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

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First let us recall some properties of the Gödel metric [9] which is of
great importance in constructing several types of exact LFFS. For the
signature (+ – – –) and after some other transformations, the Gödel
metric reads (cf. [22, 24])
or in “cylindrical” coordinates,
the same form in [20] was put as
EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
It is interesting that if (1) is rewritten in the form
it becomes clear that either of and may be interpreted as a time
coordinate (obviously related to the existence of closed timelike lines in
the Gödel universe). Then, interchanging and in this expression, we
get
(the three–dimensional sector taking here the Cartesian form). Of course,
all coordinates used above are, as it is admitted traditionally in the
Gödel metric, dimensionless. One can however incorporate into (at
least, some) coordinates variables and redefine so that,
for example, (1) takes the form
then the limit yields the Minkowski spacetime.
The Gödel spacetime admits five Killing vectors which we give here
in the coordinates used in (2), up to the positive constant factor
occurring in the covariant components: