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Multipole moments. II: Curved space. (English) Zbl 1107.83312
Summary: Multipole moments are defined for static, asymptotically flat, source-free solutions of Einstein’s equations. The definition is completely coordinate independent. We take one of the 3-surfaces $$V$$, orthogonal to the time-like Killing vector, and add to it a single point $$\Lambda$$ at infinity. The resulting space inherits a conformal structure from $$V$$. The multipole moments of the solution emerge as a collection of totally symmetric, trace-free tensors $$P,P_a,P_{ab},\cdots$$ at $$\Lambda$$. These tensors are obtained as certain combinations of the derivatives of the norm of the time-like Killing vector. (For static space-times, this norm plays the rôle of a “Newtonian gravitational potential”.) The formalism is shown to yield the usual multipole moments for a solution of Laplace’s equation in flat space, the dependence of these moments on the choice of origin being reflected in the conformal behavior of the $$P$$’s. As an example, the moments of the Weyl solutions are discussed.
Part I, cf. ibid. 11, 1955–1961 (1970; Zbl 1107.83313).

##### MSC:
 83C99 General relativity
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##### References:
 [1] DOI: 10.1002/andp.19173591804 · JFM 46.1303.01 [2] DOI: 10.1063/1.1665348 · Zbl 1107.83313 [3] DOI: 10.1098/rspa.1965.0058 · Zbl 0129.41202 [4] DOI: 10.1063/1.1664599 · Zbl 0172.27905 [5] Janis A., J. Math. Phys. 34 pp 317– (1964) [6] DOI: 10.1007/BF02734579 · Zbl 0124.22201
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