# Holographic Thermodynamics of Accelerating Black Holes

###### Abstract

We present a careful study of accelerating black holes in anti-de Sitter spacetime, formulating the thermodynamics and resolving discrepancies that have appeared in previous investigations of the topic. We compute the dual stress-energy tensor for the spacetime and identify the energy density associated with a static observer at infinity. The dual energy-momentum tensor can be written as a three-dimensional perfect fluid plus a non-hydrodynamic contribution with a universal coefficient which is given in gauge theory variables. We demonstrate that both the holographic computation and the method of conformal completion yield the same result for the mass. We compare to previous work on black funnels and droplets, showing that the boundary region can be endowed with non-compact geometry, and comment on this novel holographic dual geometry.

###### pacs:

04.20.Jb, 04.70.-sThe importance of black holes in advancing our understanding of physics cannot be underestimated. They provide a setting for testing our most fundamental ideas about gravity under extreme conditions and offer us insight into the underlying microscopic degrees of freedom that may associated with quantum gravity. The subject of black hole thermodynamics Bekenstein:1973ur ; Bekenstein:1974ax ; Hawking:1974sw has proven to be an invaluable tool to this end, and broad classes of black holes have been shown to exhibit a rich and varied range of thermodynamic behaviour, particularly in anti-de Sitter spacetime Kubiznak:2016qmn .

Within this framework, accelerating black holes have presented a
unique challenge. The idealized solution is described by the
*C-metric* Kinnersley:1970zw ; Plebanski:1976gy ; Dias:2002mi ; Griffiths:2005qp , whose spacetime has a
string-like singularity along one polar axis attached to the black hole.
We can think of this conical singularity as a cosmic string (indeed, the
conical singularity can be replaced by a finite width topological
defect Gregory:1995hd , or a magnetic flux tube Dowker:1993bt )
with the tension providing the force driving the acceleration.
Surprisingly, even though these black holes are not isolated because of
the “cosmic strings” it is possible to derive sensible looking thermodynamics,
although recent studies have apparently conflicting results
Appels:2016uha ; Appels:2017xoe ; Gregory:2017ogk ; Astorino:2016ybm .

We consider here the interpretation of an accelerating black hole in anti-de Sitter (AdS) spacetime, with a focus on a holographic interpretation of the thermodynamics. We resolve conflicting issues that exist in the literature, obtain a distinct set of thermodynamic variables that are now consistent with the gravitational action, and agree with both the conformal and holographic methods for computing conserved charges. To this end, we focus our attention to black holes with no acceleration horizon Podolsky:2002nk , so that there is no ambiguity as to which horizon temperature should be considered, or as to whether there is an equilibrium thermodynamics for the system. In addition, as we discuss, the holographic computation and interpretation are also unambiguous and straightforward. We also comment on the cases when the acceleration horizons appear and provide a novel interpretation of the boundary geometry.

An accelerating black hole in AdS can be described by the metric Podolsky:2002nk ; Hong:2003gx ; Griffiths:2005qp

(1) |

where

(2) | ||||

The potential shows clearly the black hole nature of the solution, as well as the effects of acceleration () and cosmological constant (). Note that we require to preserve the metric signature. The parameter encodes information about the conical deficits on the north and south poles that have tensions given by Appels:2017xoe

(3) |

The absence of an acceleration horizon yields the constraint , in turn constraining the parameter space to the white region bounded by the blue and red lines in figure 1. It is straightforward to show via a linear transformation Hong:2003gx on the coordinates that the latter bound is equivalent to the absence of black droplets Hubeny:2009kz .

As discussed in Appels:2017xoe ; Gregory:2017ogk , setting removes the black hole horizon, and leaves pure AdS spacetime in Rindler-type coordinates. Performing the coordinate transformation Podolsky:2002nk :

(4) |

recovers AdS in global coordinates:

(5) |

but with the notable feature that the time coordinate is not the expected AdS time, but is rescaled by a factor of . Conventionally, we choose the normalisation of our time coordinate so that it corresponds to the “time” of an asymptotic observer. While this is potentially a slightly slippery concept in AdS, taken together with the spherical asymptotic spatial coordinates, this scaling suggests that the correct time coordinate is not in fact , but rather , giving a rescaling of the time-coordinate in (1). If we now proceed with this metric, and compute the temperature associated with the black hole (also the temperature of the boundary field theory), then we obtain

(6) |

where .

