Temperature of McVittie Black Holes in Cavities

TL;DR

This paper investigates the temperature of dynamical McVittie black holes within isothermal cavities, deriving a simpler method to calculate their temperature that aligns with existing results for de Sitter black holes.

Contribution

It introduces a new, easier approach to determine the temperature of McVittie black holes in cavities, overcoming challenges of traditional methods for dynamical black holes.

Findings

Derived cavity temperature from McVittie black hole temperature and Tolman relation.

Found that cavity temperature matches de Sitter black hole results when setting H^2 to Λ/3.

Proposed a simpler method for temperature calculation of dynamical black holes.

Abstract

In this paper, we study the temperature of the McVittie black holes in cavities. The cavities are isothermal boxes that can make the McVittie black holes thermodynamically stable. The McVittie black holes are dynamical black holes, so the traditional action method can hardly be used to calculate the cavity temperature. We find that the cavity temperature $T_c$ can be derived from the McVittie black hole temperature $T_+$ and the Tolman relation. By setting $H^2$ to be $\Lambda/3$, we also get the cavity temperature of de Sitter black holes, which are interestingly the same with the results from the action method, but our method is much more easier.

Equations

$$\mathrm{d}{s}^2=-\left(1-\frac{2m}{r}-H^2{r}^2\right)\mathrm{d}{t}^2-\frac{2Hr}{\sqrt{1-\frac{2m}{r}}}\mathrm{d}t\mathrm{d}r+\frac{1}{1-\frac{2m}{r}}\mathrm{d}{r}^2+r^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\phi }^2),$$

$$\displaystyle(h^{ab})=\left(\begin{array}[]{cc}-\frac{1}{1-\frac{2m}{r}},&-\frac{Hr}{\sqrt{1-\frac{2m}{r}}}\\ \\ -\frac{Hr}{\sqrt{1-\frac{2m}{r}}},&1-\frac{2m}{r}-H^{2}r^{2}\end{array}\right),$$

$${\nabla }^{a}r{\nabla }_{a}r=0,$$

$$1-\frac{2m}{r}-H^2{r}^2=0.$$

$${\dot{r}}_{+}=\frac{2H\dot{H}{r}_{+}^3}{1-3H^2{r}_{+}^2}.$$

$$\kappa :=\frac{1}{2\sqrt{-h}}\frac{\partial }{\partial x^a}\left(\sqrt{-h}{h}^{ab}\frac{\partial r}{\partial x^b}\right),$$

$$\kappa =\frac{m}{r^2}-\frac{\dot{H}r}{2\sqrt{1-\frac{2m}{r}}}-H^{2}r.$$

$${\kappa }_{+}=\frac{1}{2r_{+}}\left(1-3H^2{r}_{+}^2\right)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right),$$

$$T_{+}=\frac{1}{4\pi r_{+}}(1-3H^2{r}_{+}^2)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right),$$

$$T=\frac{T_H}{||\xi ||},$$

$$K^a\equiv$$

$$||K^a||=\sqrt{|K_a{K}^a|}=\sqrt{-h_{00}}=\sqrt{1-\frac{2m}{r}-H^2{r}^2},$$

$$T_c=\frac{T_{+}}{||K^a|{|}_c}=\frac{(1-3H^2{r}_{+}^2)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right)}{4\pi r_{+}\sqrt{1-\frac{2m}{r_c}-H^2{r}_c^2}}=\frac{(1-3H^2{r}_{+}^2)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right)}{4\pi r_{+}\sqrt{\left(1-\frac{r_{+}}{r_c}\right)[1-H^2(r_{+}^2+r_{+}{r}_c+r_c^2)]}}.$$

$$T_c=\frac{1-\Lambda r_{+}^2}{4\pi r_{+}\sqrt{\left(1-\frac{r_{+}}{r_c}\right)\left[1-\frac{\Lambda }{3}(r_{+}^2+r_{+}{r}_c+r_c^2)\right]}},$$

