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

On maximally symmetric and totally geodesic spaces 37
Consequently, the metric (75), allows locally for the locally
maximally symmetric subspace BTZ, but because of the presence
of the factor in equation (75) in front of this subspace metric, the space
BTZ is not a totally geodesic hypersurface.
On may consider the BTZ metric as a spatial slice of the W0PC metric
given as
in the sense of limiting transition, namely by setting tends to zero
in (77), in the slice the metric becomes the metric.
8.1. THE STATIC BTZ SOLUTION AS ANTI–
DE–SITTER METRIC
Consider the BTZ metric (76), equate in it J to zero, and replace
by the resulting metric is the so called static BTZ:
Accomplishing in the above expression the coordinate transformations
one arrives at
The inverse transformations are given by
where and are restricted to the values