Abstract
Recent investigations of the initial-value problem of general relativity have shown that the initial-value constraints can be formulated in all cases as a system of elliptic equations with well-defined physical and mathematical properties. The solutions of these equations can be regarded as generalized gravitational potentials. These potentials are interrelated and depend on their sources quasilinearly. They are particularly useful in analyzing asymptotically flat solutions of Einstein's equations. We have found from these results (1) a technique for constructing physically meaningful initial data in the integration of Einstein's equations, and (2) a method for characterization and analysis of the spacelike mass, momentum, angular momemtum, and multipole moments of gravitational fields.
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A foliation (of codimension 1) is defined by a one-form u with the properties u Λ du=0 and g(u, u)=−1, where the space-time metric has signature (−+++).
Ó Murchadha, N., and York, J. W. (1974).Phys. Rev. D,10, 428; (1974).ibid.,10, 437. These articles contain references to our earlier work.
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L. Smarr is applying these equations in the numerical investigation of the collision of two black holes (personal communication).
A detailed investigation of gravitational energy is given by the authors in (1974).Phys. Rev. D,10, 2345.
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Exact expressions are given in reference 2.
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Since the three-slice is Euclidean at infinity, there exists a quasi-Cartesian coordinate system in the far-field, and we definer with respect to this coordinate system. It can be precisely defined by the monopole harmonic function of 13-01 on the manifold.
Ó Murchadha, N. and York, J. W. “Gravitational Mass, Momentum, and Multipoles” (to be submitted for publication).
This particular choice of spacelike hypersurfaces and time four-vectors is discussed in reference 11; also York, J. W., and Ó Murchadha, N. (1974).Bull. Am. Phys. Soc.,19, 509; York, J. W., and Smarr, L. (1975).ibid.,20, 544; Smarr, L., and York, J. W.
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One also has inevitable problems with numerical instabilities, as found by DeWitt, B., Smarr, L., and Eppley, K. These problems are being overcome using again the techniques the authors have developed in the initial-value analysis [Smarr, L., and Eppley, K. (personal communication)].
Smarr, L., and York, J. W. reference 12.
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This essay received the third award from the Gravity Research Foundation for the year 1975. (Editor.)
The research for this paper was partially supported by National Science Foundation grant GP43909 awarded to the University of North Carolina, Chapel Hill, North Carolina.
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Murchadha, N.ó., York, J.W. Gravitational potentials: A constructive approach to general relativity. Gen Relat Gravit 7, 257–261 (1976). https://doi.org/10.1007/BF00768526
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DOI: https://doi.org/10.1007/BF00768526