Distributional EnergyMomentum Densities of Schwarzschild SpaceTime
Abstract
For Schwarzschild spacetime, distributional expressions of energymomentum 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 energymomentum density ~{T}_{μ}^{ν}(x) of the gravitational source and the gravitational energymomentum pseudotensor density ~{t}_{μ}^{ν} have the expressions ~{T}_{μ}^{ν}(x) =  Mc^{2}δ_{μ}^{0}δ_{0}^{ν}δ^{(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 wellknown expression R^{ρσμν}()R_{ρσμν}() = 48G^{2}M^{2}/c^{4}r^{6} is not correct, if we define 1/r^{6} = lim_{ɛ>0}1/(r^{2} + ɛ^{2})^{3}.
 Publication:

Progress of Theoretical Physics
 Pub Date:
 July 1997
 DOI:
 10.1143/PTP.98.69
 arXiv:
 arXiv:grqc/9707029
 Bibcode:
 1997PThPh..98...69K
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 21 pages, LaTeX, uses amssymb.sty. To appear in Prog. Theor. Phys. 98 (1997)