It is worth pausing to reflect on this result. In past work Appels:2016uha ; Appels:2017xoe ; Gregory:2017ogk , the standard time coordinate appearing in the AdS C-metric was used to derive the temperature of the black hole horizon. This appeared to be a natural approach as the blackening factor of the metric was in its canonical form. However, as pointed out in Gibbons:2004ai , normalising the time and timelike Killing vector is key to obtaining the correct thermodynamics, although the method of obtaining this correct normalisation was less transparent. Here, having uncovered this suggestive result, we now proceed carefully with considering thermodynamics of the accelerating black hole. As usual, we will take the entropy to be one quarter of the horizon area:

(7) |

The remaining task is to correctly identify the black hole mass, often the biggest challenge in studying thermodynamics of black holes with non-trivial asymptotics. In what follows, we will provide two independent arguments, beginning with the conformal completion method Ashtekar:1999jx ; Das:2000cu . Although consistency of the thermodynamic relations is a common method for deriving thermodynamics (used for example in Astorino:2016ybm ), we do not consider this sufficient; hence we return to our theme of holography, computing the holographic stress tensor of the boundary theory, thereby confirming our result. As an ancillary argument, we finally check consistency with a computation of the free energy.

The first argument uses the Ashtekar–Das definition of conformal mass Ashtekar:1999jx ; Das:2000cu , which extracts the mass via conformal regularisation of the AdS C-metric near the boundary. The idea is to perform a conformal transformation on (1), , to remove the divergence near the boundary, then obtain a conserved charge by integrating the conserved current

(8) |

composed of the Weyl tensor of the conformal metric, , the normal to the boundary, , and a suitable Killing vector for the mass, . Even though the conformal completion is not unique, the charge thus obtained is independent of the choice of conformal completion. We pick , which provides a smooth conformal completion in the limit . The spacelike surface element tangent to is

(9) |

obtained by inserting into the metric and computing the relevant determinant. This yields

(10) |

for the mass, in agreement with the temperature (6), but in contrast to previous results Appels:2016uha ; Astorino:2016ybm . The absence of acceleration horizons ensures that vanishes in the limit only for and is positive otherwise.

It is now straightforward to verify the first law and Smarr Smarr:1972kt relation

(11) |

using (6), (7), and (10), provided

(12) | ||||

where is the thermodynamic pressure associated with the cosmological constant Kubiznak:2016qmn , and are the thermodynamic lengths introduced in Appels:2017xoe ; Gregory:2017ogk that are conjugate to the tensions. We have included the possibility that the tensions vary, as otherwise the system is constrained and identification of the correct parameters can be misleading.

We now turn to another method for deriving the thermodynamic mass, by computing the holographic stress tensor. This provides an alternate and completely independent method of computation, and will reveal the dual interpretation of this system. The idea here is to perform a Fefferman–Graham expansion of the metric FG , identifying the fall-off of sub-leading terms in the metric at the boundary. These are then used to compute the dual stress-energy tensor that can be integrated to give the mass of the system.

The action, including boundary counterterms Balasubramanian:1999re ; Mann:1999pc , is

(13) | ||||

where is the extrinsic curvature of the boundary metric, evaluated asymptotically in an appropriate coordinate system, defined presently. is the intrinsic metric on , and its Ricci curvature. Varying the action gives the energy momentum tensor:

(14) |

To compute these terms requires new coordinates near the boundary of AdS, typically parametrised by Fefferman–Graham coordinates, in which

(15) |

Although often one identifies a coordinate globally, due to the complexity of (1), we instead perform an asymptotic expansion for the coordinate transformation, writing

(16) |

The functions and are fixed by the required fall-off properties of (15), apart from , that we choose to write as

(17) |

in order to elucidate the conformal degree of freedom in the boundary metric, , with . Computing this boundary metric, , we find it sufficient to truncate the series (16) at and find:

(18) |

Note that the transformation (16) is valid in general only when , which is precisely the constraint that acceleration horizons are absent.

The expectation value of the energy momentum of the CFT can then be calculated, yielding

(19) |

where

(20) | ||||

We can re-express this in the language of the fluid/gravity correspondence (for a review and references see Rangamani:2009xk ) as

(21) |

where the 4-velocity and are defined as

(22) | ||||

with the Cotton tensor Mukhopadhyay:2013gja of the metric (18); boundary indices are raised and lowered with .

The fact that the boundary is non-conformally flat leads to the inclusion of non-hydrodynamic corrections to the perfect fluid, with

(23) |

The universal (in the sense of independent of black hole mass and acceleration) coefficient is

(24) |

using the standard dictionary to identify with the level and with the rank of the gauge groups of the ABJ(M) theory. All dissipative corrections enumerated in Rangamani:2009xk are seen to vanish, ensuring the uniqueness of the decomposition given in (21).

Integrating the energy density, measured with respect to a static geodesic observer yields

(25) |

for the mass, in agreement with (10), and independent of the conformal frame (the choice of ).

