Abstract
We discuss the “quantum deformed Schwarzschild spacetime”, as originally introduced by Kazakov and Solodukhin in 1993, and investigate the precise sense in which it does and does not satisfy the desiderata for being a “regular black hole”. We shall carefully distinguish (i) regularity of the metric components, (ii) regularity of the Christoffel components, and (iii) regularity of the curvature. We shall then embed the Kazakov–Solodukhin spacetime in a more general framework where these notions are clearly and cleanly separated. Finally, we analyze aspects of the classical physics of these “quantum deformed Schwarzschild spacetimes”. We shall discuss the surface gravity, the classical energy conditions, null and timelike geodesics, and the appropriate variant of the Regge–Wheeler equation.
Highlights
The unification of general relativity and quantum mechanics is of the utmost importance in reconciling many open problems in theoretical physics today
By this they just mean “regular” in the sense of the metric components being finite for all r ∈ [ a, ∞). This is not the meaning of the word “regular” that is usually adopted in the GR community. We find it useful to carefully distinguish (i) regularity of the metric components, (ii) regularity of the Christoffel components, and (iii) regularity of the curvature
While event horizons are mathematically easy to work with, one should bear in mind that they are impractical for observational astronomers to deal with—any physical observer limited to working in a finite region of space+time can at best detect apparent horizons or trapping horizons [66]; see Reference [67]
Summary
The unification of general relativity and quantum mechanics is of the utmost importance in reconciling many open problems in theoretical physics today. The fact that the “center” has been “smeared out” to finite r was originally hoped to be a step towards rendering the spacetime regular This metric was originally derived via an action principle which has its roots in the. In Kazakov and Solodukhin’s original work [1], it is asserted that the metric (2) is “regular at r = a” By this they just mean “regular” in the sense of the metric components (in this specific coordinate chart) being finite for all r ∈ [ a, ∞). We feel that there are a number of technical issues requiring clarification in the derivation presented in Reference [1], so instead, we shall use the results of Kazakov and Solodukhin’s work as inspiration and motivation for the analysis of our general class of metrics. Schwarzschild black holes, and so potentially of interest to observational astronomers [65]
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