$$\mathrm{d}{s}^2=-\left(N^2-H^2{r}^2\right)\mathrm{d}{t}^2-\frac{2Hr}{N}\mathrm{d}t\mathrm{d}r+\frac{1}{N^2}\mathrm{d}{r}^2+r^2(\mathrm{d}{\theta }^2+{\sin }^2\theta \mathrm{d}{\phi }^2),$$

$$1-\frac{2m}{r}+\frac{Q^2}{r^2}-H^2{r}^2=0,$$

$$\dot{H}=-(r_{+}^2-3m{r}_{+}+2Q^2)\frac{{\dot{r}}_{+}}{H{r}_{+}^5}.$$

$$\kappa =\frac{m}{r^2}-\frac{Q^2}{r^3}-\frac{\dot{H}r}{2N}-H^{2}r.$$

$${\kappa }_{+}=\frac{1}{2r_{+}}\left(1-3H^2{r}_{+}^2-\frac{Q^2}{r_{+}^2}\right)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right),$$

$$T=\frac{{\kappa }_{+}}{2\pi }=\frac{1}{4\pi r_{+}}\left(1-3H^2{r}_{+}^2-\frac{Q^2}{r_{+}^2}\right)\left(1-\frac{{\dot{r}}_{+}}{2H^2{r}_{+}^2}\right).$$

$$||K^a||=\sqrt{|K_a{K}^a|}=\sqrt{-h_{00}}=\sqrt{1-\frac{2m}{r}+\frac{Q^2}{r^2}-H^2{r}^2}.$$

$$T_c=$$

$$=$$

$$T_c=\frac{1-\Lambda r_{+}^2-\frac{Q^2}{r_{+}^2}}{4\pi r_{+}\sqrt{(1-\frac{r_{+}}{r_c})\left[1-\frac{Q^2}{r_{+}{r}_c}-\frac{\Lambda }{3}(r_c^2+r_{+}^2+r_c{r}_{+})\right]}},$$

$$K^a=$$

$$=$$

Full text

I Introduction

In previous workAbdusattar:2022bpg , we investigated the thermodynamics of the McVittie black hole and found a Hawking-Page like phase transition. Found in 1933, the McVittie black hole or cosmological black holeMcVittie:1933zz ; Faraoni:2007es ; Kaloper:2010ec ; Nandra:2011ug ; Faraoni:2012gz can be regarded as a black hole embedded in the spatially flat Friedmann-Robertson-Walker universe, which is similar to the de Sitter black hole. Like the de Sitter black hole, McVittie black hole has two horizons, the black hole horizon and the cosmological horizon, both of which can evaporate with different temperature. One can treat the two horizons as two separate thermodynamic systems, establish the first law of thermodynamics, equation of state, etc. for the two horizons respectively. However, the two temperatures are not equal, so the McVittie black hole is not in the equilibrium state.

