In order to obtain the Mazur identity, one considers two
arbitrary hermitian matrices,
and
. The aim is to compute the Laplacian (with respect to an
arbitrary metric
) of the relative difference
, say, between
and
,
It turns out to be convenient to introduce the current
matrices
and
, and their difference
, where
denotes the covariant derivative with respect to the metric
under consideration. Using
, the Laplacian of
becomes
For hermitian matrices one has
and
, which can be used to combine the trace of the first two terms
on the RHS of the above expression. One easily finds
The above expression is an identity for the relative
difference of two arbitrary hermitian matrices. If the latter are
solutions
of a non-linear
-model with action
, then their currents are conserved [see Eq. (32
)], implying that the second term on the RHS vanishes. Moreover,
if the
-model describes a mapping with coset space
, then this is parametrized by positive hermitian matrices of
the form
.
Hence, the ``on-shell'' restriction of the Mazur identity to
-models with coset
becomes
where
.
Of decisive importance to the uniqueness proof for the
Kerr-Newman metric is the fact that the RHS 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 RHS of Eq. (35
) 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
k
:
. The reduction of Eq. (35
) with respect to the second Killing field and the integration of
the resulting expression will be discussed in Sect.
6
.
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |