given by Eqs. (3) or (4), and $p=w\rho$, where the equation-of-state parameter $w$ is a constant. In the case of LTB static universe, $A$, $\rho$ and $p$ are all independent of the time $t$, and hence they are functions only depending on the spatial coordinate $r$, namely $A_{0}(r)$, $\rho_{0}(r)$ and $p_{0}(r)$, while we also denote $K=K_{0}(r)$. Due to the non-minimal coupling between geometry and matter, $T^{\nu}{}_{\mu;\nu}=Y^{\nu}{}_{\mu;\nu}\not=0$, one can find that $\rho_{0}^{\,\prime}\not=0$ and $p_{0}^{\,\prime}\not=0$, namely they are not homogeneous. The static solutions are determined by the field equations in Eq. (41), namely

[TABLE]

Noting $R_{0}=2K_{0}/A_{0}^{2}+2K_{0}^{\prime}/(A_{0}A_{0}^{\prime})$ and $T_{0}=3p_{0}-\rho_{0}$, Eq. (45) is not independent of Eqs. (44) and (46). Multiplying Eq. (44) by $w$, and then adding Eq. (46), we have

[TABLE]

Noting $K_{0}^{\prime}/A_{0}^{\prime}=dK_{0}/dA_{0}$, Eq. (47) can be regarded as an ordinary differential equation of $K_{0}$ with respect to $A_{0}$, and its solution is given by

[TABLE]

where $\cal C$ is an integral constant, and we require $w\not=-1/2$ and $w\not=-1/3$. Obviously, $K_{0}/A_{0}^{2}$ is not a constant if $w\not=-1/3$ and ${\cal C}\not=0$ (n.b. $A_{0}=A_{0}(r)$ varies with the spatial coordinate $r$), and hence the LTB static universe does not reduce to the FRW static universe. Multiplying Eq. (46) by $2$, and then subtracting Eq. (44), we have

[TABLE]

Substituting Eq. (48) into Eq. (49), it is easy to get

[TABLE]

Eqs. (48) and (50) are the explicit expressions of the LTB static solutions. Note that if $\Lambda=0$, all the results obtained here reduce to the ones in Sec. IV.1. But a non-zero $\Lambda$ makes difference.

V.2 Stability analysis

Again, to become a successful LTB static universe, it should be stable against perturbations. Similar to Sec. IV.2, we consider the perturbed metric given by Eqs. (23) and (24). Accordingly, the perturbations $\delta T_{\mu}{}^{\nu}$, $\delta R_{\mu\sigma}$, $\delta R$ are the same given by Eqs. (25)$\,\sim\,$(27). Since $\delta\Lambda=0$, the perturbation of the field equation in Eq. (41) is still the same given in Eq. (28). Once again, its space-space “$\,i\not=j\,$” components tell us $\Phi=\Psi$ as in Eq. (29). Then, its diagonal components and the time-space “$\,0\hskip 1.19995pti\,$” components become the ones given in Eqs. (30)$\,\sim\,$(33). Of course, Eqs. (34)$\,\sim\,$(36) still hold. However, the static solutions in Eqs. (44) and (46) make difference. Substituting Eqs. (44) and (46) into Eq. (36), it becomes

[TABLE]

which is different from Eq. (37) if $\Lambda\not=0$ and $w\not=-1$. Again, considering the harmonic decomposition given in Eq. (38), we can separate Eq. (51) into two differential equations, namely

[TABLE]

where the wave number $k$ is related to the degree $n$ Clarkson:2009sc ; Harrison:1967zza ; notek2 . One can obtain the spatial part of $\Psi$ by solving Eq. (52) following e.g. Clarkson:2009sc ; Harrison:1967zza , but it does not determine the stability of the perturbation $\Psi$. In fact, the stability of the perturbation $\Psi$ depends on the temporal part, $\psi_{n}(t)$. From Eq. (53), it is easy to see that the stability condition for the LTB static universe reads

[TABLE]

The LTB static universe is stable against the homogeneous ($k^{2}=0$) scalar perturbation if

[TABLE]

On the other hand, the LTB static universe is also stable against the inhomogeneous ($k^{2}>0$) scalar perturbation if Eq. (54) is satisfied for all possible modes with $k^{2}>0$. To ensure that Eq. (54) is valid even when $k^{2}\to\infty$, it is necessary to require

[TABLE]

Combining Eqs. (55) and (56), the LTB static universe can be stable against both the homogeneous ($k^{2}=0$) and the inhomogeneous ($k^{2}>0$) scalar perturbations if

[TABLE]

From Eq. (49), we find that $w\geq 0$ and $\Lambda<0$ require a closed ($K_{0}>0$) universe. So far, we successfully get a stable LTB closed static universe on the conditions given by Eq. (57).

