Affine distance.

There is a unique affine parametrization for each lightlike geodesic through the observation
event such that and . Then the affine parameter itself
can be viewed as a distance measure. This affine distance has the desirable features that it
increases monotonously along each ray and that it coincides in an infinitesimal neighborhood
of with Euclidean distance in the rest system of . The affine distance depends on
the 4-velocity of the observer but not on the 4-velocity of the light source. It is a
mathematically very convenient notion, but it is not an observable. (It can be operationally realized in
terms of an observer field whose 4-velocities are parallel along the ray. Then the affine distance
results by integration if each observer measures the length of an infinitesimally short part of the
ray in his rest system. However, in view of astronomical situations this is a purely theoretical
construction.) The notion of affine distance was introduced by Kermack, McCrea, and Whittaker [179].

Travel time.

As an alternative distance measure one can use the travel time. This requires the choice of a time function,
i.e., of a function that slices the spacetime into spacelike hypersurfaces . (Such a time
function globally exists if and only if the spacetime is stably causal; see, e.g., [153], p. 198.) The travel time
is equal to , for each on the past light cone of . In other words, the intersection of
the light cone with a hypersurface determines events of equal travel time; we call these
intersections “instantaneous wave fronts” (recall Section 2.2). Examples of instantaneous wave fronts are
shown in Figures 13, 18, 19, 27, and 28. The travel time increases monotonously along each ray. Clearly, it
depends neither on the 4-velocity of the observer nor on the 4-velocity of the light
source. Note that the travel time has a unique value at each point of ’s past light cone,
even at events that can be reached by two different rays from . Near the travel time
coincides with Euclidean distance in the observer’s rest system only if is perpendicular to
the hypersurface with . (The latter equation is true if along the
observer’s world line the time function coincides with proper time.) The travel time is not
directly observable. However, travel time differences are observable in multiple-imaging situations
if the intrinsic luminosity of the light source is time-dependent. To illustrate this, think of a
light source that flashes at a particular instant. If the flash reaches the observer’s wordline
along two different rays, the proper time difference of the two arrival events is directly
measurable. For a time function that along the observer’s worldline coincides with proper time,
this observed time delay gives the difference in travel time for the two rays. In view of
applications, the measurement of time delays is of great relevance for quasar lensing. For the double
quasar 0957+561 the observed time delay is about 417 days (see, e.g., [275], p. 149).

Redshift.

In cosmology it is common to use the redshift as a distance measure. For assigning a redshift
to a lightlike geodesic that connects the observation event on the worldline
of the observer with the emission event on the worldline of the light source, one
considers a neighboring lightlike geodesic that meets at a proper time interval
from and at a proper time interval from . The redshift is defined as

Angular diameter distances.

The notion of angular diameter distance is based on the intuitive idea that the farther an object is away the
smaller it looks, according to the rule

Area distance.

The area distance is defined according to the idea

Corrected luminosity distance.

The idea of defining distance measures in terms of bundle cross-sections dates back to Tolman [322] and
Whittaker [351]. Originally, this idea was applied not to bundles with vertex at the observer but rather to
bundles with vertex at the light source. The resulting analogue of the area distance is the so-called
corrected luminosity distance . It relates, for a bundle with vertex at the light source, the
cross-sectional area at the observer to the opening solid angle at the light source. Owing to
Etherington’s reciprocity law (35), area distance and corrected luminosity distance are related by

Luminosity distance.

The physical meaning of the corrected luminosity distance is most easily understood in the
photon picture. For photons isotropically emitted from a light source, the percentage that hit a
prescribed area at the observer is proportional to . As the energy of each photon
undergoes a redshift, the energy flux at the observer is proportional to , where

Parallax distance.

In an arbitrary spacetime, we fix an observation event and the observer’s 4-velocity . We consider
a past-oriented lightlike geodesic parametrized by affine distance, and .
To a light source passing through the event we assign the (averaged) parallax distance
, where is the expansion of an infinitesimally thin bundle with vertex
at . This definition follows [171]. Its relevance in view of cosmology was discussed in
detail by Rosquist [288]. can be measured by performing the standard trigonometric
parallax method of elementary Euclidean geometry, with the observer at and an assistant
observer at the perimeter of the bundle, and then averaging over all possible positions of the
assistant. Note that the method refers to a bundle with vertex at the light source, i.e., to light
rays that leave the light source simultaneously. (Averaging is not necessary if this bundle is
circular.) depends on the 4-velocity of the observer but not on the 4-velocity of the light
source. To within first-order approximation near the observer it coincides with affine distance
(recall Equation (32)). For the potential obervational relevance of see [288], and [298],
p. 509.

In view of lensing, , , and are the most important distance measures because they are related to image distortion (see Section 2.5) and to the brightness of images (see Section 2.6). In spacetimes with many symmetries, these quantities can be explicitly calculated (see Section 4.1 for conformally flat spactimes, and Section 4.3 for spherically symmetric static spacetimes). This is impossible in a spacetime without symmetries, in particular in a realistic cosmological model with inhomogeneities (“clumpy universe”). Following Kristian and Sachs [189], one often uses series expansions with respect to . For statistical considerations one may work with the focusing equation in a Friedmann–Robertson–Walker spacetime with average density (see Section 4.1), or with a heuristically modified focusing equation taking clumps into account. The latter leads to the so-called Dyer–Roeder distance [86, 87] which is discussed in several text-books (see, e.g., [298]). (For pre-Dyer–Roeder papers on optics in cosmological models with inhomogeneities, see the historical notes in [173].) As overdensities have a focusing and underdensities have a defocusing effect, it is widely believed (following [344]) that after averaging over sufficiently large angular scales the Friedmann–Robertson–Walker calculation gives the correct distance-redshift relation. However, it was argued by Ellis, Bassett, and Dunsby [99] that caustics produced by the lensing effect of overdensities lead to a systematic bias towards smaller angular sizes (“shrinking”). For a spherically symmetric inhomogeneity, the effect on the distance-redshift relation can be calculated analytically [231]. For thorough discussions of light propagation in a clumpy universe also see Pyne and Birkinshaw [284], and Holz and Wald [160].

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