High dimensional AdS-like black hole and phase transition in Einstein-bumblebee gravity

Local Lorentz invariance (LI) is the most fundamental spacetime symmetry of the special/general relativity (SR/GR) and the standard model (SM) of particle physics ashtekar . LI means that the equations of motion for particles and fields are invariant under all Lorentz transformations(global for SR, local for GR) and diffeomorphism transformation for a particle. However, LI may fail to be an exact principle at very high energy scales mattingly . GR and SM are the best theories describing the four fundamental forces and there are no known experimental conflicts. Nonetheless, they are fundamentally different in that SM is a quantum theory disregarding the gravitational effects of all particles, while GR is a classical geometrical theory ignoring all the quantum features of particles. For particles with energies on the order of $10^{30}$ eV (above the Planck scale), the gravitational interactions estimated by GR are very powerful, and so one shouldn’t ignore it camelia . Thus, on the very high energy scales, one has to reconsider combining the SM with GR in a unified theory, i.e., ”quantum gravity”. An effective field theory combining GR and SM on low energy scales is called standard model extension (SME), which couples the SM to GR, and involves extra items embracing information about the violations of Lorentz invariance(Lorentz violation, LV) happening on the Planck scale kostelecky2004 .

Studying the Lorentz violation (LV) is a useful approach toward investigating the foundations of modern physics. Besides SME, string theory kostelecky198939 , noncommutative field theories carroll , spacetime-varying fields bertolami69 , loop quantum gravity theory gambini , brane world scenarios burgess03 , massive gravity fernando and Einstein-aether theory jacobson are other proposals of Lorentz violation. The SME is experimentally accessible for studying possible observable signals of violation the particle Lorentz symmetry. In this model, the local Lorentz violating is provided by a spontaneous symmetry breaking potential due to self-interacting tensor fields, whose vacuum expectation value (VEV) yields to background tensor fields. A specific theory is the bumblebee field $B^{\mu}$, an self-interacting tensor field has a nonzero VEV $b^{\mu}$ which defines a privileged direction in spacetime and spontaneously breaks the local Lorentz symmetry.

There are many forms of the bumblebee potential. One is the smooth functional bumblebee potential $V(X)=kX^{2}/2$ which has a minimum when the bumblebee field equals its VEV, i.e., $V=0$ and $V^{\prime}(X)=kX=0$ if $X=0$, where $k$ is a constant bluhm and $X=B^{\mu}B_{\mu}\pm b^{2}$. With this potential, the four dimensional static black hole solutions were found by Bertolami et al. bertolami and by Casana et al. casana ; the higher $D$-dimensional static black hole was found by Ding et al. ding2022 ; the rotating black hole solutions were built by Ding et al. ding2020 and by Jha et al. jha . An other form of bumblebee potential is a linear multiplier function $V(X)=\lambda X/2$, where $\lambda$ is a Lagrange-multiplier. With this potential, the Schwarzschild-anti-de Sitter-like and Schwarzschild-de Sitter-like black hole solutions were found by Maluf et al. maluf , wherein the radial and tangential pressures are different, like as an anisotropic fluid.

The quantum field theory in a spacetime with a horizon exhibits thermal behavior which must be thought of as an emergent phenomena padmanabhan . A black hole event horizon has the conceptions of temperature and entropy hawing1975 , and the first law of thermodynamics $dU=TdS-PdV$ can be built, where $P$ is the radial pressure of the matter, $U$ is the internal energy and $V$ is the volume of the black hole. There also has transfer of heat energy and the conception of phase transition hawking . Then the gravity system can be mapped to a thermodynamic system and studying on black hole thermodynamics can help to construct a quantum theory of gravity.

We will study the high dimensional black hole solutions and its thermodynamics and phase transition with a negative cosmological constant in the theory of Einstein-bumblebee gravity. The rest paper is organized as follows. In Sec. II we give the background for the Einstein-bumblebee theory and derive the black hole solution by solving the gravitational field equations. In Sec. III, we study its thermodynamical properties. In Sec. IV, we study the phase transitions and find some effects of the Lorentz breaking constant $\ell$. Sec. V is for a summary.

In the bumblebee gravity model, one introduces the bumblebee vector field $B_{\mu}$ which has a nonzero vacuum expectation value, leading to a spontaneous Lorentz symmetry breaking in the gravitational sector via a given potential. In the higher $D\geq 4$ dimensional spacetime, the action of Einstein-bumblebee gravity is ding2021 ,

where $R$ is Ricci scalar, $\Lambda$ is an negative cosmological constant. $\kappa=8\pi G_{D}/c^{4}$, where $G_{D}$ is the $D$-dimensional gravitational constant and has the relation to Newton’s constant $G$ as $G_{D}=G\Omega_{D-2}/4\pi$ boulware , $\Omega_{D-2}=2\sqrt{\pi}^{D-1}/\Gamma[(D-1)/2]$ is the area of a unit $(D-2)$-sphere. Here and hereafter, we take $G_{D}=1$ and $c=1$ for convenience.

