Noncommutative Black Holes, the Final Appeal to Quantum Gravity: A Review

Abstract

We present the state of the art regarding the relation between the physics of Quantum Black Holes and Noncommutative Geometry. We start with a review of models proposed in the literature for describing deformations of General Relativity in the presence of noncommutativity, seen as an effective theory of Quantum Gravity. We study the resulting metrics, proposed to replace or at least to improve the conventional black hole solutions of Einstein's equation. In particular, we analyze noncommutative-inspired solutions obtained in terms of quasi-classical noncommutative coordinates: indeed because of their surprising new features, these solutions enable us to circumvent long standing problems with Quantum Field Theory in Curved Space and to cure the singular behavior of gravity at the centers of black holes. As a consequence, for the first time, we get a complete description of what we may call the black hole SCRAM, the shut down of the emission of thermal radiation from the black hole: in place of the conventional scenario of runaway evaporation in the Planck phase, we find a zero temperature final state, a stable black hole remnant, whose size and mass are determined uniquely in terms of the noncommutative parameter θ. This result turns out to be of vital importance for the physics of the forthcoming experiments at the LHC, where mini black hole production is foreseen in extreme energy hadron collisions. Because of this, we devote the final part of this review to higher dimensional solutions and their phenomenological implications for TeV Gravity.

Quantum Harmonic Black Holes

“…The marginally binding condition (1.5), that is E 10 ≃ 0, then leads to the scaling laws 8) in perfect qualitative agreement with Eq. (1.7).…”

“…[7,8], where it was proven that such densities satisfy the Einstein field equations with a "de Sitter vacuum" equation of state, ρ = −p, where p is the pressure. Curiously, BECs can display this particular equation of state [9].…”

Mini‐review on mini‐black holes from the mini‐Big Bang

“…If we consider spacetime geometries admitting horizon extremization and remnant formation in the Schwarzschild phase, one finds that the minimum energy for BH formation is M min BH ≡ M remn. ∼ 10 PeV for d = 5 and M F = 1 TeV (Rizzo 2006;Nicolini 2008;Gingrich 2010). This is also supported by a recently proposed ghost free, singularity www.an-journal.org free higher derivative theory of gravity (Biswas et al 2012).…”

Phase Transitions of Regular Schwarzschild-anti-deSitter Black Holes

“…The regular Schwarzschild anti-deSitter (AdS) metric is a static, spherically symmetric solution of the Einstein's equations with negative cosmological constant Λ = −3/b 2 and a Gaussian matter source [1,2,3,4]. To obtain this metric we replace the vacuum with a Gaussian distribution having variance equivalent to the parameter √ θ ρ (r) ≡ M (4πθ ) 3/2 e −r 2 /4θ .…”

“…This type of matter distribution emulates non-commutativity of space-time through the parameter θ that corresponds to the area of the elementary quantum cell, accounting for a natural ultraviolet spacetime cut-off (see Ref. [2] and references therein). The resulting energy momentum tensor describes an anisotropic fluid, whose components, fixed by ∇ µ T µν = 0 and the condition g 00 = −g −1 rr , read…”