In order to obtain the Mazur identity, one considers two arbitrary Hermitian matrices,
and
. The aim is to compute the Laplacian with respect to a metric
(which in the
application of interest will be flat) of the relative difference
, say, between
and
,
where, as before, the inner product is taken with respect to the three-metric
and also involves a
matrix multiplication. For hermitian matrices one has
and
, which can
be used to combine the trace of the first two terms on the right-hand side of the above expression. One
easily finds
Of decisive importance to the uniqueness proof for the Kerr–Newman metric is the fact that the
right-hand side of the above relation is non-negative. In order to achieve this one needs two Killing fields:
The requirement that be represented in the form
forces the reduction of the EM system with
respect to a space-like Killing field; otherwise the coset is
, which is not of the
desired form. As a consequence of the space-like reduction, the three-metric
is not Riemannian, and the
right-hand side of Eq. (7.3
) is indefinite, unless the matrix valued one-form
is space-like. This is the
case if there exists a time-like Killing field with
, implying that the currents are
orthogonal to
:
. The reduction of Eq. (7.3
) with respect
to the second Killing field and the integration of the resulting expression will be discussed in
Section 8.
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
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