2.3 Optical scalars and Sachs
equations
For the calculation of distance measures, of image distortion, and
of the brightness of images one has to study the Jacobi equation (= equation of geodesic
deviation) along lightlike geodesics. This is usually done in terms
of the optical scalars which were
introduced by Sachs et al. [172
, 292
]. Related background
material on lightlike geodesic congruences can be found in many
text-books (see, e.g., Wald [341
], Section 9.2). In
view of applications to lensing, a particularly useful exposition
was given by Seitz, Schneider and Ehlers [303
]. In the following
the basic notions and results will be summarized.
Infinitesimally thin
bundles.
Let
be an affinely parametrized lightlike
geodesic with tangent vector field
. We assume that
is past-oriented, because in applications to lensing
one usually considers rays from the observer to the source. We use
the summation convention for capital indices
taking the values 1 and 2. An infinitesimally thin bundle (with elliptical
cross-section) along
is a set
Here
denotes the Kronecker delta, and
and
are two vector fields along
with
such that
,
, and
are linearly independent for almost all
.
As usual,
denotes the curvature tensor, defined
by
Equation (9) is the Jacobi
equation. It is a precise mathematical formulation of the statement
that “the arrow-head of
traces an infinitesimally
neighboring geodesic”. Equation (10) guarantees that this
neighboring geodesic is, again, lightlike and spatially related to
.
Sachs basis.
For discussing the geometry of infinitesimally thin bundles it is
usual to introduce a Sachs basis,
i.e., two vector fields
and
along
that are orthonormal, orthogonal to
, and parallelly transported,
Apart from the possibility to interchange them,
and
are unique up to transformations
where
,
, and
are constant along
. A Sachs basis determines a unique vector field
with
and
along
that is perpendicular to
, and
. As
is assumed past-oriented,
is future-oriented. In the rest system of the
observer field
, the Sachs basis spans the 2-space
perpendicular to the ray. It is helpful to interpret this 2-space
as a “screen”; correspondingly, linear combinations of
and
are often refered to as “screen
vectors”.
Jacobi
matrix.
With respect to a Sachs basis, the basis vector fields
and
of an infinitesimally thin bundle can
be represented as
The Jacobi matrix
relates the shape of the cross-section of the
infinitesimally thin bundle to the Sachs basis (see Figure 3). Equation (9) implies that
satisfies the matrix Jacobi
equation
where an overdot means derivative with respect to the affine
parameter
, and
is the optical tidal matrix, with
Here
denotes the Ricci tensor, defined by
, and
denotes the conformal
curvature tensor (= Weyl tensor). The notation in Equation (18) is chosen in
agreement with the Newman-Penrose formalism (cf., e.g., [54
]). As
,
, and
are not everywhere linearly
dependent,
does not vanish identically. Linearity
of the matrix Jacobi equation implies that
has only isolated zeros. These are the “caustic
points” of the bundle (see below).
Shape
parameters.
The Jacobi matrix
can be parametrized according to
Here we made use of the fact that any matrix can be written as the
product of an orthogonal and a symmetric matrix, and that any
symmetric matrix can be diagonalized. Note that, by our definition
of infinitesimally thin bundles,
and
are non-zero almost everywhere. Equation (19) determines
and
up to sign. The most interesting case
for us is that of an infinitesimally thin bundle that issues from a
vertex at an observation event
into the past. For such
bundles we require
and
to be positive
near the vertex and differentiable everywhere; this uniquely
determines
and
everywhere. With
and
fixed, the angles
and
are unique at all points where the
bundle is non-circular; in other words, requiring them to be
continuous determines these angles uniquely along every
infinitesimally thin bundle that is non-circular almost everywhere.
In the representation of Equation (19), the extremal points
of the bundle’s elliptical cross-section are given by the position
vectors
where
means equality up to multiples of
. Hence,
and
give the semi-axes of the elliptical cross-section
and
gives the angle by which the ellipse is rotated with
respect to the Sachs basis (see Figure 3). We call
,
, and
the shape parameters of the bundle, following
Frittelli, Kling, and Newman [121
, 120
]. Instead of
and
one may also use
and
. For the case that the
infinitesimally thin bundle can be embedded in a wave front, the
shape parameters
and
have the following
interesting property (see Kantowski et al. [173
, 84
]).
and
give the principal
curvatures of the wave front in the rest system of the observer
field
which is perpendicular to the Sachs basis. The
notation
and
, which is taken
from [84
], is convenient
because it often allows to write two equations in the form of one
equation with a
sign (see, e.g., Equation (27) or Equation (93) below). The angle
can be directly linked with observations if a light
source emits linearly polarized light (see Section 2.5). If the Sachs basis is
transformed according to Equations (13, 14) and
and
are kept fixed, the Jacobi matrix
changes according to
,
,
. This demonstrates the important fact that the shape
and the size of the cross-section of an infinitesimally thin bundle
has an invariant meaning [292
].
