Vaidya spacetime in the diagonal coordinates
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Vaidya Spacetime in the Diagonal Coordinates V. A. Berezina*, V. I. Dokuchaeva,b**, and Yu. N. Eroshenkoa*** a
Institute for Nuclear Research, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 7a, Moscow, 117312 Russia b National Research Nuclear University “MEPhI,” Kashirskoe sh. 31, Moscow, 115409 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received August 25, 2016
Abstract—We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g00 = 0 or g00 = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false firewalls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence. DOI: 10.1134/S1063776117020108
1. INTRODUCTION The Vaidya metric describes the spacetime produced by a spherically symmetric radial radiation flow. This metric has the form [1–3]
2m(z)⎤ 2 ds 2 = ⎡1 − dz + 2dzdr ⎣⎢ r ⎦⎥
(1)
−r (d θ + sin θ d ϕ ). In particular, the Vaidya metric describes a nonstationary accreting or emitting black hole. In this metric, m(z) is an arbitrary mass function that depends (in the case of accretion) on coordinate z = –v, where v is the advanced light coordinate or (in the case of emission of radiation) on coordinate z = u, where u is the retarded light coordinate. For m(z) = m0 = const, metric (1) describes a Schwarzschild black hole of mass m = m0. Here and below, we are using units of measurement in which c = 1 for the velocity of light and G = 1 for the gravitational constant. Vaidya metric (1), which is one of a few known exact solutions in the general theory of relativity, has a large number of astrophysical and theoretical applica2
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