Near horizon symmetries, emergence of Goldstone modes and thermality

Abstract

For a long time, it is believed that black hole horizon is thermal and quantum mechanical in nature. The microscopic origin of this thermality is the main question behind our present investigation, which reveals possible importance of near horizon symmetry. It is this symmetry which is assumed to be spontaneously broken by the background spacetime, generates the associated Goldstone modes. In this paper, we construct a suitable classical action for those Goldstone modes and show that all the momentum modes experience nearly the same inverted harmonic potential, leading to an instability. Thanks to the recent conjectures on the chaos and thermal quantum system, particularly in the context of an inverted harmonic oscillator system. Going into the quantum regime, the system of large number of Goldstone modes with the aforementioned instability is shown to be inherently thermal. Interestingly, the temperature of the system also turns out to be proportional to that of the well-known horizon temperature. Therefore, we hope our present study can illuminate an intimate connection between the horizon symmetries and the associated Goldstone modes as a possible mechanism of the microscopic origin of the horizon thermality.

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Notes

  1. 1.

    The choice of the inverted harmonic oscillator stems from the fact that the particle motion is unstable under this potential and hence, at the classical level, any small perturbation can lead induction of chaos in the motion (for example, see [68,69,70,71]).

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Correspondence to Bibhas Ranjan Majhi.

Appendices

Appendix A: Near horizon analysis of Schwarzschild black hole

As mentioned in the main text, in this section we will argue that same dynamical equations and solution for the Goldstone modes can be obtained starting from the near horizon metric of the Schwarzschild black hole.

Now, we can easily check that the action constructed from the near horizon metric will contain three types of terms. The ones which are independent and linear in F, can be traced back from their origin which can be transmitted to the fact that the near horizon geometry of the Schwarzschild black hole is Rindler times a sphere, and it does not satisfy the background Einstein’s equation. We, therefore, ignore those terms as they can also be made total derivative. Non-trivial dynamics of the Goldstone modes are attributed to second-order term in F in the action, and it can be easily checked that those terms are exactly the same as in (39) up to a total derivative. As a result with a proper prescription, full spacetime geometry as well as near horizon geometry of the Schwarzschild background are giving rise to the same Goldstone mode dynamics.

Appendix B: Surface Hamiltonian and heat content

In the main text, we have constructed the GHY boundary term in the action formulation which did not contribute in the dynamics of the Goldstone mode. However, an important analysis is left to be discussed there. It is well-known that the boundary term of the Einstein–Hilbert action in gravitational theory leads to surface Hamiltonian which is directly related to the heat content of the Horizon (detail discussion is given in ([75, 76])). The expression of the surface Hamiltonian comes out to be the product of temperature and entropy of the horizon. Keeping this in mind, we can write surface Hamiltonian corresponding to the GHY boundary term (18):

$$\begin{aligned} H_{sur} = -\frac{\partial S_2}{\partial v}. \end{aligned}$$
(55)

Now, substituting the solution (23) in the expression (18) and integrating the boundary term , the Hamiltonian (55) comes out:

$$\begin{aligned} H_{sur}=\frac{\bar{A}}{8 \pi G} \Big [ \alpha + f_3(\alpha ) \mathrm{e}^{(f_2(\alpha ) v)}\Big ], \end{aligned}$$
(56)

\( \bar{A}\) denotes the transverse area of the Rindler horizon. In the second part function \(f_3(\alpha )\) comes from the first and higher order time derivative of F in the boundary term of the action. Whereas \(f_2 (\alpha )\) denotes all the square, cubic and higher order terms in the expression. Now, near horizon (where \(v \rightarrow -\infty \)) the second terms in the above expression vanish. Hence, the result comes out as:

$$\begin{aligned} H_{sur} = \frac{1}{8 \pi G} \bar{A} \alpha = TS, \end{aligned}$$
(57)

where \(S = \bar{A} /4 G \) and \(T = \alpha /2\pi \) are the Horizon entropy and temperature, respectively. The result clearly indicates that irrespective of Goldstone modes \(F_{m,n}\), the GHY boundary term in the action is related to the heat content of the horizon. Similar conclusion can be drawn for Schwarzschild black hole horizon also.

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Maitra, M., Maity, D. & Majhi, B.R. Near horizon symmetries, emergence of Goldstone modes and thermality. Eur. Phys. J. Plus 135, 483 (2020). https://doi.org/10.1140/epjp/s13360-020-00451-3

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