We saw in Section 3.1.1 that in the Newtonian theory of gravity the mass of the source in can be
expressed as the flux integral of the gravitational field strength on the boundary
. Similarly, in
the weak field (linear) approximation of general relativity on Minkowski spacetime the source of the
gravitational field (i.e., the linearized energy-momentum tensor) is still analogous to charge. In fact, the
total energy-momentum and angular momentum of the source can be expressed as appropriate two-surface
integrals of the curvature at infinity [481]. Thus, it is natural to expect that the energy-momentum and
angular momentum of the source in a finite three-volume
, given by Eq. (2.5
), can also be expressed as
the charge integral of the curvature on the two-surface
. However, the curvature tensor can
be integrated on
only if at least one pair of its indices is annihilated by some tensor via
contraction, i.e., according to Eq. (3.14
) if some
is chosen and
. To
simplify the subsequent analysis,
will be chosen to be anti-self-dual:
with
.11
Thus, our goal is to find an appropriate spinor field
on
such that
Recall that the space of the contravariant valence-one twistors of Minkowski spacetime is the set of the pairs
of spinor fields, which solve the valence-one–twistor equation
.
If
is a solution of this equation, then
is a solution of the
corresponding equation in the conformally-rescaled spacetime, where
and
is the conformal factor. In general, the twistor equation has only the trivial solution, but in
the (conformal) Minkowski spacetime it has a four complex-parameter family of solutions. Its
general solution in the Minkowski spacetime is
, where
and
are constant spinors. Thus, the space
of valence-one twistors, called the twistor space, is
four–complex-dimensional, and hence, has the structure
.
admits a natural
Hermitian scalar product: if
is another twistor, then
. Its
signature is
, it is conformally invariant,
, and it is
constant on Minkowski spacetime. The metric
defines a natural isomorphism between the
complex conjugate twistor space,
, and the dual twistor space,
, by
. This makes it possible to use only twistors with unprimed indices. In
particular, the complex conjugate
of the covariant valence-two twistor
can be
represented by the conjugate twistor
. We should mention two special,
higher-valence twistors. The first is the infinity twistor. This and its conjugate are given explicitly by
The solution of the valence-two twistor equation, given by Eq. (7.5
), can always be written as a
linear combination of the symmetrized product
of the solutions
and
of the valence-one
twistor equation.
uniquely defines a symmetric twistor
(see, e.g., [426
]). Its spinor parts
are
However, Eq. (7.1) can be interpreted as a
-linear mapping of
into
, i.e., Eq. (7.1
) defines a
dual twistor, the (symmetric) kinematical twistor
, which therefore has the structure
Thus, to summarize, the various spinor parts of the kinematical twistor are the energy-momentum
and s.d. angular momentum. However, additional structures, namely the infinity twistor and the Hermitian
scalar product, are needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and,
in particular, to define the mass and express the Pauli–Lubanski spin. Furthermore, the Hermiticity
condition ensuring that
has the correct number of components (ten reals) is also formulated in terms
of these additional structures.
http://www.livingreviews.org/lrr-2009-4 |
Living Rev. Relativity 12, (2009), 4
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