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 $$2+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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 $$8\pi \ell _{P}\, n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>P</mml:mi> </mml:msub> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> , where $$n\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , and $$\ell _{P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>P</mml:mi> </mml:msub> </mml:math> 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.