Entropy of Hawking Radiation for Two-Sided Hyperscaling Violating Black Branes

It is well known that black holes can emit particles with a thermal spectrum which is called Hawking radiation Hawking:1975vcx . In this manner, a black hole loses its mass, and might be disappeared completely. Assuming that the black hole was formed from the collapse of some matter in a pure quantum state leads to the famous Hawking’s information paradox Hawking:1976ra , since the final state, i.e. the emitted radiation, is in a mixed state. Moreover, the final state cannot be obtained via a S matrix from the initial pure state. On the other hand, the fine-grained (von Neumann) entropy of the Hawking radiation $S_{\rm R}$ grows linearly in time until the black hole completely disappears. In this manner, it eventually exceeds the coarse-grained (thermodynamic or Bekenstein-Hawking) entropy $S_{\rm BH}$ of the initial black hole. On the other hand, the von Neumann entropies of the black hole and radiation are equal to each other, i.e. $S_{\rm R}=S_{\rm black\;hole}$, since the whole system is in a pure state. Moreover, the fine-grained entropy of the black hole has to be less than its coarse-grained entropy, i.e. $S_{\rm black\;hole}\leq S_{\rm BH}$ Page:2013dx . Therefore, one has Page:2013dx (See also Almheiri:2020cfm )

The time at which the saturation happens is called the ”Page time” Page:1993wv ; Page:2013dx . Moreover, since the $S_{\rm BH}$ is a decreasing function of time, $S_{\rm R}$ has to start decreasing after the Page time and goes to zero at the end of the evaporation. Consequently, the unitarity requires that the fine-grained entropy of Hawking radiation follows the so called Page curve Page:1993wv ; Page:2013dx .
Unearthing the unitary evaporation of Black holes is one of the most important puzzles in gravity. Recently, a very sophisticated resolution for information paradox which is called ”island rule” were introduced Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni ; Almheiri:2019yqk in the context of the AdS/CFT correspondence Maldacena:1997re . This rule is inspired by the concept of ”Quantum Extremal Surfaces” (QES) Engelhardt:2014gca which are the generalization of the (H)RT surfaces Ryu:2006bv ; Hubeny:2007xt applied in the calculation of holographic entanglement entropy. More precisely, a QES is a classical spacelike codimension-two surface in the bulk spacetime which minimizes the generalized entropy Engelhardt:2014gca ; Faulkner:2013ana which is composed of an area term and the von Neumann entropy of matter quantum fields in a bulk region enclosed between the QES and the corresponding boundary region. The motivation for the island rule was the observation that the entanglement wedge (EW) of an evaporating black hole at late times does not include all of its interior. Therefore, one might naturally expects that the EW of the radiation region $\mathcal{R}$, which is a spatial region far away from the black hole where the radiation is collected, contains some parts of the black hole interior dubbed ”island” $\mathcal{I}$ Penington:2019npb ; Almheiri:2019hni (See figure 1). Consequently, to calculate the von Neumann entropy of the Hawking radiation, one has to consider the contributions of islands. According to the rule, one should first calculate the generalized entropy $S_{\rm gen}$ as follows Almheiri:2019hni ; Almheiri:2019yqk

where $\partial\mathcal{I}$ is the boundary of the island and is a codimension-two surface. It is expected that for one-sided black holes $\partial\mathcal{I}$ is located inside the black hole Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni ; Gautason:2020tmk ; Hartman:2020swn . However, for two-sided black holes, it is located outside the black hole Almheiri:2019yqk ; Almheiri:2019psy ; Gautason:2020tmk . If the matter quantum field theory has a gravitational dual theory, the island is connected to the radiation region through extra dimensions Almheiri:2019hni . Moreover, $S_{\rm matter}\left(\mathcal{R}\cup\mathcal{I}\right)$ is the von Neumann entropy of matter quantum fields in the region $\mathcal{R}\cup\mathcal{I}$. In the following, we denote the endpoints of $\mathcal{R}$ and $\mathcal{I}$ by $b_{\pm}$ and $a_{\pm}$, respectively (See figure 1). Next, one has to extremize $S_{\rm gen}$ over all possible islands. If there is more than an island, one should consider the one which gives the minimum generalized entropy

In this case, $\partial\mathcal{I}$ is a minimal QES. Moreover, when there are no islands, the area term in eq. (3) vanishes, and hence the entropy of Hawking radiation is simply given by the von Neumann entropy of the matter fields on the region $\mathcal{R}$

