A radiating Kerr black hole and Hawking radiation

Yu-Ching Chou

doi:10.1016/j.heliyon.2020.e03336

Abstract

This study proposes an axisymmetric generalization of the Vaidya metric, namely the Vaidya–Kerr metric, to describe a radiating rotating black hole, and presents its Hawking radiation temperature. This study is an improved version of our previous research via ellipsoid coordinate transformation, and the Einstein field equations are solved concisely and intuitively by an orthogonal ansatz. The results demonstrate that the energy–momentum tensor of the derived radiating Kerr metric satisfies the energy-conservation law and is classified as a Petrov type II fluid, whereas the stationary Kerr metric is a Petrov type IV vacuum. The inner and outer event-horizon radii, the ergosphere radii, as well as the angular velocity at the event horizon are solved, and then, surface gravity, entropy, and Hawking radiation are derived. We estimate the Hawking-radiation temperature of the black holes with the angular momentum and the same mass of Pluto and the sun, as well as the supermassive black hole in the core of the M87 galaxy to be 9.42K, 6.08×10−8K, and 8.78×10−18K, respectively. Only the value of the rotating Pluto-mass black hole is slightly greater than the 3K cosmic microwave background radiation and may be detected by high-resolution tools in the future.

Highlights

The black hole solution of the four-dimensional spacetime Einstein– Maxwell equations of classical general relativity has the following physical characteristics: mass (M), electric charge (Q), and angular momentum (J )
The static spherically symmetry solution with electric charge is the Reissner–Nördstrom metric [2, 3], and the axisymmetric generalization of the Schwarzschild metric with angular momentum is known as the Kerr metric [4]
The axisymmetric solution of the Reissner–Nördstrom metric has been generalized by incorporating angular momentum to the Kerr–Newman metric [5]

Summary

Introduction

The black hole solution of the four-dimensional spacetime Einstein– Maxwell equations of classical general relativity has the following physical characteristics: mass (M), electric charge (Q), and angular momentum (J ). The static spherically symmetric solution is a Schwarzschild metric with mass as its only physical characteristic [1]. The static spherically symmetry solution with electric charge is the Reissner–Nördstrom metric [2, 3], and the axisymmetric generalization of the Schwarzschild metric with angular momentum is known as the Kerr metric [4]. The axisymmetric solution of the Reissner–Nördstrom metric has been generalized by incorporating angular momentum to the Kerr–Newman metric [5]. The Vaidya solution can be applied to black holes to study Hawking radiation. Celestial bodies present in the nature, which require the Vaidya–Kerr solution, always have rotational angular momentum and radiation, including radiating rotating stars and black holes

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Conclusion