 
Summary: ON STATIONARY VACUUM SOLUTIONS TO THE EINSTEIN EQUATIONS
MICHAEL T. ANDERSON
0. Introduction.
A stationary spacetime (M; g) is a 4manifold M with a smooth Lorentzian metric g, of signature
( ; +; +; +); which has a smooth 1parameter group G R of isometries whose orbits are timelike
curves in M . We assume throughout the paper that M is a chronological spacetime, i.e. M admits
no closed timelike curves, c.f. x1.1 for further discussion.
Let S be the orbit space of the action G. Then S is a smooth 3manifold and the projection
: M ! S
is a principle Rbundle, with ber G. The chronology condition implies that S is Hausdor and
paracompact, c.f. [Ha] for example. The innitesimal generator of G R is a timelike 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 nonvanishing on M , X may
be viewed as a timelike 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 dieomorphism
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 1form for the Rbundle and
