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We work with the notion of apparent/trapping horizons for spherically
symmetric, dynamical spacetimes: these are quasi-locally defined, simply based
on the behaviour of congruence of light rays. We show that the sign of the
dynamical Hayward-Kodama surface gravity is dictated by the inner/outer nature
of the horizon. Using the tunneling method to c...
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Since Stephen Hawking discovered that black holes emit thermal radiation,
black holes have become the theoretical laboratories for testing our ideas on
quantum gravity. This dissertation is devoted to the study of singularities,
the formation of black holes by gravitational collapse and the global structure
of spacetime. All our investigations are...
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... where is the Hubble parameter. The temperature of the apparent horizon [59,60] is proportional to surface gravity, , and is given by the expression ...
The profound connection between the law of emergence and the thermodynamic laws provides a new thermodynamic perspective on the accelerated expansion of the universe. In this paper, we explore this connection in the context of Tsallis entropy in both equilibrium and non-equilibrium perspectives. From an equilibrium perspective, we derive the law of emergence by considering Tsallis entropy as the horizon entropy from the unified first law of thermodynamics and the Clausius relation and is found to be consistent with the earlier proposals \cite{SHEYKHI2018118,Chen2022}. However, when one uses Tsallis entropy as the horizon entropy, the thermal evolution of the system becomes non-equilibrium \cite{10.1093/mnras/stab2671}, which results in the generation of additional entropy, and the first law of thermodynamics will get modified accordingly compared to its usual equilibrium version. We obtained the law of emergence from this modified form of the first law of thermodynamics. The law of emergence thus obtained have a simple form, which explains the emergence of the universe, in terms of the rate of change of the areal volume of the horizon, instead of the effective volume as in the equilibrium case. We have further shown that the law of emergence also satisfies the condition of the maximization of entropy; thus, the entropy of the universe evolves to a bounded value in the asymptotic future.
... In contrast with the case of Schwarzschild spacetime, there is in general no timelike or causal Killing vector field near H that could be used to define and test black hole thermodynamical quantities. Hayward [29] proposed to use the Kodama vector field as a replacement (see also [30] for a review). The Kodama vector field [39] can be defined in terms of , and r as ...
We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. While these spacetimes possess in general no timelike Killing vector field, they admit a Kodama vector field which in some ways provides a replacement. The Kodama vector field allows the definition of a surface gravity of the outer trapping horizon. Spherically symmetric spacelike cross sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time parameter of the Kodama flow, where the inverse temperature β=2π/κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta = 2\pi /\kappa $$\end{document} is related to the surface gravity κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} of the horizon cross section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunnelling probability that can be obtained in the scaling limit between in- and outgoing Fourier modes with respect to the time parameter of the Kodama flow shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. Moreover, the scaling limit 2-point function allows it to define a scaling limit theory, a quantum field theory on the ingoing lightlike congruence emanating from a horizon cross section. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling limit theory is correlation-free with respect to separation along the horizon cross section; therefore, their relative entropies behave proportional to the cross-sectional area. We thus obtain a proportionality of the relative entropy of coherent states of the scaling limit theory and the area of the horizon cross section with respect to which the scaling limit is defined. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.
... In contrast to the case of Schwarzschild spacetime, there is in general no timelike or causal Killing vector field near H that could be used to define and test black hole thermodynamical quantities. Hayward [Ha97] proposed to use the Kodama vector field as a replacement (see also [He15] for a review). The Kodama vector field [Ko80] can be defined in terms of ℓ, ℓ and r as ...
We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. These spacetimes possess in general no timelike Killing vector field, but admit a Kodama vector field which provides a replacement. Spherically symmetric spacelike cross-sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time-parameter of the Kodama flow, where the inverse temperature is related to the surface gravity of the horizon cross-section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunneling probability in the scaling limit between in- and outgoing Fourier modes with respect to the the Kodama time shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory have relative entropies behaving proportional to the cross-sectional horizon area. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking. (Extended abstract in the article.)
... Furthermore, as the asymptotic region is not flat, we have to specify how we normalize the horizontal Killing vector field. For this purpose we will follow the work of [19] and [20,21], which provides a method for computing the surface gravity and temperature of trapping horizons. While the definition of an event horizon requires the knowledge of the global causal structure of a spacetime, trapping horizons are defined quasi-locally by the existence of marginally trapped surfaces. ...
