An exact time-dependent interior Schwarzschild solution

Abstract

We present a time-dependent uniform-density interior Schwarzschild solution, an exact solution to the Einstein field equations. Our solution describes the collapse (or the time-reversed expansion) of an object from an infinite radius to an intermediate radius of 9/8 of the Schwarzschild radius, at which time a curvature singularity appears at the origin, and then continues beyond the singularity to a gravastar with radius equal to the Schwarzschild radius. Usage PACS numbers Structure I. INTRODUCTIONNearly 100 years after its original discovery, the constant density interior Schwarzschild solution [1] was analyzed in more detail and shown to behave as a gravastar in the limit that the radius approaches the Schwarzschild radius R S = 2GM [2]. If the radius reaches 9/8R S the pressure at the center diverges and the convention was this implied a static solution no longer existed [1,3]. However, the static interior Schwarzschild solution may be maintained, without modification, if one accepts a region of negative pressure [4]. While the interior Schwarzschild solution strictly speaking does not avoid singularities (it is singular where the pressure diverges [5]) it is a simple and mathematically valid solution with potential for high compactness and negative pressures that can be interesting to study in its own right. For example, slowly rotating Schwarzschild stars in the compact gravastar limit behave almost exactly as non black hole extended Kerr sources [6]. Also, if one allows for a Dirac delta function in the transverse stress at the radius of the pressure divergence, the singularity has a well-defined contribution to the Komar integral [2].In this paper, we show that if one allows for a time-dependent radius and for anisotropic stress, the interior Schwarzschild solution generalizes into a new exact solution to the Einstein field equations. Its line element has the same form as the static interior Schwarzschild * † Electronic address:

Where is Love? Tidal deformability in the black hole compactness limit

“…It is interesting to note that the analysis of axial perturbations did not show echoes in the wave form [32], as proposed for generic exotic compact objects [33]. A time-dependent model that allows for the evolution from a low compactness star to a gravastar in the black hole compactness limit was presented in [34].…”

“…where p t is the tangential pressure, see [21,22,34]. This pressure anisotropy should be taken into account in the calculation of the tidal deformability.…”

Where is Love? Tidal deformability in the black hole compactness limit

“…It is interesting to note that the analysis of axial perturbations did not show echoes in the waveform [35], as proposed for generic ECOs [36]. A time-dependent model that allows for the evolution from a low compactness star to a gravastar in the black hole compactness limit was presented in [37].…”

“…where p t is the tangential pressure. This term results in a physical surface tension associated with a positive integrable transverse pressure contribution to the Komar mass, see [23,25,37]. This is the key result that allows the ultracompact interior Schwarzschild solution to be interpreted in physical terms.…”

Detectable universes inside regular black holes

“…Black hole solutions where the singularity is avoided are called regular black holes [3,4,5,6,7,8,9,10] and may or may not involve a de Sitter core. The, so called, dark energy stars or gravastars [11,12,13,14,15,16,17,18,19] generally do not predict the presence of an event horizon.…”