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 ${\tilde{T}}_{\mu }^{\nu }(x)$ of the gravitational source and the gravitational energy-momentum pseudo-tensor density ${\tilde{t}}_{\mu }^{\nu }$ have the expressions ${\tilde{T}}_{\mu }^{\nu }(x)=-M{c}^2{\delta }_{\mu }^0{\delta }_0^{\nu }{\delta }^{(3)}x)$ 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 $[δ^{(3)}(x)}]^2$, as well as other terms, which are singular at $x=0$. It is pointed out that the well-known expression $R^{\rho \sigma \mu \nu }()R_{\rho \sigma \mu \nu }()$ $=48G^2{M}^2/c^4{r}^6$ is not correct, if we define $1/r^6={\lim }_{\epsilon \to 0}1/(r^2+{\epsilon }^2{)}^3$.}

21 pages, LaTeX, uses amssymb.sty. To appear in Prog. Theor. Phys. 98 (1997)