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ASYMPTOTIC BEHAVIOR OF STATIC AND STATIONARY VACUUM SPACE-TIMES
 

Summary: ASYMPTOTIC BEHAVIOR OF STATIC AND STATIONARY
VACUUM SPACE-TIMES
Michael T. ANDERSON
Dept. of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, NY 11794, USA
E-mail: anderson@math.sunysb.edu
Let (M; g) be an Einstein vacuum chronological space-time. The space-time is
stationary if it admits a time-like Killing eld X = @=@t, and static if the Killing eld
is hypersurface orthogonal. Let u =
p
g(X; X) > 0 and let  : (M; g) ! (S; g S )
be the projection to the orbit space of the R -action generated by X . We recall the
following classical and well-known results of Lichnerowicz [1,x85, x90], c.f. however
[2] for an earlier version.
Lichnerowicz Theorems. (I). Suppose (M; g) is a geodesically complete
static vacuum space-time, with u(x) ! 1, as x !1 in S. Then (M; g) is at.
(II). Suppose (M; g) is a geodesically complete stationary vacuum space-time
which is asymptotically at, (AF). Then (M; g) is at.
Physically, these results appear quite satisfying. If the space-time is vacuum
and geodesically complete, then there are no "matter or energy sources" contained
in the space-time. Under the asymptotic conditions at in nity, one then concludes

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics