Semianalytical approach for the Vaidya metric in double-null coordinates

Abstract

We reexamine here a problem considered in detail before by Waugh and Lake: the solution of spherically symmetric Einstein's equations with a radial flow of unpolarized radiation (the Vaidya metric) in double-null coordinates. This problem is known to be not analytically solvable, the only known explicit solutions correspond to the constant mass case (Schwarzschild solution in KruskalSzekeres form) and the linear and exponential mass functions originally discovered by Waugh and Lake. We present here a semi-analytical approach that can be used to discuss some qualitative and quantitative aspects of the Vaidya metric in double-null coordinates for generic mass functions. We present also a new analytical solution corresponding to (1/v)-mass function and discuss some physical examples.

N-dimensional Vaidya metric with a cosmological constant in double-null coordinates

“…Here, we extend the approach proposed in [26] to the n-dimensional (n > 2) case and to allow the inclusion of a cosmological constant Λ. Some new exact solutions are also presented.…”

“…Such approach consists in a qualitative study of the null-geodesics, allowing the description of light-cones and revealing many features of the underlying causal structure. It can be used also for more quantitative analyses, indeed, it has already enhanced considerably the accuracy of the quasinormal modes analysis of 4-dimensional varying mass black holes [14,15], and it can be also applied to the study of gravitational collapse [26]. We notice that another method to construct conformal diagrams based on a systematic study of the null-geodesics was also recently proposed in [27].…”

“…The resulting equations, however, have revealed to be not analytical solvable for generic mass functions. Waugh and Lake's work was recently revisited in [26], where a semianalytical approach allowing for generic mass functions was proposed. Such approach consists in a qualitative study of the null-geodesics, allowing the description of light-cones and revealing many features of the underlying causal structure.…”

Radiative spacetimes approaching the Vaidya metric

“…In fact, there are two possibly different conformal diagrams depending on the value of µ: for µ > 1/16 there is a white hole singularity at r = 0, for µ ≤ 1/16 there is also a naked singularity, see e.g. [9,13,16,36] for more details. At u = 0 all of the mass m(u) is radiated away, and we can attach Minkowski space (de Sitter space when Λ > 0, anti-de Sitter when Λ < 0; the presence of the cosmological constant would change the character of conformal infinity I which would become spacelike or timelike, respectively) in the region u > 0 along the hypersurface u = 0.…”

“…Various sandwiches and shells of null matter can thus be constructed that are bounded either by flat (m = 0) or Schwarzschild-like (m = const = 0) vacuum regions. Due to this property such solutions have been extensively used as models of spherically symmetric gravitational collapse of a star, as an exterior solution describing objects consisting of heat-conducting matter, as an interesting toy model for investigation of singularities and their possible removal by quantum effects, for studies of various formulations of the cosmic censorship conjecture on both classical and quantum level, process of black-hole evaporation, and for other purposes (see, e.g., [7][8][9][10][11][12][13][14][15][16] for more details and related references).…”