Image distortion

2.5 Image distortion

In special relativity, a spherical object always shows a circular outline on the observer’s sky, independent of its state of motion [257 , 320 ]. In general relativity, this is no longer true; a small sphere usually shows an elliptic outline on the observer’s sky. This distortion is caused by the shearing effect of the spacetime geometry on light bundles. For the calculation of image distortion we need the material of Sections 2.3 and 2.4. For an observer with 4-velocity $UO$ at an event $pO$ , there is a unique affine parametrization $s ↦−→ λ (s)$ for each lightlike geodesic through $pO$ such that $λ(0) = pO$ and $( ) g ˙λ(0),UO = 1$ . Around each of these $λ$ we can consider an infinitesimally thin bundle with vertex at $s = 0$ . The elliptical cross-section of this bundle can be characterized by the shape parameters $D+ (s)$ , $D − (s)$ and $χ(s)$ (recall Figure 3

). In the terminology of Section 2.4, $s$ is the affine distance, and $D+ (s)$ and $D − (s)$ are the extremal angular diameter distances. The complex quantity

is called the ellipticity of the bundle. The phase of $𝜖$ determines the position angle of the elliptical cross-section of the bundle with respect to the Sachs basis. The absolute value of $𝜖(s)$ determines the eccentricity of this cross-section; $𝜖(s) = 0$ indicates a circular cross-section and $|𝜖(s)| = ∞$ indicates a caustic point of multiplicity one. (It is also common to use other measures for the eccentricity, e.g., $|D+ − D − |∕|D+ + D − |$ .) From Equation (27

) with $ϱ = 𝜃$ we get the derivative of $𝜖$ with respect to the affine distance $s$ ,

The initial conditions $D ±(0) = 0$ , $D˙±(0) = 1$ imply

Equation (50

) and Equation (51

) determine $𝜖$ if the shear $σ$ is known. The shear, in turn, is determined by the Sachs equations (25

, 26

) and the initial conditions (32

, 33

) with $s0 = 0$ for $𝜃(= ϱ)$ and $σ$ .

It is recommendable to change from the $𝜖$ determined this way to $𝜀 = − 𝜖$ . This transformation corresponds to replacing the Jacobi matrix $D$ by its inverse. The original quantity $𝜖(s)$ gives the true shape of objects at affine distance $s$ that show a circular image on the observer’s sky. The new quantity $𝜀(s)$ gives the observed shape for objects at affine distance $s$ that actually have a circular cross-section. In other words, if a (small) spherical body at affine distance $s$ is observed, the ellipticity of its image on the observer’s sky is given by $𝜀(s)$ .

By Equations (50 , 51 ), $𝜖$ vanishes along the entire ray if and only if the shear $σ$ vanishes along the entire ray. By Equations (26 , 33 ), the shear vanishes along the entire ray if and only if the conformal curvature term $ψ0$ vanishes along the entire ray. The latter condition means that $K = ˙λ$ is tangent to a principal null direction of the conformal curvature tensor (see, e.g., Chandrasekhar [54 ]). At a point where the conformal curvature tensor is not zero, there are at most four different principal null directions. Hence, the distortion effect vanishes along all light rays if and only if the conformal curvature vanishes everywhere, i.e., if and only if the spacetime is conformally flat. This result is due to Sachs [291 ]. An alternative proof, based on expressions for image distortions in terms of the exponential map, was given by Hasse [148 ].

For any observer, the distortion measure $- 𝜀 = − 𝜖$ is defined along every light ray from every point of the observer’s worldline. This gives $𝜀$ as a function of the observational coordinates $(s,Ψ, Θ, τ)$ (recall Section 2.1, in particular Equation (4 )). If we fix $τ$ and $s$ , $𝜀$ is a function on the observer’s sky. (Instead of $s$ , one may choose any of the distance measures discussed in Section 2.4, provided it is a unique function of $s$ .) In spacetimes with sufficiently many symmetries, this function can be explicitly determined in terms of integrals over the metric function. This will be worked out for spherically symmetric static spacetimes in Section 4.3. A general consideration of image distortion and example calculations can also be found in papers by Frittelli, Kling and Newman [120 , 119 ]. Frittelli and Oberst [126 ] calculate image distortion by a “thick gravitational lens” model within a spacetime setting.

