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Register a new userFrom Planck area to graph theory: Topologically distinct black hole microstates
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Authors
Aharon Davidson
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We postulate a Planck scale horizon unit area, with no bits of information locally attached to it, connected but otherwise of free form, and let $n$ such geometric units compactly tile the black hole horizon. Associated with each topologically distinct tiling configuration is then a simple, connected, undirected, unlabeled, planar, chordal graph. The asymptotic enumeration of the corresponding integer sequence gives rise to the Bekenstein-Hawking area entropy formula, automatically accompanied by a proper logarithmic term, and fixes the size of the horizon unit area, thereby constituting a global realization of Wheelers it from bit phrase. Invoking Polyas theorem, an exact number theoretical entropy spectrum is offered for the 2+1 dimensional quantum black hole.
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It was argued in a number of papers that the gravitational potential calculated by using the modified QFT that follows from the Planck-length deformed uncertainty relation implies the existence of black-hole remnants of the order of the Planck-mass. Usually this sort of QFTs are endowed with two specific features, the modified dispersion relation, which is universal, and the concept of minimum length, which, however, is not universal. While the emergence of the minimum-length most readily leads to the idea of the black hole remnants, here we examine the behaviour of the potential that follows from the Planck-length deformed QFT in absence of the minimum length and show that it might also lead to the formation of the Planck mass black holes in some particular cases. The calculations are made for higher-dimensional case as well. Such black hole remnants might be considered as a possible candidates for the dark-matter.
In this article we compute the black hole entropy by finding a classical central charge of the Virasoro algebra of a Liouville theory using the Cardy formula. This is done by performing a dimensional reduction of the Einstein Hilbert action with the ansatz of spherical symmetry and writing the metric in conformally flat form. We obtain two coupled field equations. Using the near horizon approximation the field equation for the conformal factor decouples. The one concerning the conformal factor is a Liouville equation, it posses the symmetry induced by a Virasoro algebra. We argue that it describes the microstates of the black hole, namely the generators of this symmetry do not change the thermodynamical properties of the black hole.
The black hole area theorem implies that when two black holes merge, the area of the final black hole should be greater than the sum of the areas of the two original black holes. We examine how this prediction can be tested with gravitational-wave observations of binary black holes. By separately fitting the early inspiral and final ringdown stages, we calculate the posterior distributions for the masses and spins of the two initial and the final black holes. This yields posterior distributions for the change in the area and thus a statistical test of the validity of the area increase law. We illustrate this method with a GW150914-like binary black hole waveform calculated using numerical relativity, and detector sensitivities representative of both the first observing run and the design configuration of Advanced LIGO. We obtain a $sim74.6%$ probability that the simulated signal is consistent with the area theorem with current sensitivity, improving to $sim99.9%$ when Advanced LIGO reaches design sensitivity. An important ingredient in our test is a method of estimating when the post-merger signal is well-fit by a damped sinusoid ringdown waveform.
The Renyi and Tsallis entropies are discussed as possible alternatives to the Bekenstein-Hawking area-law entropy. It is pointed out how replacing the entropy notion, but not the Hawking temperature and the thermodynamical energy may render the whole black hole thermodynamics inconsistent. The possibility to relate the Renyi and Tsallis entropies with the quantum gravity corrected Bekenstein-Hawking entropy is discussed.
The no-hair theorem can be tested in the strong gravity regime by using the top-bottom approach and the bottom-top approach. The non-Kerr spacetime of the later approach is an ideal framework to do the tests in the region very close to the black holes. In this work, we propose a non-Kerr black hole metric (and its charged extension) that is accelerating as well. These new objects are studied for their basic properties and thermodynamics.
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