ar X iv : g r - qc / 9 70 70 29 v 1 1 4 Ju l 1 99 7 Distributional Energy - Momentum Densities of Schwarzschild Space - Time Toshiharu Kawai

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 KerrSchild types as special cases. The energy-momentum density T̃ ν μ (x) of the gravitational source and the gravitational energy-momentum pseudo-tensor density t̃ ν μ have the expressions T̃ ν μ (x) = −Mc2δ 0 μ δ ν 0 δ(3)(x) and t̃ ν μ = 0, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like δ(3)(x)/r3 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({})Rρσμν ({}) = 48G2M2/c4r6 is not correct, if we define 1/r6 def = limǫ→0 1/(r 2 + ǫ2)3. Electronic address: [email protected] Electronic address: [email protected]