We see from (18) that the boundary metric does not satisfy Dirichlet boundary conditions. However for arbitrary variations of the parameters and we find that provided we set . Our analysis therefore points towards the possibility of generalizing the conditions under which the conformal and holographic methods coincide for the mass computation Hollands:2005wt ; Papadimitriou:2005ii .

Finally, let us return to the computation of the action (26). We find

(26) |

using the time coordinate . Some simple algebra then yields the expected result for the free energy, which we plot in figure 2.

Although similar in form, the behaviour of the free energy no longer indicates the presence of a standard Hawking–Page transition Hawking:1982dh . As the string tension is fixed for the curves in the plot, no transition to pure radiation (with zero tension) is possible. One may, however, speculate that a transition to a different type of spacetime (for example that of the expanding spherical wave with an attached semi-infinite string of given tension, similar to Podolsky:2004bk ) may still be possible—such an investigation, however, remains to be carried out.

We can also explore the isoperimetric ratio, or the ratio of volume to areal radius: (recall here). Using (7) and (12) we find , indicating it satisfies the standard reverse isoperimetric inequality Cvetic:2010jb , not adding to the notable exceptions Hennigar:2014cfa ; Hennigar:2015cja ; Brenna:2015pqa .

Our full and consistent description of the thermodynamics of an accelerating black hole reconciles discrepancies and conflicts that have appeared in previous investigations of this system Appels:2016uha ; Appels:2017xoe ; Astorino:2016ybm . For example, while a set of thermodynamic variables for charged accelerating black holes respecting the first law was obtained Appels:2016uha ; Appels:2017xoe ; Gregory:2017ogk the computations employed an incorrectly normalized Killing vector at infinity; furthermore the resultant free energy is not consistent with the standard Euclidean action calculation. Alternate expressions for mass and temperature have been posited Astorino:2016ybm , with the tension of one deficit held fixed to zero. The other tension, while allowed to vary, was not included in the first law, which was derived by assuming integrability of a scaling of mass and temperature. However no physical interpretation was given either for this scaling or for why the energy content of the tension was thermodynamically irrelevant. Furthermore, the vacuum accelerating black hole has an acceleration horizon, akin to a Rindler horizon, and the full structure of the spacetime is that of two accelerating black holes in two Rindler regions. Whether one should be considering a single thermodynamic mass and first law with an additional horizon and black hole, or whether, as suggested in Dutta:2005iy , this should be considered as a single system with a mass dipole is an open question.

We also found a decomposition of the dual stress energy tensor for the accelerating black hole in terms of a perfect fluid plus conformal tensors. We obtained a new “universal” coefficient, that is relevant for the fluid/gravity correspondence in non-conformally flat manifolds. It is natural to expect the existence of non-hydrodynamic corrections to the energy momentum tensor for an even-dimensional CFT due to the conformal anomalies. We have shown here that a similar picture arises in the odd-dimensional case by explicitly constructing the relevant non-hydrodynamic corrections necessary to provide a complete holographic description of the system, c.f. deFreitas:2014lia .

It would also be interesting to make a connection with the weak coupling calculation of stress tensors in the presence of conical deficits Dowker:1977zj . Future work will involve investigating accelerating black holes with rotation, scalar fields Anabalon:2009qt ; Anabalon:2012ta , and charge. The latter system will be a challenge due to the asymptotic structure of the gauge field.

Since our computation is independent of the conformal frame, we can compare to investigations of holographic C-metrics with an acceleration horizon. For example, by choosing , we recover the form of the boundary metric employed in Hubeny:2009kz , and our coordinate transformation (16) is now valid throughout . However, if the condition is violated, then a black droplet/black funnel is present, and we no longer have an equilibrium temperature for the system in general. The boundary geometry corresponds to a black hole in a spatially compact universe, and so there is no spatial asymptotic region as pointed out in Hubeny:2009kz . However, with the full conformal degree of freedom present in our expression, we can easily remedy this shortcoming by, for example, multiplying the above by , giving an asymptotic region at with the and radii being equal. If we multiply by then there are actually two asymptotic regions at and yields the geometry of a wormhole when there are no horizons at the boundary. The asymptotic geometry is supersymmetric and to our knowledge has been unnoticed so far in the literature.

## Acknowledgements

We thank Alex Buchel, Rob Myers, Jiří Podolský, Mukund Rangamani and Kostas Skenderis for valuable discussions. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, a Chilean FONDECYT [Grant No. 1181047 (AA), 1170279 (AA), 1161418 (AA), 3170035 (A.Ö.)], CONYCT-RCUK [Newton-Picarte Grants DPI20140053 (AA) and DPI20140115 (AA)], by the STFC [consolidated grant ST/P000371/1 (RG)]. MA and AO would like to thank Perimeter Institute and University of Waterloo for hospitality. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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