The quantum tunneling analysis of Hawking radiation from three-dimensional black holes by Ejaz, Gohar, Lin, Saifullah and Yau (Phys. Lett. B 726 (2013) 827) left two directions open. First, charged and rotating Banados-Teitelboim-Zanelli (BTZ) black holes emit charged scalars and fermions with a tunneling probability taking the form of a thermal Boltzmann factor with chemical potentials conjugate to the particles' charge and angular momentum; since BTZ geometries also arise as near-horizon factors of higher-dimensional black holes, whether and how these three-dimensional tunneling results map to the Hawking radiation of the higher-dimensional parent solutions remains unexplored. Second, the three-dimensional topological black holes asymptotic to Sol geometry analyzed in the same paper were remarked to possess further mathematical and physical properties that remain to be investigated.

Background

Hawking radiation is the thermal quantum emission of particles from the horizon of a classical black hole. In the tunneling picture, emission is treated as a quantum tunneling process across the horizon: the wave equation of the emitted particle is solved in a WKB (Wentzel-Kramers-Brillouin) approximation with complex path integration, giving a tunneling probability Γ∼exp⁡(−2 Im I/ℏ)\Gamma\sim\exp(-2\,\mathrm{Im}\,I/\hbar) from the imaginary part of the classical action; comparing with a thermal Boltzmann factor exp⁡(−βω)\exp(-\beta\omega) yields the Hawking temperature.

In 2+12+1 spacetime dimensions with a negative cosmological constant (anti-de Sitter space, AdS3_3), Einstein gravity admits the Banados-Teitelboim-Zanelli (BTZ) black hole, which, like its 3+13+1-dimensional counterparts, has inner and outer horizons and is characterized by mass MM, angular momentum JJ, and electric charge QQ; its charged version, due to Martinez, Teitelboim and Zanelli, is a three-dimensional analogue of the Reissner-Nordström black hole, with an electrostatic potential growing logarithmically (∼ln⁡r\sim\ln r) rather than as 1/r1/r, which produces nontrivial boundary terms in the gravitational action. Ejaz, Gohar, Lin, Saifullah and Yau (2013) computed the tunneling emission of charged scalar and fermionic particles from charged BTZ black holes, with and without rotation. The tunneling probability takes the form of a Boltzmann factor with chemical potentials conjugate to the particles' charge and angular momentum; for the rotating charged case it is Γ=exp⁡[−4πf′(r+)(ω−q Qln⁡r+l−j J2r+2)]\Gamma=\exp\left[-\frac{4\pi}{f'(r_+)}\left(\omega-q\,Q\ln\frac{r_+}{l}-j\,\frac{J}{2r_+^2}\right)\right], where r+r_+ is the outer horizon, f(r)f(r) the metric (blackening) function, and ll the AdS radius. The Hawking temperature is TH=f′(r+)/4πT_H=f'(r_+)/4\pi in all cases, the entropy is the horizon circumference S=4πr+S=4\pi r_+ (in units 8G=18G=1), and the temporal contribution to the imaginary part of the action keeps Γ≤1\Gamma\le 1, so emission probabilities never violate unitarity.

The same paper analyzed a three-dimensional topological black hole whose asymptotic geometry is (the Lorentzian version of) Sol geometry, ds2=e2udx2+du2+e−2udy2ds^2=e^{2u}dx^2+du^2+e^{-2u}dy^2 — one of the eight Thurston model geometries, the simply connected homogeneous Riemannian three-geometries in terms of which Thurston's geometrization conjecture classifies three-manifolds, realized as the R2\mathbb{R}^2 bundle over R\mathbb{R} with linearly expanding and contracting fiber directions. For this object the tunneling analysis gives a temperature growing as the square root of the mass and the same entropy law S=4πr+S=4\pi r_+, and the source remarks that the laws of black hole mechanics extend to these objects as well.

Two facts motivate the first open direction. Near-horizon (decoupling) limits of higher-dimensional near-extremal black holes — for example charged black holes in AdS5_5 uplifting to type IIB string theory, and five-dimensional black rings in the extremal-vanishing-horizon limit — combine the time direction, the radial direction, and a compact circle direction of the extra dimensions into an effective BTZ geometry, as established by Balasubramanian, de Boer, Jejjala and Simon and by Fareghbal, Gowdigere, Mosaffa and Sheikh-Jabbari for the AdS5_5/IIB parents, and by Sadeghian and Yavartanoo and by Ghodsi, Golchin and Sheikh-Jabbari for black rings. The three-dimensional tunneling results above and these near-horizon BTZ sectors have developed in separate literatures; whether the three-dimensional analysis, with its chemical-potential structure, captures the Hawking radiation of the higher-dimensional parents is the unexplored connection. The second open direction is that the Sol-asymptotic topological black holes were analyzed only to the extent of their tunneling temperature, entropy, and the observation that the laws of black hole mechanics extend to them; their further mathematical and physical properties were left unstudied.

Problem Statement

The problem, as posed in the closing discussion of the source paper, comprises two linked components with one shared objective — extending the tunneling and thermodynamic analysis of these three-dimensional black holes beyond the source paper's scope, outward to higher dimensions through the near-horizon connection and deeper into the unexplored properties of the Sol-asymptotic objects.

(a) The near-horizon BTZ connection. For higher-dimensional black holes whose near-horizon decoupling limits contain a BTZ factor (such as near-extremal charged AdS5_5 black holes and five-dimensional black rings), determine how the three-dimensional tunneling results — the emission probability as a Boltzmann factor with chemical potentials conjugate to charge and angular momentum, the temperature TH=f′(r+)/4πT_H=f'(r_+)/4\pi, and the unitarity bound Γ≤1\Gamma\le 1 — map to the Hawking radiation of the parent higher-dimensional solution. Concretely: analyze the tunneling emission in the three-dimensional near-horizon BTZ sector, identify how the effective BTZ mass, charge, and spin encode the parent's charges and angular momenta under the decoupling limit, and establish which of the parent's observables (Hawking temperature, chemical potentials, emission probabilities) are reproduced by the three-dimensional analysis and where the correspondence fails or requires correction.

(b) The properties of Sol-asymptotic topological black holes. Investigate the further mathematical and physical properties of the three-dimensional topological black holes asymptotic to Sol geometry (metric (7.2) of the source, with f(r)=r2/l2−Mf(r)=r^2/l^2-M) beyond the tunneling temperature and entropy already computed — their global geometry and horizon structure, the completion and consistency of their thermodynamics, their stability, and the well-posedness of the variational principle (boundary terms of the action) for the nonstandard Sol asymptotics.

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