Distributional Energy-Momentum Densities of Schwarzschild Space-Time

For Schwarzschild space-time, distributional expressions of energy-momentumdensities and of scalar concomitants of the curvature tensors are examined fora class of coordinate systems which includes those of the Schwarzschild and ofKerr-Schild types as special cases. The energy-momentum density $\tildeT_\mu^{\nu}(x)$ of the gravitational source and the gravitationalenergy-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 squaresfor this class of coordinate systems, there are terms like$\delta^{(3)}(x)/r^3$ and $[\delta^{(3)}(x)}]^2$, as well as other terms, whichare 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 $1/r^6 = \lim_{\epsilon\to0}1/(r^2+\epsilon^2)^3$.}Comment: 21 pages, LaTeX, uses amssymb.sty. To appear in Prog. Theor. Phys. 98 (1997