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 Schwarz schild and of Kerr-Schild types as special cases. The energy-momentum density T: (x) of the gravitational source and the gravitational energymomentum pseudo-tensor density I: have the expressions T: (x) = -Mc 2 8",08 o "8(3)(X) and I: = 0, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like 8(3)(x)/r3 and [8(3) (x)f, as well as other terms, which are singular at x = o. It is pointed out that the well-known expression RPU"'''({})Rpu",,,({}) = 48C 2 M2 /c 4 r 6 is not correct, if we define l/r 6 ~f lim._o 1/(r 2 + E2)3.