Quantum gravitational corrections to a star metric and the black hole limit

Abstract

In this paper we consider the full set of quantum gravitational corrections to a star metric to second order in curvature. As we use an effective field theoretical approach, these corrections apply to any model of quantum gravity that is based on general coordinate invariance. We then discuss the black hole limit and identify an interesting phenomenon which could shed some light on the nature of astrophysical black holes: while star metrics receive corrections at second order in curvature, vacuum solutions such as black hole metrics do not. What happens to these corrections when a star collapses? 1 x.calmet@sussex.ac.uk

Bootstrapped Newtonian quantum gravity

“…We note in passing that this contribution is consistently of the same order2 p as the corrections found in Ref [17]. for the metric generated by a star.…”

Long-Range Quantum Gravity

“…We define the Schwarzschild radius as R s = 2GM/c 2 . This means that the classical criticality bound (2) is saturated, namely the black hole is considered as a sort of critical star (see also [65,66]). Putting together Equation (1) and the definition of the Schwarzschild radius, we find that the number of degrees of freedom in the black hole cavity scales as the area, i.e., it scales holographically in the sense previously described:…”

Quantum corrected equations of motion in the interior and exterior Schwarzschild spacetimes

“…In earlier work [1] we derived the leading quantum corrections to the interior and exterior region of the spacetime containing a constant and uniform density star, which are classically described by the well-known interior and vacuum Schwarzschild solutions. These calculations were done in the framework of the effective field theory for quantum gravity [2][3][4][5][6][7][8].…”

“…For example, integrating out Kaluza-Klein interactions would give rise to contact interactions. Furthermore, it was shown in [1] that corrections due to such contact interactions are subleading in the case of a star, assuming higher order curvature terms are not unnaturally large. The leading order corrections to the metric describing the spacetime around a star only depend on the nonlocal physics which is calculable from first principles and in a model independent way, without a detailed knowledge of the ultra-violet complete theory of quantum gravity.…”

“…In this paper we will use the results from Ref. [1] to derive the leading quantum corrections to the geodesics and the scalar waves in such a quantum corrected spacetime. A complication in these calculations may arise, since the metric corrections and curvature invariants, such as the Ricci scalar, diverge when the surface of the star is approached.…”