Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

The superposition of null dustbeams in General Relativity 57
(7) if the three commuting Killing vectors are hypersurface–orthogonal.
Hence we are led to the conclusion:
At least two of the three Killing vectors are twisting.
3.2. THE STATIC SOLUTION
Let us first consider the static case when the two spacelike Killing
vectors and are not hypersurface–orthogonal. The corresponding
space–time metric reads
where K, U, W and A are functions of the radial coordinate Numbering
the coordinates according to we have to demand
the relations
between the contravariant components of the null vectors and The
field equations reduce to the three conditions
The first two of these equations imply
(subscripts denoting derivatives with respect to ). These relations enable
us to express the metric functions K and A in terms of W and U
(and their derivatives). Inserting the expressions (15) and (16) into the
remaining field equation S = 0 one obtains a very complicated ordinary
differential equation for W and U. Fortunately, it factorizes in the particular
case K = U. The quantity S defined in (14) then splits into the
product of two expressions. Putting one of them equal to zero one gets
the second–order differential equation
which can be solved by introducing as a new radial coordinate.
It results