1. Thermodynamics of accelerating black holes
David Kubizňák
(Perimeter Institute)
21st International Conference on General Relativity and Gravitation
Colombia University, New York, USA
July 10-15, 2016

2. Spoiler
Thermodynamics of accelerating black holes can be consistently formulated provided one considers “physically reasonable variations” in the first law.
Based on:
Michael Appels, Ruth Gregory, DK, Thermodynamics of Accelerating Black Holes, ArXiv:
Michael Appels, Ruth Gregory, DK, in preparation.

3. Black hole thermodynamics
First law of black hole thermodynamics:
Smarr-Gibbs-Duhem relation:
Works for black holes of various asymptotic properties, horizon topologies, charges, and rotation parameters. With a remarkable exception: C-metric

4. Introducing the C-metric: part 1
Exact stationary, axisymmetric solution of Einstein equations (with EM field and cosmological constant)
Levi-Civita (1917), Ehlers & Kundt (1963), Kinnersley & Walker (1970),…
O. Dias and J. Lemos, Phys.Rev. D67 (2003)
Describes a pair of accelerated black holes that are pushed away by a strut in between them (represented by a conical singularity)

5. Introducing the C-metric: part 2
Exact radiative spacetime (Bicak, Krtous, Podolsky, Pravda, Pravdova,…)
Strut can be replaced by two cosmic strings stretching to infinity (represented by a conical deficit and positive tension)
Used for:
No known generalization to higher dimensions
Not established thermodynamics
strut (m<0)
string
string (m>0)
Black ring in 5d (“Wick rotated C-metric”)
AdS/CFT: black funnels and droplets
Black hole nucleation: instability of dS space
Counter example to “no hair theorem”

6. Simpler setup: AdS C-metric with small acceleration
(Podolsky 02)
Where:
Small acceleration: single black hole, no acceleration horizon (“static black hole at fixed radius in AdS”)
Conformal factor:
(determines conformal infinity)
Simplified thermodynamics!

7. Choice of a cosmic string configuration
determined from the behavior of function g at the poles
regularity demands ( ):
North pole:
South pole:
Regular North:
The South pole: conical deficit
Corresponds to a cosmic string with tension N S

8. III) Thermodynamics Standard thermodynamic quantities are:
(method of conformal completion)
(Bekenstein)
(Gauss)
(Wick)
(electrostatic potential on the horizon)

9. Imposing standard Smarr formula:
we recover:
(In some sense this was already known and in fact in the AF case the Smar formula was used to identify the mass M of the black hole).
What about the first law?
Key idea: system = black hole + cosmic string
To isolate the black hole and consider the associated thermodynamic processes we need to “fix” the string!

10. “Isolated black hole”: string configuration is fixed
(String cannot instantaneously change its tension)
2 constraints:
“more massive (charged) object accelerates more slowly”
Standard first law holds:
(Note that due to constraints this is only “cohomogeneity-1” relation)

11. Gibbs free energy (Q=0): superficially analogous to Hawking-Page transition
Small black holes with A>>l
No phase transition (cannot create a half-infinite cosmic string)

12. Adding rotation Standard first law holds: 2 constraints:
(Which is now “cohomogeneity-3” relation. Also an acceleration horizon may be present.)

13. Summary
C-metric is one of the more unusual and surprising spacetimes of classical general relativity that has many potential applications and interesting properties.
We showed that when considering “physically reasonable thermodynamic variations” (isolate the black hole), standard thermodynamic first law remains valid (although of smaller cohomogeneity).
One can in principle include variations of cosmic string tensions (restore the full cohomogeneity) but the physical meaning of the thermodynamically conjugate quantities is not completely clear yet.
Even the C-metric is subject to consistent thermodynamics!