Unified classical and quantum treatment of uncharged black holes with complete energy-momentum tensor

In this paper, we solve the Einstein equation for an uncharged black hole by considering the energy-momentum tensor and the energy-momentum pseudotensor to first order in the gravitational constant of 3[Formula: see text]1 dimensions. The ground state of the black hole field is a spherical wave in 3+1 dimensions. Furthermore, by solving the equation, we see that the black hole vibrates and rotates spatially, since the spatial solutions of the equation are Bessel functions of order equal to an integer plus one-half. This could be an explanation of why a black hole emits radiation known as Hawking radiation. Classical and quantum treatments of the equations yield identical results under Gupta–Bleuler quantization formalism.