Negative mode of Schwarzschild black hole from the thermodynamic instability
2011 1/21(Fri.) NTU, Taiwan Takayuki Hirayama
@ Maswaka Institute （ Kyoto Sangyo U. ）
Introduction
• Black hole can be treated as a thermodynamic system.
• Schwarzschild Black hole has a negative specific heat.
• Sch. bh is known to be classically stable.
• These two facts seem contradictory. How to resolve?
• Sch. bh is known to be quantum mechanically unstable.
Introduction
• We may conjecture a equivalence between the
stability as a thermodynamic system and the stability against small perturbations.
• Reall gave a general argument under certain
assumptions that a thermodynamic instability implies a instability against a small perturbation.
• However he assumes three assumptions.
o (I) regularity of perturbation at the horizon
o (II) normalization at the infinity
o (III) traceless condition
Introduction
• In this talk, we explicitly show (I), (II), (III) can be satisfied.
• For simplicity, we treat Schwarzschild black hole. (But our argument should be able to apply for general
black holes.)
Stability of BH
• Euclidean path integral for gravity is
where the path integral is taken over the manifolds
(M, g) which is asymptotically S^1xR^3 and S^1 has the periodicity ß=1/T and with the boundary at r=rb.
• If we expand the action around BH, we obtain
Denote the eigenvalue as lambda, classical instability
quantum instability
Stability of BH
(A perturbation must be small and the norm is finite)
Reall’s argument
• Construct a negative mode in the following way.
• Negative heat capacity. We just need to increase the black hole mass. That is and then
• However this introduces a conical singularity at (notice that the periodicity is fixed with ß=1/T)
• Since goes to infinity at , goes to infinity at
• Thus Reall uses the fact that the value of action does not depend on U(r).
If V(r) and R(r) are black hole sol.,
Reall’s argument
Reall’s argument
• Thus can be taken freely and is taken so that the conical singularity does not appear at the horizon.
• Still problematic. goes to infinity at (i). Also it is not clear if can be taken safely (ii). The traceless condition is not satisfied at the horizon (iii).
• Assuming (i), (ii) and (iii) are OK, we can compute the value of action and get
• We prepare a new radial coordinate y such that the horizon is always y=0 and the boundary (r=rb) is y=1.
• We prepare, for example, a new radial coordinate and the bh metric becomes
Improvement I
Improvement I
• In this coordinate system, we take arbitrary but chosen to satisfy the boundary condition at the horizon (y=0), and
• Then around y=0,
• However the norm is
• We should use a different radial coordinate.
Improvement I
• The previous radial coordinate is too simple.
• Stretch around r=rb and do not stretch around r=rb.
Improvement II
• Then the smallness at the horizon is OK and the norm is also finite.
• The traceless condition (iii) still remains. We add a gauge transformation.
• is free and does not change the value of action, and then using the traceless condition can be satisfied without breaking the condition (i) and (ii).
Improvement II
• Finally our satisfies (i), (ii) and (iii). We then compute the value of action. (In fact my improvement does not change the value of action.)
• Our gives a negative mode. Thus the thermodynamic instability implies the existence of negative mode.
Discussion
• We improved Reall's argument and the perturbation
satisfies the smallness, normalizability and the traceless conditions.
• Finding a good radial coordinate is important. We believe we can find a good coordinate for other black holes as well.
• How about the reverse? The existence of negative mode implies the thermodynamic instability?
In fact, for many many black holes, the reverse is correct.
Discussion
• But counter examples are known where a scalar field shows instability. Scalar fields do not posses a conserved charge.
Another counter example is a unique one. Sch. bh in 5 dim.
Einstein-Gauss-Bonnet theory.
It must be important to find when the reverse is correct and when it is not correct.