... Following [20,21], trapping horizons and their Kodama-Hayward surface gravity subdivide into four cases, as follows. One chooses a local frame containing two future-directed null vectors, N µ ± which are 'outgoing' (+) and 'intgoing' (−). ...
... the four types of horizons are determined by calculating their expansions θ ± and their Lie derivative, evaluated on the horizon. Non-trapping regions in spacetime are those where θ + θ − < 0. The convention taken in [20,21] is that θ − < 0 and θ + > 0 in all non-trapping regions so that the outgoing congruence N µ + is diverging (or expanding) while the ingoing congruence N µ − is converging (or contracting). Trapping regions are those where θ + θ − > 0, so that either both congruences expand, or both congruences contract. ...
We present a modified implementation of the Euclidean action formalism suitable for studying the thermodynamics of a class of cosmological solutions containing Killing horizons. To obtain a real metric of definite signature, we perform a `triple Wick-rotation' by analytically continuing all space-like directions. The resulting Euclidean geometry is used to calculate the Euclidean on-shell action, which defines a thermodynamic potential. We show that for the vacuum de Sitter solution, planar solutions of Einstein-Maxwell theory and a previously found class of cosmological solutions of N = 2 supergravity, this thermodynamic potential can be used to define an internal energy which obeys the first law of thermodynamics. Our approach is complementary to, but consistent with the isolated horizon formalism. For planar Einstein-Maxwell solutions, we find dual solutions in Einstein-anti-Maxwell theory where the sign of the Maxwell term is reversed. These solutions are planar black holes, rather than cosmological solutions, but give rise, upon a standard Wick-rotation to the same Euclidean action and thermodynamic relations.
... The inner character of the horizon for an expanding Universe yields a negative surface gravity, while its past nature imposes a minus sign in the relation between T and κ H . This combining sign effect is explained in details in [13]. ...
... The sign of the temperature, however, is a result of both the sign of κ and the sign entering in the definition of T , as in Eq.(40). For further details on this combining sign effect, see [13]. ...
... However, the sign of the temperature depends on both the inner/outer and the past/future character of the horizon. This is clearly established in [13]. The statement is valid for black holes, white holes, expanding cosmologies as well as contracting ones 6 We have shown in Section 1 that the Hayward-Kodama surface gravity is perfectly well-defined in a dynamical context, using the Kodama field (the generalization of the static Killing field). ...
In the context of thermodynamics applied to our cosmological apparent
horizon, we explicit in greater details our previous work which established the
Friedmann Equations from projection of Hayward's Unified First Law. In
particular, we show that the dynamical Hayward-Kodama surface gravity is
perfectly well-defined and is suitable for this derivation. We then relate this
surface gravity to a physical notion of temperature, and show this has
constant, positive sign for any kind of past-inner trapping horizons. Hopefully
this will clarify the choice of temperature in a dynamical
Friedmann-Lema\^itre-Roberston-Walker spacetime.
We propose a trick for calculating the surface gravity of the Killing horizon, especially for cases of rotating black holes. By choosing nice slices, the surface gravity and angular velocities can be directly read from relevant components of the inverse metric. We give several cases to show how to apply the trick step by step.
In the context of Tsallis entropy, we explore the connection between the law of emergence and the thermodynamic laws from a more accurate non-equilibrium perspective. Here, the equilibrium Clausius relation does not conform to the standard energy-momentum conservation. Therefore, an effective gravitational coupling is introduced to rewrite the field equation similar to general relativity, and the corresponding generalized continuity equation is obtained. As a result, thermodynamic laws were modified with the non-equilibrium energy dissipation and entropy production terms, using which we derive the law of emergence. The investigation of the law of emergence and the entropy maximization principle with Tsallis entropy in the non-equilibrium perspective shows that both result in the same constraints as obtained in other gravity theories and the equilibrium context of Tsallis entropy, except for an additional constraint on the Tsallis parameter as a result of extra entropy production. Consequently, the thermodynamic interpretation of the expansion of the universe stays valid even with quantum corrections to the horizon entropy since the correction terms in Tsallis entropy can be treated as the quantum corrections to Bekenstein-Hawking entropy.
