Spherically symmetric double layers in Weyl+Einstein gravity

Abstract

The role of exact solutions in gravitational theories is impossible to overestimate. But the intrinsically nonlinear character of gravitational equations makes solving them a very difficult and problematic task. This is why the investigations of hypersurfaces where the matter energy-momentum tensor undergoes some discontinuities are so important. The physically interesting discontinuities are jumps (they can be viewed as an idealization of the shock waves) and thin shells (i. e., δ-function distributions, describing some idealized matter sources including the potential barriers between two different phases during cosmological phase transitions). In this paper we undertook the thorough investigation of the matching conditions on the singular hypersurfaces in Weyl+Einstein gravity. Unlike in the General Relativity, where the singular hypersurface may contain, at most, the Dirac δ-function both in the matter energy-momentum tensor in the right hand side of the Einstein equations and in the second derivatives of the metric tensor in their left hand side, in the so called quadratic gravity (the gravitational action integral, except the Einstein-Hilbert and cosmological terms, has all possible quadratic combinations of the Riemann curvature scalar). there appear the possibility of the double layer, i. e., the derivative of the δ − f unction. But, this derivative is absent in the energy-momentum tensor (no mass-dipole analogous to the charge-dipoles in the classical electrodynamics), so the double layer is a purely geometrical phenomenon ant it may be treated as the purely gravitational shock wave. The mathematical formalism for the double layers was elaborated layers was elaborated by J. M. M Senovilla for the double layers was elaborated for the generic quadratic gravity. Our choice of the Weyl+Einstein gravity is motivated by thee fact that the latter differs from the generic case in some aspects December 21, 2018 1:35 WSPC/INSTRUCTION FILE Berezin2018IJMPD 2 Victor Berezin, Vyacheslav Dokuchaev and Yury Eroshenkoand requires separate consideration. Moreover, we confined ourselves by the spherically symmetry, because in such a case the theory becomes, essentially, two-dimensional. The three-dimensional hypersurface reduces to the world-line, and it becomes much easier to understand every step in the calculations and interpretations of the results. The main results are the following. We derived the matching conditions for the spherically symmetric singular hypersurface (in our case it is equivalent to the world line) in the Weyl+Einstein gravity. It was found, that the residual extrinsic curvature tensor of this surface (i. e., having only one, (00))-component after separating the spherical angular part of the metric tensor), indeed, must be continuous on the singular hypersurface. The result is the same as that found by Senovilla, it is dictated by the very possibility to have the double layer, but the jump in the normal derivative of the radius may be not zero. (and this is different from Senovilla's). It was found ...

Quadratic Gravity, Double Layers and Non-Conservative Energy-Momentum Tensor

“…This means, that 377 α 2 = −α 1 and α 3 = (1/3)α 1 according to relations α 2 + 4α 1 = 2α 1 and α 2 + 4α 3 = −(2/3)α 1 . The 378 choice stemmed from the fact that we know all spherically symmetric solutions in this theory [22,23]. 379 The equations for the double layer (only those that are needed for the determination of the singular 380 hypersurface Σ 0 , which now becomes, actually, a world line) are the following (see (63) and (64)):…”

“…with γ ij = (1/r 2 )g ij , and we can now forget about the radius, provided S nn and S ni mean now, 387 respectively, (1/r 6 )S nn and (1/r 6 )S ni (for details see [22,23]).…”

“…The gravitational field may cause the parametric resonance inducing the process of the matter 21 creation. A.D.Sakharov even suggested [1] that the gravitational field does not exist as the fundamental 22 one, but it is simply the tensions of the vacuum fluctuations of all other quantum fields.…”

“…The relevant calculations are 194 very lengthy and cumbersome, so we will present here just main steps and will extensively make use 195 of our experience in obtaining the Israel equations. For more details see [22,23].…”

“…Actually, B i j i j are not completely arbitrary, they depend on our choice of the solutions in the bulk and should be found when solving the junction equations. Now we are ready to write down the equations for the double layer in Quadratic Gravity [22,23]:…”