### 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. [171291]. 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 [302]. 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 [120119]. 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. [17284]). 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 [291].

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. [302])
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 [103] reciprocity law. The method by which this law was proven here follows Ellis [97] (cf. Schneider, Ehlers, and Falco [298]). 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 [260189]).

The results of this section are the basis for Sections 2.4, 2.5, and 2.6.