7.1 Motivations
7.1.1 How do the twistors emerge?
In the Newtonian theory of gravity the mass contained in some finite 3-volume
can be
expressed as the flux integral of the gravitational field strength on the boundary
:
where
is the gravitational potential and
is the outward directed unit normal to
. If
is
deformed in
through a source-free region, then the mass does not change. Thus the mass
is
analogous to charge in electrostatics.
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 2-surface
integrals of the curvature at infinity [349]. Thus it is natural to expect that the energy-momentum and
angular momentum of the source in a finite 3-volume
, given by Equation (5), can also be expressed as
the charge integral of the curvature on the 2-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 Equation (15) if some
is chosen and
. To
simplify the subsequent analysis
will be chosen to be anti-self-dual:
with
.
Thus our claim is to find an appropriate spinor field
on
such that
Since the dual of the exterior derivative of the integrand on the right, and, by Einstein’s equations, the dual
of the
times the integrand on the left, respectively, is
Equations (44) and (45) are equal if
satisfies
This equation in its symmetrized form,
, is the valence 2 twistor equation, a specific
example for the general twistor equation
for
. Thus, as could be
expected,
depends on the Killing vector
, and, in fact,
can be recovered from
as
. Thus
plays the role of a potential for the Killing vector
. However, as a
consequence of Equation (46),
is a self-dual Killing 1-form in the sense that its derivative is a
self-dual (or s.d.) 2-form: In fact, the general solution of equation (46) and the corresponding Killing vector
are
where
,
, and
are constant spinors, using the notation
, where
is a constant spin frame (the ‘Cartesian spin frame’) and
are the standard
Pauli
matrices (divided by
). These yield that
is, in fact, self-dual,
, and
is
a translation and
generates self-dual rotations. Then
, implying
that the charges corresponding to
are vanishing; the four components of the quasi-local
energy-momentum correspond to the real
s, and the spatial angular momentum and centre-of-mass
are combined into the three complex components of the self-dual angular momentum
, generated by
.
7.1.2 Twistor space and the kinematical twistor
Recall that the space of the contravariant valence 1 twistors of Minkowski spacetime is the set of the
pairs
of spinor fields, which solve the so-called valence 1 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 1 twistors, the so-called twistor-space, is
4-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 2 twistor
can be represented by the
so-called conjugate twistor
. We should mention two special, higher valence
twistors. The first is the so-called infinity twistor. This and its conjugate are given explicitly by
The other is the completely anti-symmetric twistor
, whose component
in an
-orthonormal basis is required to be one. The only non-vanishing spinor parts of
are those
with two primed and two unprimed spinor indices:
,
,
, …. Thus for any four twistors
,
, the determinant of the
matrix whose
-th column is
, where the
, …,
are the components of
the spinors
and
in some spin frame, is
where
is the totally antisymmetric Levi-Civita symbol. Then
and
are dual to each other
in the sense that
, and by the simplicity of
one has
.
The solution
of the valence 2 twistor equation, given by Equation (47), can always be written as
a linear combination of the symmetrized product
of the solutions
and
of the valence 1
twistor equation.
defines uniquely a symmetric twistor
(see for example [313
]). Its spinor
parts are
However, Equation (43) can be interpreted as a
-linear mapping of
into
, i.e. Equation (43)
defines a dual twistor, the (symmetric) kinematical twistor
, which therefore has the structure
Thus the quasi-local energy-momentum and self-dual angular momentum of the source are certain spinor
parts of the kinematical twistor. In contrast to the ten complex components of a general symmetric twistor,
it has only ten real components as a consequence of its structure (its spinor part
is identically zero)
and the reality of
. These properties can be reformulated by the infinity twistor and the Hermitian
metric as conditions on
: The vanishing of the spinor part
is equivalent to
and the energy momentum is the
part of the kinematical twistor, while the whole
reality condition (ensuring both
and the reality of the energy-momentum) is equivalent to
Using the conjugate twistors this can be rewritten (and, in fact, usually it is written) as
. Finally, the quasi-local mass can also be
expressed by the kinematical twistor as its Hermitian norm [307
] or as its determinant [371
]:
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, were needed to be able to ‘isolate’ its energy-momentum and angular momentum parts, and,
in particular, to define the mass. Furthermore, the Hermiticity condition ensuring
to have the
correct number of components (ten reals) were also formulated in terms of these additional
structures.