Stationary solutions and asymptotic flatness

Details Stationary solutions and asymptotic flatness Stationary solutions and asymptotic flatness

2010-02-05

arxiv.org/abs/1002.1172v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

This new version extends in 34 pages the previous one. Full technical details have been provided. Otherwise, there are no chan

Scientific paper

Consider a complete end of a four dimensional strictly stationary solution of the vacuum Einstein equations homeomorphic to $\field{R}^{3}$ minus a ball. We prove that the end is asymptotically flat with the standard Kerr-type of fall off if and only if the metric $g=|X|^{2} \g$ is complete, where $\g$ is the quotient three-metric, ${\bf g}$ is the space-time metric, $X$ is the time-like stationary Killing field and $|X|^{2}:=-_{\bf g}$. In particular the end is asymptotically flat if $|X|^{2}\geq c>0$. The proof is based on Anderson's a priori curvature estimates and a volume comparison on the conformally related space on which the non-metric part of the Einstein equations take the form of the harmonic Ernst map into the hyperbolic plane. The Kaluza-Klein monopoles show that a similar statement is false in higher dimensions. Moreover the statement is false in the presence of matter even in dimension four, for there exist spherically symmetric, static, perfect fluid solutions (with positive energy density and pressure) in hydrostatic equilibrium and whose asymptotic is isothermal and therefore not asymptotically flat. The result above originated while studying the isothermal asymptotic of dark halos in spiral galaxies. We discuss this genesis.

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