Quantum-mechanical corrections to the Schwarzschild black-hole metric

Abstract

Motivated by quantum mechanical corrections to the Newtonian potential, which can be translated into an -correction to the g00 component of the Schwarzschild metric, we construct a quantum mechanically corrected metric assuming −g00 = g rr . We show how the Bekenstein black hole entropy S receives its logarithmic contribution provided the quantum mechanical corrections to the metric are negative. In this case the standard horizon at the Schwarzschild radius rS increases by small terms proportional to and a remnant of the order of Planck mass emerges. We contrast these results with a positive correction to the metric which, apart from a corrected Schwarzschild horizon, leads to a new purely quantum mechanical horizon. PACS numbers: Valid PACS appear here I. INTRODUCTIONThe full theory of Quantum Gravity is one of the last unsolved challenges in fundamental science and is still eluding us. Nevertheless, some effects of Quantum Theory do enter the context of the theory of Quantum Gravity and can be handled in a rigorous way without the knowledge of the full fledged theory. Such is the case of the Hawking radiation [1] and/or the Unruh effect [2]. Apart from these paradigms there are some other interesting quantum effects related to gravity like the absence of stable orbits of fermions around a black hole [3], the quantum correction to the Bekenstein entropy S of black holes and of other black objects using different approaches to Quantum Gravity [4-33] and the quantum correction to the Newtonian potential or metrics [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] (for some applications of the new corrections see [53][54][55]). Indeed, the results regarding the corrected Newtonian potential Φ spread over a period of the last forty five years starting with the early seventies whereas the corrections to S are a relative new undertaking. Whichever model one uses it turns out that S receives corrections proportional to the logarithm of the black hole area and, in some models, also proportional to the square root of this quantity. This is also the finding of our approach starting from a different context. We will make a connection between the -corrected metric and the quantum mechanical corrections to the entropy. We will construct our quantum mechanically corrected metric by demanding (i) that it reproduces the corrected Newtonian limit, (ii) that it reproduces the standard result for the entropy of black hole including, in addition, the corrections which are similar to results established elsewhere and (iii) that it passes some consistency checks regarding the geodesic motion of a test particle moving in this metric. Point (i) which has to do with weak gravity, can be easily accommodated by invoking the classical connection between the g 00 metric component and the Newtonian potential. The second point requires the determination of the horizons and probes into the strong regime of gravity. In principle, we cannot infer the strong gravity effects from results zeroing around the weak regime as...

Epicyclic oscillations of charged particles in stationary solutions immersed in a magnetic field with application to the Kerr–Newman black hole

“…We fix the values of M = 1, Q = 1/10, a = 1/100 and Q = 1/500 as in (47), we solve numerically Eqs. (45) and (46), and evaluate the epicyclic frequencies ω 2 r (27) and ω 2 θ (22). The first thing we prove, as shown in figures 4 to 6, is the existence of a vertically stable, but radially unstable, prograde circular motion in the vicinity of r isco (for r h < r < r isco where r h = 1.994937 is the event horizon).…”

“…These solutions have been determined upon solving numerically Eqs. (45), (46), and (32), with ω 2 r being given by (27). Radially stable prograde circular orbits exit for the whole range of B (46), while for retrograde orbits it seems that there is some critical value B c beyond which no radially stable retrograde circular orbit exists (in the vicinity of r isco of the prograde circular orbits); however, as we shall see in Sec.…”

“…Numerical solutions to Eqs. (45), (46), and (32), with ω 2 r being given by (27), have been obtained and plots of r isco and ω isco (= u ϕ /u t ) in terms of a are depicted in Fig. 1 (the case of prograde circular orbits, u ϕ > 0, of a charged particle) and in Fig.…”

“…This constitutes an analytic solution to Eqs. (45), (46), and (32), with e = 0 and ω 2 r being given by (27). Equation (49) is known in the scientific literature [61,62]…”

“…The plots show the effect of the magnetic field B encoded in the dimensionless ratio B(44), which is proportional to eBM. These solutions have been determined upon solving numerically Eqs (45),(46),. and (32), with ω 2 r being given by(27).…”