Distributional Energy-Momentum Densities of Schwarzschild Space-Time

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 e T ¯ (x) of the gravitational source and the gravitational energy-momentum pseudo-tensor density e t ¯ have the expressions e T ¯ (x) = \GammaM c 2 ffi 0 ¯ ffi 0 ffi (3) (x) and e t ¯ = 0, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like ffi (3) (x)=r 3 and \Theta ffi (3) (x) 2 , as well as other terms, which are singular at x = 0. It is pointed out that the well-known expression R aeoe¯ (fg)R aeoe¯ (fg) = 48G 2 M 2 =c 4 r 6 is not correct, if we define 1=r 6 def = lim ffl!0 1=(r 2 + ffl 2 ) 3 . Electronic address: kawai@sci.osaka-cu.ac.jp y Electronic address: sakane@sci...