In order to make the McVittie black hole thermodynamically stable, one can place a cavity between the black hole horizon and the cosmological horizon as is often done for de Sitter black holesSimovic:2018tdy ; Marks:2021fpe . It has a fixed temperature and allows the black hole to remain in an equilibrium state. In this paper, we use this method to stabilize the McVittie black hole and study its temperature. Unlike the de Sitter black hole, we can not integrate the action of the McVittie black hole, because one can not find a static patch or coordinate system for McVittie black hole. Therefore, we can not get the temperature at the horizons from the action. We find that the Hawking temperature of McVittie black hole can be used to derive the cavity temperature. The McVittie black hole temperature $T_{+}$ is defined from the surface gravity at the black hole horizon. Start from $T_{+}$, we can get the cavity temperature $T_{c}$ by using the Tolman relation. In flat spacetime, equilibrium means temperature is the same in every position, but it is not the case for curved spacetime. It is found that in a curved spacetime or a gravitational system, temperature can be different even equilibrium is established. For a black hole, the temperature of the Hawking radiation cools down as it travels away from the horizon due to the redshift effect. It has an extreme high temperature near the horizon and is called the Hawking temperature if measured at an infinite far place. Temperature at different distance satisfies the Tolman(or Tolman-Ehrenfest) relationRovelli:2010mv , which says that the multiplication of temperature and the norm of the Killing vector field is a constant, which is actually equivalent to the redshift formula. However, McVittie black hole does not have a Killing vector field, so it is not a stationary black hole but a dynamical black hole. Fortunately, for dynamical black holes, we have the Kodama vector $K^{a}$ as a counterpart of the Killing vector. We get the Kodama vector for the McVittie black hole and find that it is just $\left(\frac{\partial}{\partial t}\right)^{a}$, which is very surprising and interesting. In the following, we put the Hawking temperature and Kodama vector into the Tolman relation and get the cavity temperature at $r_{c}$. What’s more, if the Hubble parameter is treated as a constant, i.e. $H^{2}=\Lambda/3$, the McVittie black hole becomes the de Sitter black hole. Therefore, from our results we also get the cavity temperature of the de Sitter black hole. Interestingly, the result is the same with that from the traditional action method. We further investigate the charged McVittie black holeGao:2004cr ; Faraoni:2014nba ; Guariento:2019ock and find that the Kodama vector is still $\left(\frac{\partial}{\partial t}\right)^{a}$. We also get the cavity temperature of the charged McVittie black hole from the Tolman relation and Kodama vector. As a special case of charged McVittie black hole, i.e. charged de Sitter black hole, we also get its cavity temperature by setting $H^{2}=\Lambda/3$, and the result is also the same with that from the action method. These results could be regarded as consistency checks, but our method is much more easier.

The paper is organized in the following way. In Sec.II, we introduce the McVittie black hole and calculate its temperature at the cavity. In Sec.III, we introduce the charged McVittie black hole and calculate its temperature at the cavity. In Sec.IV, we give the conclusions and discussions.

II Cavity Temperature of the McVittie Black Hole

The metric of the McVittie black hole can be written asNandra:2011ug

[TABLE]

where $H\equiv\dot{a}(t)/a(t)$ is the Hubble parameter111We use different signature with the original paper.. For convenience, we use $h_{ab}$ to denote the metric of the $(t,r)$ part and also give its inverse metric

[TABLE]

where $a,b=0,1$ and $x^{0}=t,x^{1}=r$.

The apparent horizon of a dynamical spacetime satisfies

[TABLE]

so for the McVittie black hole (1), this condition results to

[TABLE]

It has two solutions generally, and we denote the smaller one as $r_{+}$, i.e. the black hole horizon, and the larger one as $r_{cosmo}$, i.e. the cosmological horizon. From the above condition, one can get a useful relation

[TABLE]

The surface gravity at radius $r$ is defined asAbdusattar:2022bpg

[TABLE]

where $h$ is the determinant of $h_{ab}$, and inserting the metric (1), one can get

[TABLE]

At the black hole horizon $r_{+}$, the surface gravity can be written as

[TABLE]

so one can get the Hawking temperature of the black hole horizon

[TABLE]

where (7) is used.

At the cavity with radius $r_{+}<r_{c}<r_{cosmo}$, the temperature $T_{c}$ is fixed with the temperature of the Hawking radiation there, which is different with $T_{+}$ but can be obtained from the Tolman relation. Tolman relationRovelli:2010mv says that at a finite place of a stationary black hole, its temperature has the following relation with the Hawking temperature

[TABLE]

where $\xi$ is the Killing vector of the black hole. However, the McVittie black hole is not a stationary black hole and does not have a Killing vector, so this relation can not be directly used to calculate the temperature at the cavity of the Mcvittie black hole. Fortunately, we have a counterpart for the Killing vector, i.e. the Kodama vector, which plays similar role for dynamical black holes like the McVittie black hole. We can put it in the Tolman relation and get the cavity temperature. The Kodama vectorCai:2008gw is defined and got as follows

