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 $\tilde T_\mu^{\nu}(x)$ of the gravitational source and the gravitational energy-momentum pseudo-tensor density $\tilde t_\mu^{\nu}$ have the expressions $\tilde T_\mu^{\nu}(x) =-Mc^2\delta_\mu^0\delta_0^{\nu} \delta^{(3)}x)$ and $\tilde t_\mu^{\nu}=0$, respectively. In expressions of the curvature squares for this class of coordinate systems, there are terms like $\delta^{(3)}(x)/r^3$ and $[\delta^{(3)}(x)}]^2$, as well as other terms, which are singular at $x=0$. It is pointed out that the well-known expression $R^{\rho\sigma\mu\nu}({}) R_{\rho\sigma\mu\nu}({})$ $=48G^{2}M^{2}/c^{4}r^{6}$ is not correct, if we define $1/r^6 = \lim_{\epsilon\to 0}1/(r^2+\epsilon^2)^3$.}Comment: 21 pages, LaTeX, uses amssymb.sty. To appear in Prog. Theor. Phys. 98 (1997
In an extended, new form of general relativity, which is a teleparallel theory of gravity, we examine the energy-momentum and angular momentum carried by gravitational wave radiated from Newtonian point masses in a weak-field approximation. The resulting wave form is identical to the corresponding wave form in general relativity, which is consistent with previous results in teleparallel theory. The expression for the dynamical energy-momentum density is identical to that for the canonical energy-momentum density in general relativity up to leading order terms on the boundary of a large sphere including the gravitational source, and the loss of dynamical energy-momentum, which is the generator of internal translations, is the same as that of the canonical energy-momentum in general relativity. Under certain asymptotic conditions for a non-dynamical Higgs-type field ψ k , the loss of "spin" angular momentum, which is the generator of internal SL(2, C) transformations, is the same as that of angular momentum in general relativity, and the losses of canonical energy-momentum and orbital angular momentum, which constitute the generator of Poincaré coordinate transformations, are vanishing. 1The results indicate that our definitions of the dynamical energy-momentum and angular momentum densities in this extended new general relativity work well for gravitational wave radiations, and the extended new general relativity accounts for the Hulse-Taylor measurement of the pulsar PSR1913+16.
In Poincar\'e gauge theory of gravity and in $\overline{\mbox{Poincar\'e}}$ gauge theory of gravity, we give the necessary and sufficient condition in order that the Schwarzschild space-time expressed in terms of the Schwarzschild coordinates is obtainable as a torsionless exact solution of gravitational field equations with a spinless point-like source having the energy-momentum density $\widetilde{\mbox{\boldmath $T$}}_\mu^{~\nu}(x) = - Mc^2 \delta_\mu^{~0} \delta_0^{~\nu} \delta^{(3)}(\mbox{\boldmath $x$})$. Further, for the case when this condition is satisfied, the energy-momentum and the angular momentum of the Schwarzschild space-time are examined in their relations to the asymptotic forms of vierbein fields. We show, among other things, that asymptotic forms of vierbeins are restricted by requiring the equality of the active gravitational mass and the inertial mass. Conversely speaking, this equality is violated for a class of vierbeins giving the Schwarzschild metric.Comment: 26 pages, LaTeX, uses amssymb.sty. To appear in Prog. Theor. Phys. 99 (1998
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