Optical scalars and Sachs equations

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. [171

, 291

]. 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 $s ↦−→ λ (s)$ be an affinely parametrized lightlike geodesic with tangent vector field $K = ˙λ$ . We assume that $λ$ 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 $A,B, ...$ taking the values 1 and 2. An infinitesimally thin bundle (with elliptical cross-section) along $λ$ is a set

Here $δAB$ denotes the Kronecker delta, and $Y1$ and $Y2$ are two vector fields along $λ$ with

such that $Y (s) 1$ , $Y (s) 2$ , and $K (s )$ are linearly independent for almost all $s$ . As usual, $R$ 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 $YA$ traces an infinitesimally neighboring geodesic”. Equation (10

) guarantees that this neighboring geodesic is, again, lightlike and spatially related to $λ$ .

Sachs basis.
For discussing the geometry of infinitesimally thin bundles it is usual to introduce a Sachs basis, i.e., two vector fields $E1$ and $E2$ along $λ$ that are orthonormal, orthogonal to $K = λ˙$ , and parallelly transported,

Apart from the possibility to interchange them, $E1$ and $E2$ are unique up to transformations

where $α$ , $a1$ , and $a2$ are constant along $λ$ . A Sachs basis determines a unique vector field $U$ with $g(U, U ) = − 1$ and $g(U,K ) = 1$ along $λ$ that is perpendicular to $E1$ , and $E2$ . As $K$ is assumed past-oriented, $U$ is future-oriented. In the rest system of the observer field $U$ , 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 $E1$ and $E2$ are often refered to as “screen vectors”.

Jacobi matrix.
With respect to a Sachs basis, the basis vector fields $Y 1$ and $Y 2$ of an infinitesimally thin bundle can be represented as

The Jacobi matrix $D = (DB ) A$ relates the shape of the cross-section of the infinitesimally thin bundle to the Sachs basis (see Figure 3

). Equation (9

) implies that $D$ satisfies the matrix Jacobi equation

where an overdot means derivative with respect to the affine parameter $s$ , and

is the optical tidal matrix, with

Here $Ric$ denotes the Ricci tensor, defined by $Ric(X, Y) = tr(R (⋅,X, Y ))$ , and $C$ 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 $Y1$ , $Y2$ , and $K$ are not everywhere linearly dependent, $det(D )$ does not vanish identically. Linearity of the matrix Jacobi equation implies that $det(D )$ has only isolated zeros. These are the “caustic points” of the bundle (see below).

Shape parameters.
The Jacobi matrix $D$ 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, $D+$ and $D −$ are non-zero almost everywhere. Equation (19

) determines $D+$ and $D −$ 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 $pO$ into the past. For such bundles we require $D +$ and $D −$ to be positive near the vertex and differentiable everywhere; this uniquely determines $D+$ and $D −$ everywhere. With $D+$ and $D−$ fixed, the angles $χ$ and $ψ$ 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 $K$ . Hence, $|D+ |$ and $|D − |$ give the semi-axes of the elliptical cross-section and $χ$ gives the angle by which the ellipse is rotated with respect to the Sachs basis (see Figure 3

). We call $D+$ , $D −$ , and $χ$ the shape parameters of the bundle, following Frittelli, Kling, and Newman [120

, 119

]. Instead of $D+$ and $D−$ one may also use $D+D −$ and $D+ ∕D −$ . For the case that the infinitesimally thin bundle can be embedded in a wave front, the shape parameters $D+$ and $D −$ have the following interesting property (see Kantowski et al. [172

, 84

]). $˙ D+ ∕D+$ and $˙ D − ∕D −$ give the principal curvatures of the wave front in the rest system of the observer field $U$ which is perpendicular to the Sachs basis. The notation $D +$ and $D −$ , 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 $χ$ 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 $Y1$ and $Y2$ are kept fixed, the Jacobi matrix changes according to $&tidle; D ± = D ±$ , $χ&tidle;= χ + α$ , $&tidle; ψ = ψ$ . 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

