It is recommendable to change from the
determined this way to
. This transformation
corresponds to replacing the Jacobi matrix
by its inverse. The original quantity
gives the true shape of objects at affine distance
that show a circular image on the observer’s sky. The
new quantity
gives the observed shape for objects at
affine distance
that actually have a circular
cross-section. In other words, if a (small) spherical body at
affine distance
is observed, the ellipticity of its
image on the observer’s sky is given by
.
By Equations (50, 51
),
vanishes along the entire ray if and only if the shear
vanishes along the entire ray. By Equations (26
, 33
), the shear vanishes
along the entire ray if and only if the conformal curvature term
vanishes along the entire ray. The latter condition
means that
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 [292]. An alternative
proof, based on expressions for image distortions in terms of the
exponential map, was given by Hasse [149].
For any observer, the distortion measure is defined along every light ray from every point of
the observer’s worldline. This gives
as a function of the
observational coordinates
(recall
Section 2.1, in particular Equation (4
)). If we fix
and
,
is a function on the observer’s sky.
(Instead of
, one may choose any of the distance
measures discussed in Section 2.4, provided it is a unique
function of
.) 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 [121
, 120
]. Frittelli and
Oberst [127] calculate image
distortion by a “thick gravitational lens” model within a spacetime
setting.
In cases where it is not possible to determine
by explicitly integrating the relevant differential
equations, one may consider series expansions with respect to the
affine parameter
. This technique, which is of particular
relevance in view of cosmology, dates back to Kristian and
Sachs [190] who introduced image
distortion as an observable in cosmology. In lowest non-vanishing
order,
is quadratic with respect to
and completely determined by the conformal curvature tensor at the
observation event
, 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
, where all four principal null directions coincide,
the distortion pattern is shown in Figure 5
.
|
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 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 [299], 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.,
[225
], p. 577). We
may, thus, use the polarization vector as a realization of the
Sachs basis vector
. 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.,
. A
distortion effect is verified if the observed polarization direction is not
aligned with an axis of the image,
. 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 [318].