KMI-IEEC Joint International Workshop“Inflation, Dark Energy and Modified Gravity in the PLANCK Era”

Black hole solutions
and entropy in bigravity
Taishi Katsuragawa, QG-lab. Nagoya Univ.
Based on Phys. Rev. D 87, 104032 (2013) arXiv: 1304.3181 TK and Shin’ichi Nojiri

In this talk
• We consider black hole solutions in bigravity for a minimalmodel.
• For the black hole solution, we evaluate the black hole entropy.
• In calculation for entropy, we use a recently proposed approachand we make it simple and useful to evaluate the entropy inbigravity.

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Outline• Introduction• Black hole solutions in bigravity• Black hole entropy in bigravity• Summary

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Outline• Introduction• Black hole solutions in bigravity• Black hole entropy in bigravity• Summary

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Introduction
What is bigravity?

Generalization of massive gravityLinear massive gravity
Massive spin-2

Fierz and Pauli (1939)

Vainshtein mechanism, Λ 3 theory etc.

Non-linear massive gravity de Rham, Gabadadze and Tolley (2011)
General covariance (dynamical reference metric)
Massive spin-2
Massless spin-2

Bigravity Hassan and Rosen(2012)

If we consider the massless spin-2 field as graviton,
bigravity describes massive spin-2 field coupled to gravity.
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Introduction
Five 𝛽𝑛 sThe action

Kinetic term of 𝑔・Mass : 1・Cosmologicalconstants : 2・Free para. : 2Kinetic term of 𝑓

Planck massInteraction term between 𝑔 and 𝑓

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

6Introduction
Bigravity describes massive spin-2 field coupled to gravity.

Considering the black hole solution in bigravity, we obtain theblack hole with massive spin-2 field.
cf. Charged black hole in Einstein-Maxwell system.

• Massive spin-2 field can be “new hair” of black hole.
• How the massive spin-2 field near the horizon affects theblack hole entropy?
Motivation
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

7Outline• Introduction• Black hole solutions in bigravity• Black hole entropy in bigravity• Summary

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Black hole solutions in bigravity
In this work, we consider the minimal model of bigravity

withEquations of motion

It’s hard to solve!
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

9Black hole solutions in bigravityWe impose the condition.

On this condition, two e.o.m.s form

Cosmological constants

For the consistency, both equations should be identicalto each other.
We obtain asymptotically flat solutions with 𝑓𝜇𝜈 = 𝑔𝜇𝜈 .
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

10Outline• Introduction• Black hole solutions in bigravity• Black hole entropy in bigravity• Summary

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Black hole entropy in bigravity
We use a recently proposed approach.

Majhi and Padmanabhan (2011)

The basic idea :Noether current from the surface term

Brown andHenneaux (1986)

Virasoro algebra of the conserved charges

Cardy (1986)
Carlip (1999)

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Using the Cardy formula

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Black hole entropy in bigravity
The surface term

𝐾 is Gibbons-Hawking term

N.B. We can ignore the interaction term, because it doesn’t
include derivatives and doesn’t contribute to surface term.

It is simple and useful for calculation.
We use the Schwarzschild solutions.

Diffeomorphism
𝑇 is arbitrary function.

N.B. In this approach, we choose the diffeomorphism
to keep the horizon structure invariant.
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Black hole entropy in bigravity
We expand 𝑇.
𝑎 obeys theAnd we choose 𝑇𝑚 so that the resulting 𝜉𝑚algebra isomorphic to Diff 𝑆 1 .
{ , } is the Lie bracket.

Such a 𝑇𝑚 is given by
𝛼 is a constant, 𝑝 is an integer and 𝑔(𝜌) is a function that is regular at the horizon

Substituting 𝑇𝑚 into the Noether charges, we obtain theFourier mode of the charges
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Black hole entropy in bigravity
We obtain

Zero-mode eigenvalue

𝐴 is area and 𝜅 is the surface gravity

Virasoro algebra
Central charge

From the Cardy formula
We obtain a double portion of Bekenstein-Hawking entropy.
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Outline• Introduction• Black hole solutions in bigravity• Black hole entropy in bigravity• Summary

2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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Summary• We have shown that the bigravity for the minimal model has
asymptotically flat solutions with 𝑓𝜇𝜈 = 𝑔𝜇𝜈 .
• We have evaluated the black hole entropy for
the Schwarzschild solution.
• We find that the obtained entropy is twice as much as theBekenstein-Hawking entropy in the Einstein gravity.
• It is interesting that our approach may be generalized to thecase of the other models.
2013/8/3

KMI-IEEC Joint International Workshop Taishi Katsuragawa, QG-lab. Nagoya Univ.

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