The island rule were verified for two-dimensional Jackiw-Teitelboim (JT) black holes Jackiw:1984je ; Teitelboim:1983ux in refs. Penington:2019kki ; Almheiri:2019qdq ; Goto:2020wnk ²²2See also Marolf:2020rpm ; Bousso:2021sji for related discussions. Moreover, a new proof was recently introduced in ref. Pedraza:2021ssc which is based on minimizing the microcanonical action of an entanglement wedge. by calculating the Euclidean gravitational path integral via replica trick in gravity Lewkowycz:2013nqa ; Dong:2016hjy . It was observed that the gravitational path integral has two saddle points: First, Hawking saddle which gives the usual entropy for Hawking radiation that grows linearly with time. Second, replica wormhole saddle which is a wormhole connecting various copies of the original black hole. It was observed that the replica saddle leads to the contribution of the islands and enforces the entropy of radiation to saturate. It should be pointed out that the Hawking saddle is dominant before the Page time. However, the replica wormhole saddle is dominant after the Page time.
Furthermore, the island rule have been explored extensively for various black holes in flat and AdS spacetimes such as: JT gravity Penington:2019kki ; Almheiri:2019qdq ; Almheiri:2019yqk ; Hollowood:2020cou ; Chen:2020jvn ; Goto:2020wnk ; Chen:2019uhq ; Balasubramanian:2021xcm , two-dimensional dilaton gravity Gautason:2020tmk ; Anegawa:2020ezn ; Hartman:2020swn ; Wang:2021mqq ; He:2021mst , in higher dimensions Almheiri:2019psy ; Hashimoto:2020cas ; Karananas:2020fwx ; He:2021mst ; Chen:2020uac ; Chen:2020hmv ; Krishnan:2020fer ; Geng:2020qvw ; Matsuo:2020ypv ; Ghosh:2021axl ; Arefeva:2021kfx ; Saha:2021ohr , higher derivative gravity theories Alishahiha:2020qza , charged black holes Ling:2020laa ; Wang:2021woy ; Kim:2021gzd ; Ahn:2021chg ; Yu:2021cgi , pure BTZ black hole microstates ³³3These geometries are obtained by excising a two-sided BTZ black hole with a timelike dynamical brane, dubbed End-of-the-World (ETW) brane Kourkoulou:2017zaj ; Almheiri:2018ijj ; Cooper:2018cmb . The action-complexity of this model was studied in ref Omidi:2020oit . Balasubramanian:2020hfs ; Anderson:2020vwi ; Fallows:2021sge , and black holes coupled to gravitating baths Balasubramanian:2021wgd ; Anderson:2020vwi ; Geng:2020fxl ; Geng:2021iyq . See also Almheiri:2020cfm for a very nice review on the topic. Furthermore, the rule were applied in de Sitter spacetime Hartman:2020khs ; Balasubramanian:2020xqf ; Aalsma:2021bit ; Sybesma:2020fxg ; Geng:2021wcq ; Chen:2020tes , flat-space cosmology ⁴⁴4It is a solution of the three dimensional Einstein gravity with no cosmological constant. Azarnia:2021uch and separate universes Balasubramanian:2021wgd ; Balasubramanian:2020coy ; Miyata:2021ncm ; Fallows:2021sge ; Miyata:2021qsm .
In this paper, we study the von Neumann entropy of the Hawking radiation for a two-sided Hyperscaling Violating (HV) black brane which is coupled to two thermal baths that are Minkowski spacetimes. We assume that the matter fields in the black brane geometry as well as in the baths are described by the same QFT. We consider two different scenarios for the QFT: First, it is a d+2 dimensional CFT which does not necessarily have a dual gravity theory. Second, it is a d+2 dimensional QFT which has a dual gravity theory that is a zero-temperature HV geometry (See eq. (88)). We dub it ”HV QFT”. We calculate $S_{\rm R}$ for the two cases and verify that it obeys eq. (2). It is observed that at early times there are no islands. However, at late times there is an island which leads to the saturation of $S_{\rm R}$ at twice the Bekenstein-Hawking entropy of the black brane. Moreover, we study the Page curve and Page time for different values of the dynamical exponent $z$ and hyperscaling violation exponent $\theta$ of the black brane. For the case where the matter is described by a HV QFT, we also study the behavior of the Page time and Page curve with respect to the dynamical and hyperscaling violation exponents of the HV QFT which are shown by $z_{m}$ and $\theta_{m}$, respectively. We study the two cases $\theta_{m}=0$ and $\theta_{m}\neq 0$, separately.
The organization of the paper is as follows: in Section 2, we briefly review the HV black brane geometry. In Section 3, we calculate the entropy of Hawking radiation for the case where the matter fields are described by a $CFT_{d+2}$. We also find the Page time and study its behavior as a function of the dynamical exponent $z$ and hyperscaling violation exponent $\theta$ of the black brane geometry. In Section 4, we do the same calculations for the case where the matter fields are described by a d+2 dimensional HV QFT. In section 5, we summarize our results and briefly address a few interesting future directions.

