Since is boost-gauge dependent and Eq. (10.20
) in itself does not yield, e.g., the correct ADM
energy in asymptotically flat spacetime, a boost gauge and a restriction on the vector field
and/or a
‘renormalization’ of Eq. (10.20
) (in the form of an appropriate reference term) must be given. Wang and
Yau suggest that one determine these by embedding the spacelike two-surface
isometrically into the
Minkowski spacetime in an appropriate way.
Thus, suppose that there is an isometric embedding , and let us fix a constant
future-pointing unit timelike vector field
in
. This
defines a global orthonormal frame field
in the normal bundle of
by requiring
, and let us denote
the mean extrinsic curvature vector of this embedding by
. Then, supposing that the
mean extrinsic curvature vector
of
in the physical spacetime is spacelike, there is a
uniquely-determined global orthonormal frame field
in the normal bundle of
such that
. This fixes the boost gauge in
, and, in addition, makes it possible to identify the
normal bundle of
in
and the normal bundle of
in
via the identification
,
. This, together with the natural identification of the tangent bundle
of
and the tangent bundle
of
yields a natural identification of the
Lorentzian vector bundles over
in
and over
in
. Therefore, any vector (and
tensor) field on
yields a vector (tensor) field on
. In particular, if
,
then
is a tangent of
, and hence, there is a uniquely determined tangent
of
such that
. Consequently
can be identified with the vector field
on
. Similarly, the connection one-form
on the normal bundle (in the
boost gauge
) can be pulled back along
to a one-form
on
. Then,
denoting by
and
the mean curvature of
and
in the direction
and
,
respectively, Wang and Yau [544
] define the quasi-local energy with respect to the pair
by
To prove, e.g., the positivity of this energy, or to ensure that in flat spacetime the energy be zero, further
conditions must be satisfied. Wang and Yau formulate these conditions in the notion of admissible pairs
:
should have a convex shadow in the direction
,
must be the boundary of some
spacelike hypersurface in
on which the Dirichlet boundary value problem for the Jang equation can
be solved with the time function
discussed in Section 4.1.3, and the connection 1-form and the
mean curvature in a certain gauge must satisfy an inequality. (For the precise definition of
the admissible pairs see [544
]; for the geometrical background see [543] and Section 4.1.3.)
Then it is shown that if the dominant energy condition holds and
has a spacelike mean
curvature vector, then for the admissible pairs the quasi-local energy (10.21
) is non-negative.
Therefore, if the set of the admissible pairs is not empty (e.g., when the scalar curvature of
is positive), then the infimum
of
among all admissible pairs is
non-negative, and is called the quasi-local mass. If this infimum is achieved by the pair
, i.e.,
by an embedding
and a timelike
, then
is called the quasi-local
energy-momentum, which is then future pointing and timelike. It is still an open question that if
the quasi-local mass
is vanishing, then the domain of dependence
of the
spacelike hypersurface
with boundary
can be curved (e.g., a pp-wave geometry with
pure radiation) or not. If not, then the quasi-local energy-momentum would be expected to be
null.
The quasi-local energy-momentum associated with any two-surface in Minkowski spacetime with a
convex shadow in some direction is clearly zero. The mass has been calculated for round spheres in the
Schwarzschild spacetime. It is , and hence, for the event horizon it gives
.
has been calculated for large spheres and it has the expected limits at the spatial and null
infinities [545, 142]. Also, it has the correct small sphere limit both in nonvacuum and vacuum [544].
Upper and lower estimates of the Wang–Yau energy are derived, and its critical points are investigated in
[362] and [363], respectively. On the applicability of
in the formulation and potential proof of
Thorne’s hoop conjecture see Section 13.2.2. A recent review of the results in connection with the
Wang–Yau energy see [541].
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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