2.4 Distance measures
In this section we summarize various distance measures that are
defined in an arbitrary spacetime. Some of them are directly
related to observable quantities with relevance for lensing. The
material of this section makes use of the results on
infinitesimally thin bundles which are summarized in
Section 2.3. All of the distance measures to
be discussed refer to a past-oriented lightlike geodesic
from an observation event
to an emission event
(see Figure 4). Some of them depend
on the 4-velocity
of the observer at
and/or on the 4-velocity
of the light source
at
. If a vector field
with
is distinguished on
, we can choose for
the observer an integral curve of
and for the light
sources all other integral curves of
. Then each of the
distance measures becomes a function of the observational
coordinates
(recall Section 2.1).
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 [180
].
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., [154
], 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
If
is affinely parametrized with
and
, one finds that
is given by
This general redshift formula is due
to Kermack, McCrea, and Whittaker [180]. Their proof is
based on the fact that
is a constant for all Jacobi
fields
that connect
with an infinitesimally
neighboring lightlike geodesic. The same proof can be found, in a
more elegant form, in [41] and
in [312
], p. 109. An
alternative proof, based on variational methods, was given by
Schrödinger [300].
Equation (37) is in agreement with
the Hamilton formalism for photons. Clearly, the redshift depends
on the 4-velocity
of the observer and on the 4-velocity
of the light source. If a vector field
with
has been distinguished on
, we may choose one integral curve of
as the observer and all other integral curves of
as the light sources. Then the redshift becomes a
function of the observational coordinates
. For
, the redshift goes to 0,
with a (generalized) Hubble parameter
that depends on spatial direction and on time. For
criteria that
and the higher-order coefficients are
independent of
and
(see [152]). If the redshift is known
for one observer field
, it can be calculated for
any other
, according to Equation (37), just by adding the
usual special-relativistic Doppler factors. Note that if
is given, the redshift can be made to zero along any
one ray
from
by choosing the 4-velocities
appropriately. This shows that
is a reasonable distance measure only for special situations, e.g.,
in cosmological models with
denoting the mean flow of
luminous matter (“Hubble flow”). In any case, the redshift is
directly observable if the light source emits identifiable spectral
lines. For the calculation of Sagnac-like effects, the redshift
formula (37) can be evaluated
piecewise along broken lightlike geodesics [23].
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
The formal definition needs the results of Section 2.3 on infinitesimally thin bundles.
One considers a past-oriented lightlike geodesic
parametrized by affine distance, i.e.,
and
, and along
an infinitesimally thin bundle with vertex at the observer, i.e.,
at
. Then the shape parameters
and
(recall Figure 3) satisfy the initial
conditions
and
. They have
the following physical meaning. If the observer sees a circular
image of (small) angular diameter
on his or her sky,
the (small but extended) light source at affine distance
actually has an elliptical cross-section with extremal diameters
. It is therefore reasonable to call
and
the extremal
angular diameter distances.
Near the vertex,
and
are monotonously
increasing functions of the affine distance,
. Farther away from the vertex, however,
they may become decreasing, so the functions
and
need
not be invertible. At a caustic point of multiplicity one, one of
the two functions
and
changes sign; at a
caustic point of multiplicity two, both change sign (recall
Section 2.3). The image of a light source at
affine distance
is said to have even parity if
and odd parity if
. Images with odd parity show the neighborhood of the
light source side-inverted in comparison to images with even
parity. Clearly,
and
are reasonable
distance measures only in a neighborhood of the vertex where they
are monotonously increasing. However, the physical relevance of
and
lies in the fact that they relate
cross-sectional diameters at the source to angular diameters at the
observer, and this is always true, even beyond caustic points.
and
depend on the 4-velocity
of the observer but not on the 4-velocity
of the source. This reflects the fact that the
angular diameter of an image on the observer’s sky is subject to
aberration whereas the cross-sectional diameter of an
infinitesimally thin bundle has an invariant meaning (recall
Section 2.3). Hence, if the observer’s
worldline
has been specified,
and
are well-defined functions of the
observational coordinates
.
