Embedding with Vaidya geometry

A V Nikolaev,S D Maharaj

doi:10.1140/epjc/s10052-020-8231-0

Abstract

The Vaidya metric is important in describing the exterior spacetime of a radiating star and for describing astrophysical processes. In this paper we study embedding properties of the generalized Vaidya metric. We had obtained embedding conditions, for embedding into 5-dimensional Euclidean space, by two different methods and solved them in general. As a result we found the form of the mass function which generates a subclass of the generalized Vaidya metric. Our result is purely geometrical and may be applied to any theory of gravity. When we apply Einstein’s equations we find that the embedding generates an equation of state relating the null string density to the null string pressure. The energy conditions lead to particular metrics including the anti/de Sitter spacetimes.

Highlights

The local embedding of a four-dimensional pseudo-Riemannian spacetime can be isometrically embedded in a higher dimensional Euclidean space
Examples of explicit embeddings and solutions of the Einstein field equations which are embeddable are given in the works of Rosen [2], Stephani [3] and Collinson [4]
Two different approaches of embedding the generalized Vaidya metric into the 5-dimensional Euclidean geometry are considered in Sect

Summary

Introduction

The local embedding of a four-dimensional pseudo-Riemannian spacetime can be isometrically embedded in a higher dimensional Euclidean space. A general algorithm to describe embeddable anisotropic compact stellar models was discovered by Murad [30] These astrophysical applications apply to static spherically symmetric fields. An interesting nonstatic embeddable stellar interior was produced for the first time by Naidu et al [31] which satisfies the Karmarkar condition In this nonstatic model the star is radiating and the exterior atmosphere is given by the Vaidya spacetime. We note that Naidu et al [31], generated a specific model of a relativistic radiating star for a particular spacetime metric matching to the generalized Vaidya spacetime which satisfies the Karmarkar embedding condition. Two different approaches of embedding the generalized Vaidya metric into the 5-dimensional Euclidean geometry are considered in Sect.

Vaidya geometry

Embedding condition

Gauss–Coddazzi–Ricci equations

The mass function

Application to general relativity

The weak and strong energy conditions

The dominant energy condition:

Radiating stars

B2 A B 2A 2B