Suppose that the spacetime is asymptotically flat at future null infinity, and the closed spacelike
2-surface can be joined to future null infinity by a smooth null hypersurface
. Let
, the cut defined by the intersection of
with the future null infinity. Then
the null geodesic generators of
define a smooth bijection between
and the cut
(and hence, in particular,
). We saw in Section 4.2.4 that on the cut
at the
future null infinity we have the asymptotic spin space
. The suggestion of Ludvigsen
and Vickers [259
] for the spin space
on
is to import the two independent
solutions of the asymptotic twistor equations, i.e. the asymptotic spinors, from the future null
infinity back to the 2-surface along the null geodesic generators of the null hypersurface
.
Their propagation equations, given both in terms of spinors and in the GHP formalism, are
Before discussing the usual questions about the properties of the construction (positivity, monotonicity, the
various limits, etc.), we should make some general remarks. First, it is obvious that the Ludvigsen-Vickers
energy-momentum in its form above cannot be defined in a spacetime which is not asymptotically flat at
null infinity. Thus their construction is not genuinely quasi-local, because it depends not only on the
(intrinsic and extrinsic) geometry of , but on the global structure of the spacetime as well. In addition,
the requirement of the smoothness of the null hypersurface
connecting the 2-surface to the null infinity
is a very strong restriction. In fact, for general (even for convex) 2-surfaces in a general asymptotically
flat spacetime conjugate points will develop along the (outgoing) null geodesics orthogonal
to the 2-surface [304, 175
]. Thus either the 2-surface must be near enough to the future null
infinity (in the conformal picture), or the spacetime and the 2-surface must be nearly spherically
symmetric (or the former cannot be ‘very much curved’ and the latter cannot be ‘very much
bent’).
This limitation yields that in general the original construction above does not have a small sphere
limit. However, using the same propagation equations (59, 60
) one could define a quasi-local
energy-momentum for small spheres [259
, 66
]. The basic idea is that there is a spin space at the
vertex
of the null cone in the spacetime whose spacelike cross section is the actual 2-surface,
and the Ludvigsen-Vickers spinors on
are defined by propagating these spinors from the
vertex
to
via Equations (59
, 60
). This definition works in arbitrary spacetime, but the
2-surface cannot be extended to a large sphere near the null infinity, and it is still not genuinely
quasi-local.
Once the Ludvigsen-Vickers spinors are given on a spacelike 2-surface of constant affine
parameter
in the outgoing null hypersurface
, then they are uniquely determined on any
other spacelike 2-surface
in
, too, i.e. the propagation law (59
, 60
) defines a natural
isomorphism between the space of the Ludvigsen-Vickers spinors on different 2-surfaces of
constant affine parameter in the same
. (
need not be a Bondi-type coordinate.) This
makes it possible to compare the components of the Ludvigsen-Vickers energy-momenta on
different surfaces. In fact [259
], if the dominant energy condition is satisfied (at least on
),
then for any Ludvigsen-Vickers spinor
and affine parameter values
one has
, and the difference
can be interpreted as the
energy flux of the matter and the gravitational radiation through
between
and
.
Thus both
and
are increasing with
(‘mass-gain’). A similar monotonicity
property (‘mass-loss’) can be proven on ingoing null hypersurfaces, but then the propagation
law (59
, 60
) should be replaced by
and
. Using these
equations the positivity of the Ludvigsen-Vickers mass was proven in various special cases
in [259
].
Concerning the positivity properties of the Ludvigsen-Vickers mass and energy, first it is obvious by the
remarks on the nature of the propagation law (59, 60
) that in Minkowski spacetime the Ludvigsen-Vickers
energy-momentum is vanishing. However, in the proof of the non-negativity of the Dougan-Mason energy
(discussed in Section 8.2) only the
part of the propagation equations is used. Therefore, as
realized by Bergqvist [61
], the Ludvigsen-Vickers energy-momenta (both based on the asymptotic and the
point spinors) are also future directed and nonspacelike if
is the boundary of some compact spacelike
hypersurface
on which the dominant energy condition is satisfied and
is weakly future
convex (or at least
). Similarly, the Ludvigsen-Vickers definitions share the rigidity
properties proven for the Dougan-Mason energy-momentum [354
]: Under the same conditions the
vanishing of the energy-momentum implies the flatness of the domain of dependence
of
.
In the weak field approximation [259] the difference
is just the integral of
on the portion of
between the two 2-surfaces, where
is the linearized
energy-momentum tensor: The increment of
on
is due only to the flux of the matter
energy-momentum.
Since the Bondi-Sachs energy-momentum can be written as the integral of the Nester-Witten 2-form on
the cut in question at the null infinity with the asymptotic spinors, it is natural to expect that the first
version of the Ludvigsen-Vickers energy-momentum tends to that of Bondi and Sachs. It was shown
in [259, 340
] that this expectation is, in fact, correct. The Ludvigsen-Vickers mass was calculated for large
spheres both for radiative and stationary spacetimes with
and
accuracy, respectively,
in [338, 340].
Finally, on a small sphere of radius in non-vacuum the second definition gives [66
] the expected
result (28
), while in vacuum [66, 360
] it is
![]() |
http://www.livingreviews.org/lrr-2004-4 |
© Max Planck Society and the author(s)
Problems/comments to |