Hawking radiation from black holes in $$2+1$$ dimensions

Abstract

The paper develops a model to understand the effective quantum geometry of a black hole horizon and the emission of Hawking spectrum in

dimensions. Using the algebra of Hamiltonian charges on the horizon, we establish that one should view the black hole horizon as formed out of quantised lengths of elementary quanta of value

$$8\pi \ell _{P}\, n$$ 8 π ℓ P n

, where

$$n\in \mathbb {N}$$ n ∈ N

, and

$$\ell _{P}$$ ℓ P

is the Planck length. We determine the black hole entropy using this equidistant length spectrum in the microcanonical ensemble and show that its value is close to the Bekenstein-Hawking entropy. To evaluate the Hawking spectrum, we note that, to an observer near the black hole horizon, the entropy (or length of horizon cross-section) is related to the black hole energy. Hence, one may develop a formulation of length ensemble (similar to the area canonical ensemble of Krasnov) from which the black body spectrum may be obtained directly. This local observer perceives a Hawking spectrum whose temperature is modified by the Tolman factor.