and Einstein's equations, implying
.
For a stationary and axisymmetric spacetime with Killing fields
(one-forms)
k
and
m, Eq. (49
) implies
and similarly for
.
By virtue of Eq. (50
) - and the fact that the Frobenius condition
can be written as
- the circularity problem is reduced to the following two
tasks:
(i) Since
is a
function, it must be constant if its derivative vanishes. As
m
vanishes on the rotation axis, this implies
in every domain of spacetime intersecting the axis. (At this
point it is worthwhile to recall that the corresponding step in
the staticity theorem requires more effort: Concluding from
that
vanishes is more involved, since
is a
one-form
. However, using Stoke's theorem to integrate an identity for the
twist [88
] shows that a strictly stationary - not necessarily simply
connected - domain of outer communication must be static if
is closed.
)
(ii) While
follows from the symmetry conditions for electro-magnetic fields
[26] and for scalar fields [86], it cannot be established for non-Abelian gauge fields [88]. This implies that the usual foliation of spacetime used to
integrate the stationary and axisymmetric Maxwell equations is
too restrictive to treat the Einstein-Yang-Mills (EYM) system.
This is seen as follows: In Sect.(4.4) we have derived the formula (22
). By virtue of Eq. (10
) this becomes an expression for the derivative of the twist in
terms of the electric Yang-Mills potential
(defined with respect to the stationary Killing field
k) and the magnetic one-form
:
Contracting this relation with the axial Killing field
m, and using again the fact that the Lie derivative of
with respect to
m
vanishes, yields immediately
The difference between the Abelian and the non-Abelian case
lies in the circumstance that the Maxwell equations automatically
imply that the (km)-component of
vanishes,
whereas this does not follow from the Yang-Mills equations.
Moreover, the latter do not imply that the Lie algebra valued
scalars
and
are orthogonal. Hence, circularity is a generic property of the
Einstein-Maxwell (EM) system, whereas it imposes additional
requirements on non-Abelian gauge fields.
Both the staticity and the circularity theorems can be
established for scalar fields or, more generally, scalar mappings
with arbitrary target manifolds: Consider a self-gravitating
scalar mapping
with Lagrangian
. The stress energy tensor is of the form
where the functions
and
P
may depend on
,
, the spacetime metric
and the target metric
. If
is invariant under the action of a Killing field
- in the sense that
for each component
of
- then the one-form
becomes proportional to
:
. By virtue of the Killing field identity (49
), this implies that the twist of
is closed. Hence, the staticity and the circularity issue for
self-gravitating scalar mappings reduce to the corresponding
vacuum problems. From this one concludes that stationary
non-rotating black hole configuration of self-gravitating scalar
fields are static if
, while stationary and axisymmetric ones are circular if
.
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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 |