[TABLE]

which is exactly $\partial_{t}$ and very surprising, see Appendix for the calculations. We have $K_{a}K^{a}=h_{ab}K^{a}K^{b}=h_{00}$, so it is timelike in $r_{+}<r<r_{cosmo}$, null in $r=r_{+}$ or $r_{cosmo}$, and spacelike in $r<r_{+}$ or $r>r_{cosmo}$, respectively. In the timelike region, its norm is

[TABLE]

so the Temperature at the cavity $r=r_{c}$ is

[TABLE]

The McVittie black hole (1) can reduce to de Sitter black hole by fixing $H^{2}=\Lambda/3$, and in this case we have $\dot{r}_{+}=0$ from (8). Therefore, from the above result one can also get the cavity temperature of de Sitter black hole,

[TABLE]

which is exactly the same with the result by the action methodSimovic:2018tdy . This could be regarded as a consistency check and shows that the above method is reasonable.

III Cavity Temperature of the Charged McVittie Black Hole

Similar to the charged de Sitter black hole, the solution of the charged McVittie has also been found. The metric of the charged McVittie black hole can be written asGuariento:2019ock

[TABLE]

where $N\equiv\sqrt{1-2m/r+Q^{2}/r^{2}},H\equiv\dot{a}(t)/a(t)$, and $Q$ is the charge.

The apparent horizon of the charged McVittie black hole also satisfies (6), which in this case can be written as

[TABLE]

and the smaller solution is the black hole horizon and denoted by $r_{+}$. From the above condition, we can also get

[TABLE]

Insert the metric (18) into the definition (9) of the surface gravity, one gets the surface gravity at radius $r$ of the charged McVittie black hole,

[TABLE]

By using (7), the surface gravity at the black hole horizon can be written as

[TABLE]

so the Hawking temperature can be obtained

[TABLE]

It can be proved that the Kodama vector for the charged McVittie spacetime is still $K^{a}=(\frac{\partial}{\partial t})^{a}$, which is very shocking, see Appendix for the calculations. In the timelike region, its norm is

[TABLE]

Therefore, using the Tolman relation, the temperature at the cavity radius $r_{c}$ for the charged McVittie black hole is given by

[TABLE]

If one sets $H^{2}=\Lambda/3$, the charged McVittie black hole reduces to the charged de Sitter black hole, so we can get the cavity temperature of the charged de Sitter black hole

[TABLE]

which is still equivalent to the result from the action methodSimovic:2018tdy . Again, it shows that the above method is consistency and reasonable.

IV Conclusions and Discussions

In this paper, we get the temperature of the McVittie black hole and the charged McVittie black hole in a cavity with radius $r_{c}$. The cavity temperature is derived from the black hole Hawking temperature and Kodama vector by using the Tolman relation. We also get the cavity temperature of the de Sitter black hole and the charged de Sitter black hole respectively by setting $H^{2}=\Lambda/3$ in the results, which are the same with those from the action method. By the way, we also find that the Kodama vector of the McVittie and charged McVittie black hole is just $\frac{\partial}{\partial t}$, which is very surprising and interesting.

It shows that this method to get the cavity temperature is consistency and applicable at least. It is obvious that this method is much easier than the action method and can be used to similar investigations. It also shows that Kodama vector is very important and usedful for dynamical black holes.

Acknowledgments

This work is supported by National Natural Science Foundation of China (NSFC) under grant 12465011 and East China University of Technology (ECUT) under grant DHBK2023002.

Appendix A Kodama Vector

The following calculations apply to both the McVittie and charged McVittie cases.

[TABLE]

where $\epsilon_{cd}=\sqrt{|h_{ab}|}(\mathrm{d}t)_{c}\wedge(\mathrm{d}r)_{d}$.