VI Concluding remarks

In this work, we try to obtain a stable LTB static universe, which is spherically symmetric and radially inhomogeneous. However, this is not an easy task, and fails in GR and various modified gravity theories, because the corresponding LTB static universes must reduce to the FRW static universes. We find a way out in a new type of modified gravity theory, in which the conservation of energy and momentum is broken. In this work, we have proposed a novel modification to the original Rastall gravity. In some sense, our Rastall-like gravity is essentially different from GR and the original Rastall gravity (see below). In this Rastall-like gravity, LTB static solutions have been found. The stability of LTB static universe against both the homogeneous and the inhomogeneous scalar perturbations is also discussed in details. We show that a LTB static universe can be stable in this Rastall-like gravity.

As is well known, the stability conditions for many FRW (Einstein) static universes in various modified gravity theories are fairly complicated, and usually require exotic matter with $w<0$, in particular dark energy ($w<-1/3$) or even phantom ($w<-1$). On the contrary, the stability conditions for the LTB static universe given in Eq. (57) is very simple. Obviously, the condition $w\geq 0$ can be easily satisfied by using ordinary matter, such as radiation ($w=1/3$) or dust matter ($w=0$). On the other hand, a negative cosmological constant ($\Lambda<0$) is also welcome in e.g. string theory. Although the current accelerated expansion of the universe requires dark energy ($w<-1/3$) or a positive cosmological constant ($\Lambda>0$), this is not the case of LTB static universe (as the initial state for the past-eternal early universe in the emergent universe scenario).

It is worth noting that the (effective) gravitational forces provided by the ordinary matter ($w\geq 0$) and a negative cosmological constant ($\Lambda<0$) are attractive. This means that the effective force contributed by the non-minimal coupling between matter and geometry $Y^{\nu}{}_{\mu;\nu}$ is repulsive, which can also be seen from the first equation of $T^{\nu}{}_{\mu;\nu}=X_{\mu}=Y^{\nu}{}_{\mu;\nu}$, namely

[TABLE]

with a negative effective pressure $p_{\rm eff}<0$ coming from $X_{\mu}=Y^{\nu}{}_{\mu;\nu}$ (note that Eq. (58) in the LTB cosmology corresponds to the familiar $\dot{\rho}+3H\left(\rho+p+p_{\rm eff}\right)=0$ in the FRW cosmology). So, when the (effective) attractive forces are stably balanced by the effective repulsive force, the LTB static universe can be accomplished.

In this work, we assume that the universe contains a perfect fluid. Actually, one can also extend our discussions to a non-perfect fluid, for example, van der Waals fluid, viscous fluid, and Newtonian fluid. Of course, the role of matter can also be played by a scalar field or a vector field. In Rastall-like gravity, we expect that a stable LTB static universe can also be accomplished in these cases.

For simplicity, in this work we only consider the simplest choices of $Y^{\nu}{}_{\mu}$, as given in Eqs. (15) and (43). Actually, there is a large room for other reasonable choices. In the case without a cosmological constant, one might instead consider, for example,

[TABLE]

or even a more general choice

[TABLE]

where $\alpha$ and $\beta$ are both constants. Further, if we are willing to involve the other two spatial components, it is also possible to choose

[TABLE]

where $\alpha_{i}$ and $\beta_{i}$ are arbitrary constants. Obviously, Eqs. (59) and (60) are just special cases of Eq. (61). It is worth noting that the general $\kappa Y^{\nu}{}_{\mu}$ in Eq. (61) can be recast as

[TABLE]

where $\eta^{\nu}{}_{\mu}={\rm diag}\left(-1,\,1,\,1,\,1\right)$, and ${\cal J}^{\nu}{}_{\mu}$ is defined in Eq. (15). Of course, in the case with a cosmological constant $\Lambda$, the choices are quite similar, while one should appropriately insert $\Lambda$ into Eqs. (59)$\,\sim\,$(62). Note that all the choices mentioned above satisfy the conditions (C1)$\,\sim\,$(C3) to obtain a successful LTB static universe, as in Sec. III.2 or Sec. V. However, when we consider other topics rather than static universe, the conditions (C1) and (C3) could be abandoned, and we can then adopt other suitable choices of $Y^{\nu}{}_{\mu}$, while the trace condition (C2) is still required.

It is of interest to discuss the key difference between our Rastall-like gravity proposed in this work and the original Rastall gravity proposed in Rastall:1973nw . Actually, it is argued in e.g. Visser:2017gpz that one can alternatively regard the original Rastall gravity proposed in Rastall:1973nw as GR with a hypothetical matter sector. Contracting Eq. (10) gives $\left(4\kappa\lambda-1\right)R=\kappa T$. Then, the field equations of the original Rastall gravity, namely Eq. (10), could be recast as (see e.g. Batista:2011nu )

[TABLE]

where the effective energy-momentum tensor is given by

[TABLE]

in which we have used the relation $\left(4\kappa\lambda-1\right)R=\kappa T$. Note that $\tilde{T}^{\nu}{}_{\mu}$ is completely determined by the usual matter $T^{\nu}{}_{\mu}$ (n.b. Eq. (64) and $T=T^{\mu}{}_{\mu}$). This is the key point of e.g. Visser:2017gpz arguing that the original Rastall gravity proposed in Rastall:1973nw is equivalent to GR with a hypothetical matter sector. However, in e.g. Moradpour:2017tbp , it is argued that the same logic can also be applied to any modified gravity theory. In fact, one can always recast the field equations of modified gravity theory as the form of Eq. (7), where $M^{\nu}{}_{\mu}$ is the modification term with respect to GR. Similarly, the field equations of any modified gravity theory, namely Eq. (7), could also be recast as

