Distributional Energy-Momentum Densities Schwarzschild Space-Time : Astrophysics and Relativity

概要

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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 T^^~_μ^ν(x) of the gravitational source and the gravitational energy-momentum pseudo-tensor density t^^~_μ^ν have the expressions T^^~_μ^ν(x) = -Mc^2δ_μ^0δ_0^νδ^<(3)>(x) and 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^2M^2/c^4r^6 is not correct, if we define 1/r^6 =^^<def> lim_<ε→0>1/(r^2 + ε^2)3.
理論物理学刊行会の論文
1997-07-25