Optical scalars.
Along each infinitesimally thin bundle one defines the deformation matrix
by
This reduces the second-order linear differential equation (16) for
to a first-order non-linear differential equation
for
,
It is usual to decompose
into antisymmetric,
symmetric-tracefree, and trace parts,
This defines the optical scalars
(twist),
(expansion), and
(shear). One usually combines them into two
complex scalars
and
. A change (13, 14) of the Sachs basis
affects the optical scalars according to
and
. Thus,
and
are invariant. If
rewritten in terms of the optical scalars, Equation (23) gives the Sachs equations
One sees that the Ricci curvature term
directly produces
expansion (focusing) and that the conformal curvature term
directly produces shear. However, as the shear
appears in Equation (25), conformal curvature
indirectly influences focusing (cf. Penrose [260
]). With
written in terms of the shape parameters and
written in terms of the optical scalars,
Equation (22) results in
Along
, Equations (25, 26) give a system of 4
real first-order differential equations for the 4 real variables
and
; if
and
are known, Equation (27) gives a system of 4
real first-order differential equations for the 4 real variables
,
, and
. The twist-free solutions
(
real) to Equations (25, 26) constitute a
3-dimensional linear subspace of the 4-dimensional space of all
solutions. This subspace carries a natural metric of Lorentzian
signature, unique up to a conformal factor, and was nicknamed Minikowski space in [20].
Conservation
law.
As the optical tidal matrix
is symmetric, for any two
solutions
and
of the matrix Jacobi
equation (16) we have
where
means transposition. Evaluating the case
shows that for every infinitesimally thin bundle
Thus, there are two types of infinitesimally thin bundles: those
for which this constant is non-zero and those for which it is zero.
In the first case the bundle is twisting (
everywhere) and its cross-section nowhere collapses
to a line or to a point (
and
everywhere). In the second case the bundle must be
non-twisting (
everywhere), because our definition of
infinitesimally thin bundles implies that
and
almost everywhere. A quick
calculation shows that
is exactly the integrability
condition that makes sure that the infinitesimally thin bundle can
be embedded in a wave front. (For the definition of wave fronts see
Section 2.2.) In other words, for an
infinitesimally thin bundle we can find a wave front such that
is one of the generators, and
and
connect
with infinitesimally neighboring
generators if and only if the bundle is twist-free. For a
(necessarily twist-free) infinitesimally thin bundle, points where
one of the two shape parameters
and
vanishes are called caustic
points of multiplicity one, and
points where both shape parameters
and
vanish are called caustic
points of multiplicity two.
This notion coincides exactly with the notion of caustic points, or
conjugate points, of wave fronts as introduced in Section 2.2. The behavior of the optical
scalars near caustic points can be deduced from Equation (27) with
Equations (25, 26). For a caustic point
of multiplicty one at
one finds
By contrast, for a caustic point of multiplicity two at
the equations read (cf. [303])
Infinitesimally thin bundles with
vertex.
We say that an infinitesimally thin bundle has a vertex at
if the Jacobi
matrix satisfies
A vertex is, in particular, a caustic point of multiplicity two. An
infinitesimally thin bundle with a vertex must be non-twisting.
While any non-twisting infinitesimally thin bundle can be embedded
in a wave front, an infinitesimally thin bundle with a vertex can
be embedded in a light cone. Near the vertex, it has a circular
cross-section. If
has a vertex at
and
has a vertex at
, the conservation law (28) implies
This is Etherington’s [104]
reciprocity law. The method by which this law was proven here
follows Ellis [97]
(cf. Schneider, Ehlers, and Falco [299
]). Etherington’s
reciprocity law is of relevance, in particular in view of
cosmology, because it relates the luminosity distance to the area
distance (see Equation (47)). It was
independently rediscovered in the 1960s by Sachs and Penrose
(see [260, 190
]).
The results of this section are the basis for
Sections 2.4, 2.5, and 2.6.