Distributional Energy-Momentum Densities of Schwarzschild Space-Time

TL;DR

This paper investigates distributional energy-momentum densities and curvature invariants in Schwarzschild space-time, revealing singularities and correcting a common expression for the curvature squared invariant.

Contribution

It provides new distributional expressions for energy-momentum densities and curvature invariants in Schwarzschild space-time, including a correction to a known curvature invariant formula.

Findings

Energy-momentum density is localized at the source as a delta function.

The curvature invariant expression $R^{ ho\sigma\mu\nu} R_{\rho\sigma\mu\nu}$ is corrected for distributional definitions.

Singular terms like $\delta^{(3)}(x)/r^3$ appear in curvature squares.

Abstract

For Schwarzschild space-time, distributional expressions of energy-momentum densities and of scalar concomitants of the curvature tensors are examined for a class of coordinate systems which includes those of the Schwarzschild and of Kerr-Schild types as special cases. The energy-momentum density $\tilde T_\mu^{\nu}(x)$ of the gravitational source and the gravitational energy-momentum pseudo-tensor density $\tilde t_\mu^{\nu}$ have the expressions $\tilde T_\mu^{\nu}(x) =-Mc^2\delta_\mu^0\delta_0^{\nu} \delta^{(3)}x)$ and $\tilde t_\mu^{\nu}=0$, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like $\delta^{(3)}(x)/r^3$ and $[\delta^{(3)}(x)}]^2$, as well as other terms, which are singular at $x=0$. It is pointed out that the well-known expression $R^{\rho\sigma\mu\nu}({}) R_{\rho\sigma\mu\nu}({})$ $=48G^{2}M^{2}/c^{4}r^{6}$ is not correct, if we define $1/r^6 = \lim_{\epsilon\to 0}1/(r^2+\epsilon^2)^3$.}