Sources of Gravitational Fields

Abstract

In this chapter we shall first find a general expression of the acceleration of gravity due to a mass distribution. Then we shall deduce the solution of Einstein’s field equations inside an incompressible star—the internal Schwarzschild solution . Furthermore we shall present Israel’s formalism for describing singular mass shells in the general theory of relativity, and apply this first to a shell consisting of dust particles, and then to find a source of the conformally flat, spherically symmetric Levi-Civita—Bertotti—Robinson metric , and finally to the Kerr spacetime . Lastly we shall introduce a river model of space .

Exercises

• 11.1 The Schwarzschild–de Sitter metric

ShowSchwarzschild–de Sitter metric that the solution of Einstein’s field equations with a cosmological constant in a static, spherically symmetric space is

$$ {\text{d}}s^{2} = - \left( {1 - \frac{{R_{S} }}{r} + \frac{{r^{2} }}{{R_{H}^{2} }}} \right)c^{2} {\text{d}}t^{2} + \frac{{dr^{2} }}{{1 - \frac{{R_{S} }}{r} + \frac{{r^{2} }}{{R_{H}^{2} }}}} + r^{2} {\text{d}}\Omega ^{2} , $$

(11.172)

where $R_{S} = 2GM/c^{2}$ is the Schwarzschild radius of the central mass, and $R_{H} = \sqrt {3/\Lambda }$ is the de-Sitter horizon radius which is the horizon radius in the case that there is no central mass, $R_{S} = 0$.

• 11.2 A spherical domain wall described by the Israel formalism

Consider a static, spherically symmetric domain wall in empty space with mass density $\sigma$ and radius R. Show that the mass M of the Schwarzschild spacetime outside the wall is

$$ M = \left( {1 - 2\pi \sigma R} \right)4\pi \sigma R^{2} . $$

(11.173)