Double Layer in the Quadratic Gravity and the Least Action Principle
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uble Layer in the Quadratic Gravity and the Least Action Principle V. A. Berezina, *, V. Yu. Dokuchaeva, Yu. N. Eroshenkoa, and A. L. Smirnova aInstitute
for Nuclear Research, Russian Academy of Sciences, Moscow, 117312 Russia *e-mail: [email protected]
Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—The Israel equations for thin shells in General Relativity are derived directly from the least action principle. The method is elaborated for obtaining the equations for double layers in quadratic gravity from the least action principle. DOI: 10.1134/S1063779620040139
INTRODUCTION Any relativistic gravitation theory should be described by nonlinear equations, since the gravitational field has energy and consequently is a source itself. The solution of these equations is extremely challenging even in the case of vacuum. Therefore, singular distributions of the energy-momentum tensor of material (nongravitational) sources are of special interest. We mean primarily the distributions described by the Dirac δ function, i.e., thin shells. More specifically, we consider the action integral in the form of a sum of gravitational action S grav , and the actions for the matter fields, Sm ,
Stot = Sgrav + Sm .
(1)
According to the definition, the variation of the matter action as a metric tensor gμν (ds 2 = gμνdx μdx ν ) gives us the energy-momentum tensor Tμν : def
δSm = 1 Tμνδg μν − gd 4 x = 1 T μνδgμν − gd 4 x, (2) 2 2
∫
∫
where g is the determinant of tensor gμν . It is assumed that there is a singular surface Σ0 inside the integration domain, on which the energymomentum tensor is localized. We are interested only in this singular part. The three-dimensional hypersurface Σ0 divides the four-dimensional spatiotemporal integration domain into two parts, conventionally internal (“–”) and external (“+”). The transformation of coordinates in each of these domains can provide the way to attain the metric tensor continuity on Σ0 (this is the only thing that connects them). Note that except this continuity, the internal and external parts should be considered absolutely separate and unrelated manifolds. In particular, it is possible to
introduce the Gaussian normal coordinate system related with Σ0 : x μ = (n, x i ) , x i ∈ Σ0 , (3) ds 2 = edn2 + γ ij dx i dx j , where the coordinate n is directed along the external normal, and e = ±1 depending on whether Σ0 is a space-like or a time-like hypersurface (we use signature (+ − −−)), and the equation for Σ0 is just n = 0 . The Σ0 enclosure into the four-dimensional volume is described by the extrinsic curvature tensor K ij : def ∂γ K ij = − 1 ij . 2 ∂n Σ0
(4)
In these coordinates Tμν = Sμνδ(n) + … and
δSm = 1 (Snnδg 2 Σ0
∫
nn
+ 2Sni δg + Sij δg ) γ d x. ni
ij
3
(5)
Tensor Sμν is the energy-momentum surface tensor on the mass shell. Note that, although g nn = g nn = e, g ni = g ni = 0 , their variations are not necessarily equal to zero on Σ0 . We shall work with the Riemann geometry, i.e., the λ connectivity coefficients Γμν can
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