In this section, we briefly review Hyperscaling Violating (HV) black branes. These are solutions to the Einstein-Maxwell-Dilaton gravity with the following action Alishahiha:2012qu

The gauge field $A$ breaks Lorentz invariance and introduces the dynamical exponent $z$. ⁵⁵5It should be pointed out that the action is Lorentz invariant and the solution does not respect the Lorentz symmetry. More precisely, under the coordinate transformation Dong:2012se $\displaystyle t\rightarrow\lambda^{z}t\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x^{i}\rightarrow\lambda x^{i}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;r\rightarrow\lambda r$ the metric (11) is covariant, i.e. $ds\rightarrow\lambda^{\frac{\theta}{d}}ds$. It is clear that there is an anisotropy among the $t$ and $x^{i}$ coordinates in the above transformation. Therefore, the Lorentz symmetry is broken when $z\neq 1$, and it is a consequence of having a non-zero gauge field in eq. (9) for $z\neq 1$. Moreover, the non-trivial potential $V(\phi)$ breaks the scaling symmetry and introduces the Hyperscaling Violation exponent $\theta$. The dilaton field $\phi$, its potential $V(\phi)$ and the field strength of the gauge field are given by (See ref. Alishahiha:2012qu for more details)

where the constant parameters are defined as follows

On the other hand, the metric of the black brane is given by Alishahiha:2012qu ⁶⁶6We set the AdS radius to one, i.e. $R=1$. Moreover, we also set the dynamical scale $r_{F}$ to one.

where the emblankening factor is as follows

Here we defined $\theta_{e}=\frac{\theta}{d}$ and an effective dimension $d_{e}=d-\theta$. Furthermore, the null energy condition puts the following constraints on the values of $z$ and $\theta$

In the following, we restrict ourselves to the case $z\geq 1$ and $d_{e}>0$. ⁷⁷7As mentioned in ref. Alishahiha:2012qu , this solution is not valid for the case $d_{e}=0$. Moreover, for $d>\theta$ the geometry is unstable Dong:2012se . It should also be pointed out that for $\theta=0$ and $z=1$, the scaling and Lorentz symmetries are restored in the dual HV QFT. In this case, the scalar field becomes a constant and the gauge field equals to zero. Moreover, $V(\phi)$ plays the rule of the cosmological constant $\Lambda$, and hence the solution reduces to a d+2 dimensional AdS black brane. On the other hand, the temperature and thermal (Bekenstein-Hawking) entropy of the black brane are given by Alishahiha:2012qu

Here $V_{d}$ is the volume in the transverse directions $x^{i}$. On the other hand, the tortoise coordinate is as follows

In this section, we study the entropy of Hawking radiation for a two-sided HV black brane geometry. To allow the black brane to evaporate, we couple it to two thermal baths which are two Minkowski spacetimes on the left and right hand side of the Penrose diagram (See figure 1). Similar to ref. Almheiri:2019hni we add some matter fields in the bulk spacetime. For convenience, we assume that the matter fields in the HV black brane geometry and inside the two baths are described by the same QFT. Moreover, one can impose transparent boundary conditions Almheiri:2019qdq for the matter fields on the boundary of the HV black brane geometry. Furthermore, we assume that the matter fields are described by a $\rm CFT_{d+2}$. Then the whole action in the gravity region (see the purple region in figure 1) is given by

The CFT may Almheiri:2019hni ; Almheiri:2019psy ; Gautason:2020tmk ; Chen:2019uhq ; Chen:2020uac ; Chen:2020jvn ; Chen:2020hmv ; Ling:2020laa or may not Almheiri:2019yqk ; Alishahiha:2020qza ; Hashimoto:2020cas have a dual gravity theory. Moreover, we assume that its central charge is very large, i.e. $c\gg 1$. Therefore, the contributions of the matter fields of the $CFT_{d+2}$ to the EE are dominant over those of the graviton and the matter fields in $I_{\rm HV}$ Almheiri:2019hni ; Alishahiha:2020qza ; Azarnia:2021uch . Moreover, we assume that