[TABLE]

which has the same form of GR. However, as is well known, most of the modified gravity theories in the literature (e.g. $f(R)$ theories) are essentially different from GR, and have richer phenomena than GR. They are not equivalent to GR in fact. Therefore, the arguments of e.g. Visser:2017gpz are disagreed by e.g. Moradpour:2017tbp . Here, let us push these discussions further. Actually, in most of the modified gravity theories in the literature, the modification term $M^{\nu}{}_{\mu}$ is a function of geometric quantities. For example, in the well-known $f(R)$ theories (we adopt the form of ${\cal S}_{\rm grav}\propto\int d^{4}x\sqrt{-g}\left(R+f(R)\right)$ in the metric formalism), the modification term with respect to GR is given by (see e.g. Seahra:2009ft )

[TABLE]

where $f_{,R}=df/dR$. Clearly, it is essentially a geometric quantity. So, there exists a key difference between the original Rastall gravity proposed in Rastall:1973nw and most of the modified gravity theories in the literature. If the usual mater sector $T^{\nu}{}_{\mu}$ is given, in the original Rastall gravity Rastall:1973nw , the corresponding $\tilde{T}^{\nu}{}_{\mu}$ in Eq. (64) is completely determined, and it is a matter quantity essentially. However, this is not the case in most of the modified gravity theories in the literature (e.g. $f(R)$ theories), because $M_{\mu\nu}$ is a geometric quantity, or $M_{\mu\nu}$ depends on both the geometric and the matter sectors, and hence the corresponding $\tilde{T}^{\nu}{}_{\mu}$ in Eq. (65) is not a pure matter quantity. Even if the usual mater sector $T^{\nu}{}_{\mu}$ is given, this $\tilde{T}^{\nu}{}_{\mu}$ still cannot be explicitly determined. In this sense, most of the modified gravity theories in the literature (e.g. $f(R)$ theories) are essentially different from GR and the original Rastall gravity theory Rastall:1973nw . We hope this insight could reconcile the disputation between e.g. Visser:2017gpz and Moradpour:2017tbp . Now, let us turn to our Rastall-like gravity proposed in this work. Clearly, our $M^{\nu}{}_{\mu}=\kappa Y^{\nu}{}_{\mu}$ (n.b. Eq. (13)) depends on both the geometric and the matter sectors. Actually, in all choices given in Eqs. (15), (43), and (59)$\,\sim\,$(62), the corresponding $M^{\nu}{}_{\mu}=\kappa Y^{\nu}{}_{\mu}$ depend on both $R$ and $T$. Unlike the original Rastall gravity proposed in Rastall:1973nw , the relation $\left(4\kappa\lambda-1\right)R=\kappa T$ is not valid in our Rastall-like gravity. Due to the lack of the explicit relation between $R$ and $T$, our $M^{\nu}{}_{\mu}=\kappa Y^{\nu}{}_{\mu}$ certainly cannot be a pure matter quantity, and at least part of $M^{\nu}{}_{\mu}=\kappa Y^{\nu}{}_{\mu}$ comes from geometry. Even if the usual mater sector $T^{\nu}{}_{\mu}$ is given, the corresponding $\tilde{T}^{\nu}{}_{\mu}$ in Eq. (65) still cannot be explicitly determined. In this sense, our Rastall-like gravity proposed in this work is essentially different from GR and the original Rastall gravity proposed in Rastall:1973nw , and actually it is somewhat similar to most of the modified gravity theories in the literature (e.g. $f(R)$ theories). Without the need of exotic matter, the modification term coming from geometry makes difference, as in the cases of other modified gravity theories in the literature (e.g. $f(R)$ theories).

In a successful emergent universe scenario, the universe must exit the static state and then enter an expansion phase (we thank the referee for pointing out this issue). For example, in Eddington-inspired Born-Infeld (EiBI) theory Banados:2010ix , the universe can be past-eternal (see Li:2017ttl for stability analysis), and then expands in the late time (see the bottom panel ($\kappa>0$) of Fig. 2 in Banados:2010ix ). Noting that the ultimate theory of gravity is not available so far, the existing theories (including our Rastall-like gravity theory) might be just approximations of a fundamental theory. In the past ($t\to-\infty$), the difference between Rastall-like gravity theory and the fundamental theory can be negligible, and hence the static universe is past-eternal. However, in the late time, the difference between Rastall-like gravity theory and the fundamental theory becomes significant. In this case, the universe accordingly becomes unstable in the presence of perturbations. Then, the universe exits the static state and enters an expansion phase.

ACKNOWLEDGMENTS

We thank the anonymous referee for useful comments and suggestions, which helped us to improve this work. We are grateful to Zhao-Yu Yin, Da-Chun Qiang, Hua-Kai Deng and Shu-Ling Li for kind help and discussions. This work was supported in part by NSFC under Grants No. 11975046 and No. 11575022.