Integral conservation laws for isolated systems by the method of privileged vector fields in the general theory of relativity and the external Kerr metric

On the basis of the differential conservation laws by the method of privileged vector fields in the general theory of relativity integral conserved quantities are defined and investigated for isolated systems. The existence of these quantities and their equivalence impose certain conditions on the vector fields or on the structure of the energy-momentum tensor of the matter distribution Tsub(ik). A necessary and sufficient condition for the existence of Tsup(i)sub(j)xisup(j) as a conserved guantity is given. The equivalence between the conserved quantities of Komar's and Trautman's type are investigated for a simple structure of Tsub(ik) (Tsup(a)sub(a) = 0), and applied to the case of the external Kerr metric for the calculation of the energy Esub(g) (Esub(g) = ∫T44df4) and the momentum P3 (P3=(1/c) ∫ T43df4) of the gravitational source. (Auth.)