Stationary Axially Symmetric Fields and the Kerr Metric

Abstract

In this note the axially symmetric metric for stationary gravitational field, in a slightly general form, is discussed. The vacuum field equations for this metric are given. Specialization of this metric leads to a different form of field equations previously discussed in literature. In particular, the Kerr metric is given in a new form. A justification for interpreting the Kerr metric as an exterior solution corresponding to a spinning rod or a rotating spherical body is given.

Bonnor-Type Black Dihole Solution in Brans–dicke–maxwell Theory

“…(21). Prior to this, however, it seems necessary to discuss the peculiar geometric structure of this BD-Bonnor solution that appears to inherit essentially the same generic features of the BD gravity empty space-time solution we studied earlier in the previous subsection.…”

“…(3) by performing a coordinate transformation (of r alone) suggested by Misra and Pandey. 21 Namely, we start with Bonnor's magnetic dipole (dihole) solution of Einstein-Maxwell theory written in Boyer-Lindquist coordinates 6,13…”

“…(21) for ω-parameter of the BD theory taking values in the range, −5/2 ≤ ω < −3/2. Just as we expected, we encounter the seemingly pathological "2π-angle deficit" again.…”

“…13 Lastly, therefore we need to ask, prior to its black dihole interpretation, whether or not the BD-Bonnor solution given in Eq. (21) is qualified at all to be endowed with any meaning as a physical space-time solution. To answer this, recall that for a value of the BD ω parameter in the range of interest −5/2 ≤ ω < −3/2, the BD gravity empty space-time solution was given an acceptable physical meaning or interpretation in terms of the space-time-dependent behavior of the effective Newton's constant.…”

“…(A.1) are defined differently from those with the same names appeared in the BD-Bonnor solution in Eq. (21) in the text. Indeed, they are related by a → ia.…”