to be able to neglect the backreaction of the matter on the HV black brane geometry Alishahiha:2020qza ; Azarnia:2021uch .
Furthermore, it is believed that the generalized entropy is finite and independent of the UV cutoff of the theory Bousso:2015mna ; Susskind:1994sm ; Engelhardt:2014gca ; Almheiri:2020cfm . On the other hand, there are some UV divergent terms in the entanglement entropy (EE) of a QFT, and the leading divergent term follows the area law Bombelli:1986rw ; Srednicki:1993im . Notice that our entangling regions are in the shape of strips on a constant time slice such that they are extended in the transverse directions $x^{i}$ (See figure 1). Moreover, from holographic calculations, it is known that the EE of a strip with width $\ell$ and lengths $L$ in a holographic $CFT_{d+2}$ with $d\geq 1$ which is in its vacuum state, is given by Ryu:2006ef ⁸⁸8For spherical entangling regions there are subleading UV divergent terms Ryu:2006ef .

Here the first term is proportional to the area $L^{d}$ of the strip and the second term is a constant universal term. Inspired by this observation, one might expect that

where $\epsilon$ is the UV cutoff of the matter $CFT_{d+2}$. Therefore, we renormalize Newton’s constant as follows Susskind:1994sm ; Almheiri:2020cfm ; Hashimoto:2020cas ; Alishahiha:2020qza ; Azarnia:2021uch ⁹⁹9It was shown in ref. Susskind:1994sm that in four dimensions, i.e. for d=2, the loop diagrams for matter fields lead to the renormalization of the gravitational coupling $G_{N}$ as in eq. (20). Moreover, this renormalization cancel the area law divergent term in eq. (19) such that the generalized entropy is finite. See Bousso:2015mna for a nice review on this topic.

to absorb the UV divergent term in eq. (19). Consequently, we merely consider the finite part of the EE of the matter fields in the following, and for an arbitrary interval $[m_{1},m_{2}]$ in the bulk spacetime, we denote it by $S^{\rm f}_{\rm matter}(m_{1},m_{2})$. Having said this, we rewrite the generalized entropy in eq. (3) as follows Hashimoto:2020cas ; Alishahiha:2020qza ; Azarnia:2021uch

Another important point is that the metric in eq. (11), has translational symmetries along the transverse directions $x^{i}$. Therefore, one might expect that the locations of the endpoints of the island, and hence the generalized entropy are independent of the coordinates $x_{i}$. Consequently, one might expect that the problem effectively reduces to two dimensions with coordinates $(t,r)$. Moreover, by compactification of the $x_{i}$ directions, one might consider the transverse manifold $\mathbb{R}^{d}$ which is parametrized by $x_{i}$’s, as a d dimensional sphere with a very large radius. In this manner, one might assume that the s-wave approximation mentioned in refs. Hashimoto:2020cas ; Alishahiha:2020qza ; He:2021mst ; Penington:2019npb can be applied. ¹⁰¹⁰10We would like to thank Mohsen Alishahiha and Ali Naseh for their very helpful comments on this topic. In other words, after the expansion of the matter fields in terms of the spherical harmonics on the large sphere, one can have both massless and massive Kaluza-Klein modes. Then, each mode behaves as an independent free field in two dimensions (with coordinates $t$ and $r$) whose mass is proportional to the inverse of the radius of the sphere Penington:2019npb . Since the radius of the sphere is very large, the masses of the Kaluza-Klein modes have to be very small. Moreover, since the endpoints $b_{\pm}$ of the radiation region are very far from the endpoints $a_{\pm}$ of the islands ¹¹¹¹11We are assuming that $a\approx r_{h}$ and $b\gg r_{h}$, where $a$ and $b$ are the radial coordinates of the endpoints $a_{\pm}$ and $b_{\pm}$, respectively. , we assume that only massless modes can reach the radiation region (See also Azarnia:2021uch ; Penington:2019npb ; Hashimoto:2020cas ). Therefore, the contribution of the massless modes to $S_{\rm matter}(\mathcal{R}\cup\mathcal{I})$ is dominant over that of the massive modes. ¹²¹²12We would like to thank Ali Naseh for bringing this point to our attention. Consequently, one may apply the formula for a two dimensional CFT in its vacuum state to calculate $S_{\rm matter}(\mathcal{R}\cup\mathcal{I})$, which for an interval of length $\ell$ is given by Holzhey:1994we ; Calabrese:2004eu

where $c$ is the central charge and $\epsilon$ is the UV cutoff.