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ON STATIONARY VACUUM SOLUTIONS TO THE EINSTEIN EQUATIONS MICHAEL T. ANDERSON
 

Summary: ON STATIONARY VACUUM SOLUTIONS TO THE EINSTEIN EQUATIONS
MICHAEL T. ANDERSON
0. Introduction.
A stationary space-time (M; g) is a 4-manifold M with a smooth Lorentzian metric g, of signature
( ; +; +; +); which has a smooth 1-parameter group G  R of isometries whose orbits are time-like
curves in M . We assume throughout the paper that M is a chronological space-time, i.e. M admits
no closed time-like curves, c.f. x1.1 for further discussion.
Let S be the orbit space of the action G. Then S is a smooth 3-manifold and the projection
 : M ! S
is a principle R-bundle, with ber G. The chronology condition implies that S is Hausdor and
paracompact, c.f. [Ha] for example. The in nitesimal generator of G  R is a time-like Killing vector
eld X on M , so that
LX g = 0:
The metric g = g M restricted to the horizontal subspaces of TM , i.e. the orthogonal complement of
< X >  TM then induces a Riemannian metric g S on S. Since X is non-vanishing on M , X may
be viewed as a time-like coordinate vector eld, i.e. X = @=@t; where t is a global time function on
M . The time function t gives a global trivialization of the bundle  and so induces a di eomorphism
from M to R  S. The metric g M on M may be then written globally in the form
g M = u 2 (dt + ) 2 +   g S ; (0.1)
where  is a connection 1-form for the R-bundle  and

  

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

 

Collections: Mathematics