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

Lorentz force free charged fluids in general relativity 313
The Killing vectors I, II and IV are everywhere timelike, III spacelike,
and the Killing vector V has a sign-indefinite square, thus in some
parts of the space it is timelike and in other parts, spacelike. Due to
the commutation properties of the three isometries corresponding to I,
II and IV, only two of them (always including I) can be simultaneously
used to introduce coordinates of which the metric coefficients will not depend
(the coefficients of their squared differentials will be therefore both
positive – timelike axes). The third Killing coordinate available simultaneously
with these two, is clearly spacelike. Of course, it seems
strange to use two timelike axes at once, though this is always exactly
the case in dealing with the Gödel spacetime. What is not less exotic,
there exists the third alternative in introduction of one of this pair of
coordinates: it can involve the Killing coordinate corresponding to V
which naturally excludes the possibility of simultaneous use of the coordinates
which are Killingian with respect to II and IV. The situation
really is in its full extent a rare exception in the theory of spacetime.
The real physical signature (+ – – –) of the Gödel spacetime is easily
seen from the existence of the orthonormal tetrad (with the normalization
properties corresponding to this signature) in this spacetime, for
example, the natural tetrads of (1) and (5). The very same consideration
saves time in the calculation of the metric tensor determinant: e.g.,
in the case of (2), which
yields simply
The fact that there exists only one Killing vector which is spacelike in
all spacetime, does not mean, of course, absence of spatial homogeneity
in the Gödel universe. There are linear combinations (with constant coefficients)
of Killing vectors being as well spacelike, though this property
is limited to certain regions of spacetime (similar to the situation with
the Killing vector V), the region depending of the choice of constants
and never embracing the whole spacetime. Thus the homogeneity property
of the Gödel universe is strictly local, but it covers all the spacetime
piecewise. Similarly, the local essentially stationary property is known
for the ergosphere region of the Kerr solution (see [10], p. 331).
Many authors subdivide the multitude of exact solutions into “physical”
and “unphysical” subsets. My opinion is that the point of view
that in the well established physical theories there could be present unphysical
solutions and effects alongside with the “good”, physical ones,