Distributional Energy-Momentum Densities of Schwarzschild Space-Time

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 $\tilde T_μ^ν(x)$ of the gravitational source and the gravitational energy-momentum pseudo-tensor density $\tilde t_μ^ν$ have the expressions $\tilde T_μ^ν(x) =-Mc^2δ_μ^0δ_0^ν δ^{(3)}x)$ and $\tilde t_μ^ν=0$, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like $δ^{(3)}(x)/r^3$ and $[δ^{(3)}(x)}]^2$, as well as other terms, which are singular at $x=0$. It is pointed out that the well-known expression $R^{ρσμν}({}) R_{ρσμν}({})$ $=48G^{2}M^{2}/c^{4}r^{6}$ is not correct, if we define $1/r^6 = \lim_{ε\to 0}1/(r^2+ε^2)^3$.}