Figure 3: Cross-section of an infinitesimally thin bundle. The Jacobi matrix (19

) relates the Jacobi fields $Y1$ and $Y2$ that span the bundle to the Sachs basis vectors $E1$ and $E2$ . The shape parameters $D+$ , $D −$ , and $χ$ determine the outline of the cross-section; the angle $ψ$ that appears in Equation (19

) does not show in the outline. The picture shows the projection into the 2-space (“screen”) spanned by $E 1$ and $E 2$ ; note that, in general, $Y 1$ and $Y 2$ have components perpendicular to the screen.

Optical scalars.
Along each infinitesimally thin bundle one defines the deformation matrix $S$ by

This reduces the second-order linear differential equation (16

) for $D$ to a first-order non-linear differential equation for $S$ ,

It is usual to decompose $S$ into antisymmetric, symmetric-tracefree, and trace parts,

This defines the optical scalars $ω$ (twist), $𝜃$ (expansion), and $(σ1,σ2 )$ (shear). One usually combines them into two complex scalars $ϱ = 𝜃 + iω$ and $σ = σ + iσ 1 2$ . A change (13

, 14

) of the Sachs basis affects the optical scalars according to $&tidle;ϱ = ϱ$ and $−2iα &tidle;σ = e σ$ . Thus, $ϱ$ and $|σ |$ are invariant. If rewritten in terms of the optical scalars, Equation (23

) gives the Sachs equations

One sees that the Ricci curvature term $Φ00$ directly produces expansion (focusing) and that the conformal curvature term $ψ0$ directly produces shear. However, as the shear appears in Equation (25

), conformal curvature indirectly influences focusing (cf. Penrose [260

]). With $D$ written in terms of the shape parameters and $S$ written in terms of the optical scalars, Equation (22

) results in

Along $λ$ , Equations (25

, 26

) give a system of 4 real first-order differential equations for the 4 real variables $ϱ$ and $σ$ ; if $ϱ$ and $σ$ are known, Equation (27

) gives a system of 4 real first-order differential equations for the 4 real variables $D ±$ , $χ$ , and $ψ$ . The twist-free solutions ( $ϱ$ 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 $R$ is symmetric, for any two solutions $D 1$ and $D 2$ of the matrix Jacobi equation (16 ) we have

where $T ( )$ means transposition. Evaluating the case $D1 = D2$ 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 ( $ω ⁄= 0$ everywhere) and its cross-section nowhere collapses to a line or to a point ( $D+ ⁄= 0$ and $D− ⁄= 0$ everywhere). In the second case the bundle must be non-twisting ( $ω = 0$ everywhere), because our definition of infinitesimally thin bundles implies that $D+ ⁄= 0$ and $D − ⁄= 0$ almost everywhere. A quick calculation shows that $ω = 0$ 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 $λ$ is one of the generators, and $Y1$ and $Y2$ connect $λ$ 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 $D+$ and $D −$ vanishes are called caustic points of multiplicity one, and points where both shape parameters $D+$ and $D −$ 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 $s = s0$ one finds

----1---- 𝜃(s) = 2(s − s ) (1 + 𝒪 (s − s0)), (30 ) 0 |σ(s)| = ----1----(1 + 𝒪 (s − s0)). (31 ) 2(s − s0)

By contrast, for a caustic point of multiplicity two at $s = s0$ the equations read (cf. [302 ])

1 𝜃 (s) = ------+ 𝒪 (s − s0), (32 ) s − s0 1- ( 2) σ (s) = 3ψ0(s0)(s − s0) + 𝒪 (s − s0) . (33 )

Infinitesimally thin bundles with vertex.
We say that an infinitesimally thin bundle has a vertex at $s = s0$ 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 $D1$ has a vertex at $s1$ and $D2$ has a vertex at $s2$ , 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 [260 , 189

]).

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