The coupling constant $\varrho$ dominates the non-minimal gravity interaction to bumblebee field $B_{\mu}$. The term $\mathcal{L}_{M}$ represents possible interactions with matter or external currents. The potential $V(B_{\mu}B^{\mu}\mp b^{2})$ triggers Lorentz and/or $CPT$ (charge, parity and time) violation, where the field $B_{\mu}$ acquires a nonzero VEV, $\langle B^{\mu}\rangle=b^{\mu}$, satisfying the condition $B^{\mu}B_{\mu}\pm b^{2}=0$. The constant $b$ is a real positive constant. Another vector $b^{\mu}$ is a function of the spacetime coordinates and has a constant value $b_{\mu}b^{\mu}=\mp b^{2}$, where $\pm$ signs mean that $b^{\mu}$ is timelike or spacelike, respectively. It gives a nonzero vacuum expectation value (VEV) for bumblebee field $B_{\mu}$ indicating that the vacuum of this model obtains a prior direction in the spacetime. The bumblebee field strength is

This antisymmetry of $B_{\mu\nu}$ implies the constraint bluhm

Varying the action (1) with respect to the metric yields the gravitational field equations

where $G_{\mu\nu}=R_{\mu\nu}-g_{\mu\nu}R/2$, and the bumblebee energy momentum tensor $T_{\mu\nu}^{B}$ is

The prime denotes differentiation with respect to the argument,

Varying instead with respect to the the bumblebee field generates the bumblebee equations of motion (supposing that there is no coupling between the bumblebee field and $\mathcal{L}_{M}$),

The contracted Bianchi identities ($\nabla^{\mu}G_{\mu\nu}=0$) lead to conservation of the total energy-momentum tensor

We suppose that there is no matter field and the bumblebee field is frozen at its VEV like in Refs casana ; bertolami , i.e., it is

For the forms of potential $V$, there are both classes: the smooth functionals of $x$, i.e., $\kappa x^{2}/2$, that minimized by the condition $x=0$ and, the Lagrange-multiplier form $\lambda x/2$, where $\lambda$ is a non-zero constant and deserves as a Lagrange-multiplier field bluhm . The smooth potential does not admit new black hole solution with nonzero cosmological constant $\Lambda$ maluf , one can also see the proof in the appendix for more details.

In order to generate a black hole solution in the case of non-zero cosmological constant $\Lambda$, we should apply a linear function form of the Lagrange-multiplier potential

This potential is $V=0$ under the condition (9) and its derivative is $V^{\prime}=\lambda/2$ which can modify the Einstein equations. However, this additional degree of freedom of the $\lambda$ field is auxiliary. Then the first two terms in Eq. (II) are like those of the electromagnetic field, the only distinctiveness are the coupling items to Ricci tensor and this $\lambda$ term. Under this condition, Eq. (4) leads to gravitational field equations ding2021

with

The static spherically symmetric black hole metric in the $D$ dimensional spacetime has the form

where $\Omega_{D-2}$ is a standard $D-2$ sphere. In this static spherically symmetric spacetime, the most general form for the bumblebee field would be $b_{\mu}=(b_{t},b_{r},0,0)$, where $b_{t}$ and $b_{r}$ are functions of $r$ subject to the constraint $-b_{t}^{2}e^{-2\phi}+b_{r}^{2}e^{-2\psi}=b^{2}$, here $b$ is a positive constant. In this general case, the bumblebee field has both radial and a time component for the vacuum expectation value. In the purely radial case $b_{t}=0$, the authors in Refs. bertolami ; casana obtained new black hole solutions indeed. But for the general case(temporal and radial), the authors in Ref. bertolami obtain a slightly perturbed metric, where one cannot constrain the physical parameters from the observed limits on the PPN(parameterized post-Newtonian) parameters. Hence here we consider only the purely radial case to get a black hole solution and let the general case for the future work.

We pay attention to that the bumblebee field has a radial vacuum energy expectation because that the spacetime curvature has a strong radial variation, on the contrary that the temporal changes are very slow. Now the bumblebee field is supposed to be spacelike as that

Then the bumblebee field strength is

whose components are all zero. And their divergences are all zero, i.e.,

From the bumblebee field motion equation (7), we have the projection of the Ricci tensor along the bumblebee field is

As to gravitational field equation (11), one can obtain the following three component equations

where we have redefined the Lorentz-violating parameter $\ell=\varrho b^{2}$ and, the prime ^′ is the derivative with respect to the corresponding argument, respectively. From the Eq. (18), one can obtain a metric function as

where

and $M$ is the mass of the black hole. In order to recover the 4-dimensional Schwarzschild-like solution casana in the cases of $\Lambda=0$, then we let another function to

It is easy to prove that the above results of $e^{2\phi}=f(r),e^{2\psi}=(1+\ell)/f(r)$ can meet the gravitational Eq. (20). Then the bumblebee field is