In cases where it is not possible to determine $𝜀$ by explicitly integrating the relevant differential equations, one may consider series expansions with respect to the affine parameter $s$ . This technique, which is of particular relevance in view of cosmology, dates back to Kristian and Sachs [189 ] who introduced image distortion as an observable in cosmology. In lowest non-vanishing order, $𝜀(s,Ψ, Θ, τO)$ is quadratic with respect to $s$ and completely determined by the conformal curvature tensor at the observation event $pO = γ (τO)$ , as can be read from Equations (50 , 51 , 33 ). One can classify all possible distortion patterns on the observer’s sky in terms of the Petrov type of the Weyl tensor [56 ]. As outlined in [56 ], these patterns are closely related to what Penrose and Rindler [262 ] call the fingerprint of the Weyl tensor. At all observation events where the Weyl tensor is non-zero, the following is true. There are at most four points on the observer’s sky where the distortion vanishes, corresponding to the four (not necessarily distinct) principal null directions of the Weyl tensor. For type $N$ , where all four principal null directions coincide, the distortion pattern is shown in Figure 5 .

Figure 5: Distortion pattern. The picture shows, in a Mercator projection with $Φ$ as the horizontal and $Θ$ as the vertical coordinate, the celestial sphere of an observer at a spacetime point where the Weyl tensor is of Petrov type $N$ . The pattern indicates the elliptical images of spherical objects to within lowest non-trivial order with respect to distance. The length of each line segment is a measure for the eccentricity of the elliptical image, the direction of the line segment indicates its major axis. The distortion effect vanishes at the north pole $Θ = 0$ which corresponds to the fourfold principal null direction. Contrary to the other Petrov types, for type N the pattern is universal up to an overall scaling factor. The picture is taken from [56 ] where the distortion patterns for the other Petrov types are given as well.

The distortion effect is routinely observed since the mid-1980s in the form of arcs and (radio) rings (see [298 , 275 , 343 ] for an overview). In these cases a distant galaxy appears strongly elongated in one direction. Such strong elongations occur near a caustic point of multiplicity one where $|𝜀| → ∞$ . In the case of rings and (long) arcs, the entire bundle cannot be treated as infinitesimally thin, i.e., a theoretical description of the effect requires an integration. For the idealized case of a point source, images in the form of (1-dimensional) rings on the observer’s sky occur in cases of rotational symmetry and are usually called “Einstein rings” (see Section 4.3). The rings that are actually observed show extended sources in situations close to rotational symmetry.

For the majority of galaxies that are not distorted into arcs or rings, there is a “weak lensing” effect on the apparent shape that can be investigated statistically. The method is based on the assumption that there is no prefered direction in the universe, i.e., that the axes of (approximately spheroidal) galaxies are randomly distributed. So, without a distortion effect, the axes of galaxy images should make a randomly distributed angle with the $(Ψ,Θ )$ grid on the observer’s sky. Any deviation from a random distribution is to be attributed to a distortion effect, produced by the gravitational field of intervening masses. With the help of the quasi-Newtonian approximation, this method has been elaborated into a sophisticated formalism for determining mass distributions, projected onto the plane perpendicular to the line of sight, from observed image distortions. This is one of the most important astrophysical tools for detecting (dark) matter. It has been used to determine the mass distribution in galaxies and galaxy clusters, and more recently observations of image distortions produced by large-scale structure have begun (see [22 ] for a detailed review). From a methodological point of view, it would be desirable to analyse this important line of astronomical research within a spacetime setting. This should give prominence to the role of the conformal curvature tensor.

Another interesting way of observing weak image distortions is possible for sources that emit linearly polarized radiation. (This is true for many radio galaxies. Polarization measurements are also relevant for strong-lensing situations; see Schneider, Ehlers, and Falco [298 ], p. 82 for an example.) The method is based on the geometric optics approximation of Maxwell’s theory. In this approximation, the polarization vector is parallel along each ray between source and observer [88 ] (cf., e.g., [226 ], p. 577). We may, thus, use the polarization vector as a realization of the Sachs basis vector $E1$ . If the light source is a spheroidal celestial body (e.g., an elliptic galaxy), it is reasonable to assume that at the light source the polarization direction is aligned with one of the axes, i.e., $2χ (s)∕π ∈ ℤ$ . A distortion effect is verified if the observed polarization direction is not aligned with an axis of the image, $2χ(0)∕π ∈∕ ℤ$ . It is to be emphasized that the deviation of the polarization direction from the elongation axis is not the result of a rotation (the bundles under consideration have a vertex and are, thus, twist-free) but rather of successive shearing processes along the ray. Also, the effect has nothing to do with the rotation of an observer field. It is a pure conformal curvature effect. Related misunderstandings have been clarified by Panov and Sbytov [254 , 255 ]. The distortion effect on the polarization plane has, so far, not been observed. (Panov and Sbytov [254 ] have clearly shown that an effect observed by Birch [31 ], even if real, cannot be attributed to distortion.) Its future detectability is estimated, for distant radio sources, in [317 ].