Area
distance.
The area distance
is defined according to the idea
As a formal definition for
, in terms of the extremal
angular diameter distances
and
as functions of affine distance
,
we use the equation
indeed relates, for a bundle with
vertex at the observer, the cross-sectional area at the source to
the opening solid angle at the observer. Such a bundle has a
caustic point exactly at those points where
. The area distance is often called “angular diameter
distance” although, as indicated by Equation (41), the name “averaged
angular diameter distance” would be more appropriate. Just as
and
, the area distance depends on the
4-velocity
of the observer but not on the
4-velocity
of the light source. The area distance
is observable for a light source whose true size is known (or can
be reasonably estimated). It is sometimes convenient to introduce
the magnification or amplification factor
The absolute value of
determines the area distance, and the
sign of
determines the parity. In Minkowski spacetime,
and, thus,
. Hence,
means that a (small but extended) light source at
affine distance
subtends a larger solid angle on the
observer’s sky than a light source of the same size at the same
affine distance in Minkowski spacetime. Note that in a
multiple-imaging situation the individual images may have different
affine distances. Thus, the relative magnification factor of two
images is not directly observable. This is an important difference
to the magnification factor that is used in the quasi-Newtonian
approximation formalism of lensing. The latter is defined by
comparison with an “unlensed image” (see, e.g., [299
]), a notion that
makes sense only if the metric is viewed as a perturbation of some
“background” metric. One can derive a differential equation for the
area distance (or, equivalently, for the magnification factor) as a
function of affine distance in the following way. On every
parameter interval where
has no zeros, the real part
of Equation (27) shows that the area
distance is related to the expansion by
Insertion into the Sachs equation (25) for
gives the focusing
equation
Between the vertex at
and the first conjugate
point (caustic point),
is determined by
Equation (44) and the initial
conditions
The Ricci term in Equation (44) is non-negative if
Einstein’s field equation holds and if the energy density is
non-negative for all observers (“weak energy condition”). Then
Equations (44, 45) imply that
i.e.,
, for all
between the vertex at
and the first conjugate point. In Minkowski
spacetime, Equation (46) holds with equality.
Hence, Equation (46) says that the
gravitational field has a focusing, as opposed to a defocusing,
effect. This is sometimes called the focusing theorem.
Corrected luminosity
distance.
The idea of defining distance measures in terms of bundle
cross-sections dates back to Tolman [323] 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
The redshift factor has its origin in the fact that the definition
of
refers to an affine parametrization adapted to
, and the definition of
refers to an
affine parametrization adapted to
. While
depends on
but not on
,
depends on
but not on
.
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
Thus,
is the relevant quantity for calculating the
luminosity (apparent brightness) of pointlike light sources (see
Equation (52)). For this reason
is called the (uncorrected) luminosity distance. The observation that
the purely geometric quantity
must be modified
by an additional redshift factor to give the energy flux is due to
Walker [342].
depends on the 4-velocity
of the observer
and of the 4-velocity
of the light source.
and
can be viewed as functions
of the observational coordinates
if a
vector field
with
has
been distinguished, one integral curve of
is chosen as the observer, and the other integral
curves of
are chosen as the light sources. In
that case Equation (38) implies that not only
but also
and
are of the form
. Thus, near
the observer all three distance measures coincide with Euclidean
distance in the observer’s rest space.
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 [172]. Its relevance in
view of cosmology was discussed in detail by Rosquist [289
].
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 [289],
and [299
], 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 [190
], 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., [299
]). (For
pre-Dyer-Roeder papers on optics in cosmological models with
inhomogeneities, see the historical notes in [174].) 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 [230]. For
thorough discussions of light propagation in a clumpy universe also
see Pyne and Birkinshaw [283], and Holz and
Wald [161].