Schwarzschild geometry counterpart in semiclassical gravity

Abstract

We investigate the effects of vacuum polarization on vacuum static sphericallysymmetric spacetimes. We start from the Polyakov approximation to the renormalized stress-energy tensor (RSET) of a minimally coupled massless scalar field. This RSET is not regular at r = 0, so we define a regularized version of the Polyakov RSET. Using this Regularized RSET, and under the previous symmetry assumptions, we find all the solutions to the semiclassical field equations in vacuum. The resulting counterpart to the Schwarzschild classical geometry substitutes the presence of an event horizon by a wormhole throat that connects an external asymptotically flat region with an internal asymptotic region possessing a naked singularity: there are no semiclassical vacuum solutions with well-defined Cauchy surfaces. We also show that the Regularized Polyakov RSET allows for wormhole geometries of arbitrarily small throat radius. This analysis paves the way to future investigations of proper stellar configurations with an internal non-vacuum region.

Vacuum Semiclassical Gravity Does Not Leave Space for Safe Singularities

“…with a 0 , b 0 being dimensionless constants. The structure of wormhole solutions with a mass much greater than the Planck mass is qualitatively the same independently of the particular choice of regulator parameter α; in fact, even different approximations for the RSET [23,25,26] lead to equivalent solutions. Only Planck-sized solutions differ significantly from one another depending on these choices, but these are discarded as they fall outside the range of validity of the semiclassical approximation itself.…”

“…If one tries to generate a solution in semiclassical gravity similar to the eternal blackhole configuration, one encounters a well-known obstacle: the RSET in the Boulware vacuum state, the only genuine vacuum consistent with staticity and asymptotic flatness, diverges at the Schwarzschild horizon. Recently, some of the present authors performed a detailed analysis of the form of the eternal semiclassically self-consistent counterpart of the Schwarzschild solution [25]. Elaborating on previous works [23,26], and with the choice of a regularised function…”

“…Nonetheless, the singular character of the Polyakov RSET can be remedied by deforming the function f (r) into one regular at r = 0, at the cost of breaking the conservation of the RSET. In a static situation, where C(u, v) can be expressed as a function of r only, conservation is recovered simply by adding non-zero angular components to the RSET, as discussed in [25]. There, a simple regularising term was added in f (r) which removes the divergence at the origin while preserving the standard Polyakov approximation at large radii.…”

Semiclassical constant-density spheres in a regularized Polyakov approximation

“…In a previous work [35] we presented a regularization scheme for the Polyakov RSET. Following [42,43] we constructed a Regularized Polyakov RSET devoid of the r = 0 singularity, thus allowing for a self-consistent treatment of the field equations in vacuum.…”

“…Before turning to more refined analyses, we decided to exhaust this framework to clearly see its scope and limitations. In a previous paper we analyzed the form of the self-consistent vacuum solutions of semiclassical gravity [35]. In the current paper, we add a classical perfect fluid of constant energy density to the semiclassical vacuum.…”