With the bumblebee field motion equation (17) or the gravitational Eq. (19), a solution of this type will be possible if and only if

which is the constraint on the additional field $\lambda$ coming from the potential (10). It is easy to see that if the Lagrange-multiplier field $\lambda=0$, then $\Lambda=0$ and the metric (22) is the $D$-dimensional Schwarzschild-like black hole ding2022 . So $\lambda$ isn’t a new freedom and we can define the effective cosmological constant $\Lambda_{e}$ as following to absorb it,

Then $\Lambda=(1+\ell)\Lambda_{e}$ and, the present black hole metric is

with the metric function $f(r)$ as

When LV constant $\ell\rightarrow 0$, the effective cosmological constant $\Lambda_{e}\rightarrow\Lambda$ and it recovers the Schwarzschild-AdS black hole. When $r\rightarrow\infty$, the metric function

and the metric (27) becomes a $D$-dimensional AdS-like spacetime. The black hole horizon locates at the largest real root of the following equation

It is easy to see that it can give the black hole mass $M$ from the horizon radius,

It shows that the bumblebee field affects the location of the event horizons, contrary to the Schwarzschild-like solution in which the horizon radius is the same as of the Schwarzschild black hole.

A black hole is not only a gravity system, but also a special thermodynamic system due to that its surface gravity $\kappa$ and horizon area $A$ are the close similarity to the temperature $T$ and entropy $S$, $T=\kappa/2\pi$, $S=A/4$. Four laws of black hole thermodynamics were established in Ref. bardeen . LV can modify the geometry of a black hole, so in this kind of black hole, can the thermodynamical laws still hold? In this section we study the Smarr formula and the first law of this LV black hole.

A phase transition is a discontinuous change in the properties of a substance, as its environment is changed only infinitesimally. An isolated Schwarzschild black hole is thermodynamically unstable due to its negative heat capacity. However for a Schwarzschild AdS black hole, besides the negative heat phase, a phase of positive heat capacity will occur. At a given temperature and pressure, the stable phase is always the one with the lower Gibbs free energy. In an AdS space, the Gibbs energy of the thermal radiation phase is zero wei . So the negative free energy of the positive heat capacity black hole phase is stable phase. In this section, we study this two phase transitions.

From the Eq. (30), we can express the mass $M$ as

$M$ can be interpreted as an enthalpy kastor $H=U+PV$, so the black hole internal energy $U$ is

Then the first law (84) can be rewritten as following

which is the same as the fundamental thermodynamical equation of a simple gas system. The temperature (44) can be rewritten as

which is shown in Fig. 1 when $D=4$.

There exist a minimum temperature

when $r_{h}=\sqrt{(D-2)(D-3)/[16(1+\ell)\pi P]}$. Note that $T_{min}$ isn’t dependent on $\ell$. From the left panel of the Fig. 1, when $T>T_{min}$, there are two possible black hole masses hawking which can be in equilibrium with thermal radiation, i.e., $M>M_{d}$ or $M<M_{d}$, where $M_{d}$ is the black hole mass whose heat capacity is divergent (66). Fig. 1 shows that the temperature of the large black hole ($M>M_{d}$) increases when the LV coupling constant $\ell$ becomes bigger. At a fixed pressure $P$, the black hole heat capacity $C_{P}$ is

which is also shown in Fig. 1 when $D=4$. From the right panel of the Fig. 1, one can see that the small black hole mass has negative specific heat, so it is unstable and will decay. While the large black hole mass has positive heat capacity and therefore it is thermally stable. It is easy to see that the heat capacity at constant pressure is divergent at the black hole mass $M_{d}$,

It is easy to see that the LV constant $\ell$ decreases the mass $M_{d}$, showing that the phase transition from a small black hole to large one will more easily occur. At this divergent point, the black hole entropy is

Fig. 2 shows the entropy with the temperature at unit pressure when $D=4$.

The phase transition between these two phases is called small-large black hole phase transition.

There is another kind of phase transition called Hawking-Page(HP) phase transition, which is between a metastable large black hole phase and stable large black hole phase in an anti-de Sitter spacetime. Witten witten explained that this HP phase transition is a confinement/deconfinement phase transition in the AdS/CFT correspondence. It can also be understood as a solid/liquid phase transition in black hole chemistry kubiznak2015 . HP phase transition occurs when Gibbs free energy is zero. The black hole Gibbs free energy can be obtained

which is shown in Fig. 3 when $D=4$. It can be seen that there are two branches, the above parts corresponding to small black hole phase whose free energy are positive for any temperature; the below ones corresponding to metastable large black hole phase when $G>0$ and stable large black hole phase when $G<0$. When the temperature $T<T_{min}$, the spacetime has only one phase—thermal radiation, whose free energy is zero.

At the temperature $T=T_{min}$, its free energy has a maximum value $G_{max}$,

The temperature at Hawking-Page phase transition point can be obtained when $G=0$,

which is also independent on $\ell$ like the minimum temperature $T_{min}$. At this point $r_{h}=\sqrt{(D-1)(D-2)/[16(1+\ell)\pi P]}$ and the black hole mass is,