8.1 Integrability properties of Killing fields
Our aim here is to discuss the circularity problem in some more detail. The task is to use the symmetry
properties of the matter model in order to establish the orthogonal-integrability conditions for the Killing
fields. The link between the relevant components of the stress-energy tensor and the integrability
conditions is provided by a general identity for the derivative of the twist of a Killing field
, say,
and Einstein’s equations, implying
. This follows from the definition of the twist
and the Ricci identity for Killing fields,
, where
is the one-form with components
; see, e.g., [151], Chapter 2. For a stationary and axisymmetric spacetime with Killing
fields (one-forms)
and
, Eq. (8.1) implies
and similarly for
. Eq. (8.2) is an identity up to a term involving the Lie derivative of the twist of
the first Killing field with respect to the second one (since
). In order to
establish
, it is sufficient to show that
and
commute in an asymptotically-flat
spacetime. This was first achieved by Carter [44] and later, under more general conditions, by
Szabados [304].
The following is understood to also apply for
: By virtue of Eq. (8.2) – and the fact that the
condition
can be written as
– the circularity problem is reduced to the
following two tasks:
-
(i)
- Show that
implies
.
-
(ii)
- Establish
from the stationary and axisymmetric matter equations.
(i) Since
is a function, it is locally constant if its derivative vanishes. As
vanishes on the
rotation axis, this implies
in every connected 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
the Stokes’ theorem to integrate an identity for the twist [152
] shows that a strictly stationary – not
necessarily simply connected – domain of outer communication must be static if
is closed. While this
proves the staticity theorem for vacuum and self-gravitating scalar fields [152
], it does not solve
the electrovacuum case. It should be noted that in the context of the proof of uniqueness the
strictly stationary property follows from staticity [72] and not the other way around (compare
Figure 3).
(ii) While
follows from the symmetry conditions for electro-magnetic fields [43] and
for scalar fields [150], it cannot be established for non-Abelian gauge fields [152]. This implies that the
usual foliation of spacetime used to integrate the stationary and axisymmetric Maxwell equations is too
restrictive to treat the EYM system. This is seen as follows: In Section 6.3 we have derived the
formula (6.17). By virtue of Eq. (6.3) 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
) and the
magnetic one-form
:
where
is a suitably normalized trace (see Eq. (6.15)). Contracting this relation with the axial Killing
field
, and using again the fact that the Lie derivative of
with respect to
vanishes, yields
immediately
The difference between the Abelian and the non-Abelian case is due to the fact that the Maxwell equations
automatically imply that the
-component of
vanishes, whereas this does not follow from the
Yang–Mills equations. In fact, the Maxwell equation
and the symmetry property
imply the existence of a magnetic potential,
, thus,
. Moreover, the latter do not imply that the Lie algebra valued scalars
and
are orthogonal. Hence, circularity is an intrinsic property of the EM system,
whereas it imposes additional requirements on non-Abelian gauge fields.
Both staticity and circularity theorems can be established for scalar fields or, more generally, scalar
mappings with arbitrary target manifolds: Consider, for instance, a self-gravitating scalar mapping
with Lagrangian
. The stress energy tensor is of the form
where the functions
and
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 (8.1), this implies that the twist of
is closed. Hence, the staticity and
the circularity issue for self-gravitating scalar mappings can be established, under appropriate conditions, as
in the vacuum case. From this one concludes that (strictly) stationary non-rotating black-hole configuration
of self-gravitating scalar fields are static if
, while stationary and axisymmetric ones are circular
if
.