The results discussed in this thesis are based on the works in [166–169]. In this section, we have summarized the research guidelines of the thesis as follows,
1.3.1 Chapter 2:
To understand the underlying degrees of freedom, near horizon symmetry analysis of a black hole has gained significant interest in the recent past. In this chapter, we generalize that analysis first by taking into account a generic null surface carrying U(1) electromagnetic charge. With the appropriate boundary conditions near the surface under study, we identify the symmetry algebra among the subset of diffeomorphism and gauge generators which preserve the metric of the null surface and the form of the gauge field configuration. With the knowledge of those symmetries, we further derive the algebra among the associated charges considering general Lanczos-Lovelock gravity theory and also non-linearU(1) gauge theory. Importantly while computing the charges, not only we consider general theory, but also used off-shell formalism which is believed to play crucial role in understanding quantum gravity. Both the nonextremal and extremal cases are addressed here.
1.3.2 Chapter 3:
For a long time, it is believed that black hole horizon are thermal and quantum mechanical in nature. The microscopic origin of this thermality is the main ques- tion behind our present investigation in this chapter, which reveals the 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 chapter, 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 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 a 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. There- fore, we hope that our present study in this chapter can illuminate an intimate
1.3. Chapter-wise description of the thesis connection between the horizon symmetries and the associated Goldstone modes
as a possible mechanism of the microscopic origin of the horizon thermality.
1.3.3 Chapter 4:
In the previous chapter, we have discussed how the near horizon Bondi-Metzner- Sachs (BMS) like symmetry is spontaneously broken by the black hole background itself and hence gives rise to Goldstone mode. The associated Goldstone mode for the near horizon BMS-like symmetry of a Schwarzschild black hole was found to behave like inverted harmonic oscillators, which has been further shown to lead to thermodynamic temperature in the semi-classical regime. Here we investigate the generalization of these previous findings for the Kerr black hole. The analysis is being performed for two different situations. Firstly, we analyze Goldstone mode dynamics considering slowly rotating Kerr. In another case, the problem is solved in the frame of zero angular momentum observer (ZAMO) with an arbitrary value of rotation. In both analyses, the effective semi-classical temperature of Goldstone modes turns out to be proportional to that of Hawking temperature.
Due to such similarity and generality, we feel that these Goldstone modes may play an important role in understanding the underlying microscopic description of horizon thermalization.
1.3.4 Chapter 5:
Recently symmetries of gravity and gauge fields in the asymptotic regions of spacetime have been shown to play a vital role in their low energy scattering phenomena. Further, for the black hole spacetime, near horizon symmetry has been observed to play a possible role in understanding the underlying degrees of freedom for thermodynamic behaviour of horizon. Following the similar idea, in this chapter, we analyzed the symmetry and associated algebra near a time- like surface that is situated at any arbitrary radial position and is embedded in black hole spacetime. Here we consider both Schwarzschild and Kerr black hole spacetimes. The families of hypersurfaces with constant radial coordinate (out- side the horizon) in these spacetimes are timelike in nature and divide the space into two distinct regions. The symmetry algebra turned out to be reminiscent of Bondi-Metzner-Sach (BMS) symmetries found in the asymptotic null boundaries.
1.3.5 Chapter 6:
Here we have presented the conclusion of the thesis and also discussed some interesting scope for future work.
From the next chapter onwards, we have the detail analysis of the thesis. Each chapter of the thesis contains several appendices that are added at the end of the respective chapter. Here we will consider the signature of the Lorentzian metric to be(−,+,+,+).
Symmetries near a generic charged null
surface and associated algebra: An off-shell
analysis 2
2.1 Introduction
In the introduction chapter of the present thesis, we have widely discussed that for a generic diffeomorphism invariant gravity theory, the Noether current and charge are very important in understanding the thermodynamic properties of black holes [28,54,55,61–63,97,104,105]. In this regard, one of the significant results is the commutator algebra among the charges associated with the asymptotic symmetries of spacetime under study. Those generally lead to Virasoro algebra with a central charge [56]. This central charge is found to be intimately connected with the entropy of black holes through well known Cardy formula [58].
The asymptotic symmetries near the null infinities of asymptotically flat space- time and the horizon lead to an infinite-dimensional BMS group, which is a semidirect product of usual Poincare symmetry and the infinite-dimensional su- pertranslation symmetry transformation. Later this idea has been extended to different situations; among them, one crucial extension includes gauge fields in the exploration of boundary symmetries [66, 129–132]. For instance, the elab- orate symmetry structure of three-dimensional Einstein-Maxwell systems with non-trivial asymptotics at null infinity has been explored in [131] which leads to Virasoro-Kac-Moody algebra, which is an extension ofBMS3algebra of pure
gravitational case. In [132] the analysis for gravity in four spacetime dimensions at null infinity has been extended to include Yang-Mills fields. Most recently, in [170] the unified treatment of asymptotic symmetries for the Einstein-Maxwell system has been discussed for Kerr-Newman (A)dS black hole horizon. Therefore asymptotic symmetry analysis for Einstein-Maxwell theory was extended and explored near a static or stationary horizon (see [91,171,172] for different cases), which are solutions of Einstein’s equations of motion. The general strategy is to choose a subset of diffeomorphism such that under those transformations, the solutions of the field equation must remain invariant near the null boundaries.
So far, the boundary diffeomorphism symmetries have been explored at first near null infinity for asymptotically flat spacetime, later at the black hole horizon, which acts as another null boundary of spacetime. However, one can have a generic null surface that will serve as a horizon for a class of observers in any spacetime. It has been observed that not only black hole horizon has a thermo- dynamics interpretation, but also any generic null surface in gravity theory has this property [108]. The idea stems from the equivalence principle - locally, an accelerated frame known as the Rindler frame can mimic gravity. Hence, it can be a good candidate for exploring various properties of gravity. Therefore, an accelerated observer in flat spacetime background is equivalent to a static observer in curved spacetime. This stimulates us to think the gravity as an “emergent phenomenon” [111]. Nevertheless, from the above discussions, one would tend to believe that understanding the behavior of a generic null surface not only can provide the desired results of on-shell properties of the theory under study but also can shed light on the off-shell behavior which naturally appears in quantum theory.