can be explicitly calculated in
spacetimes where the Jacobi fields along lightlike geodesics can be
explicitly determined. This is true, e.g., in spherically symmetric
and static spacetimes where the extremal angular diameter distances
and
can be calculated in terms of integrals
over the metric coefficients. The resulting formulas are given in
Section 4.3 below. Knowledge of
and
immediately gives the area distance
via Equation (41
).
together with the redshift determines
via Equation (48
). Such an explicit
calculation is, of course, possible only for spacetimes with many
symmetries.
By Equation (48), the zeros of
coincide with the zeros of
, i.e., with the caustic points. Hence, in the
ray-optical treatment a point source is infinitely bright
(magnitude
) if it passes through the caustic of the
observer’s past light cone. A wave-optical treatment shows that the
energy flux at the observer is actually bounded by diffraction. In
the quasi-Newtonian approximation formalism, this was demonstrated
by an explicit calculation for light rays deflected by a spheroidal
mass by Ohanian [245]
(cf. [299
], p. 220).
Quite generally, the ray-optical calculation of the energy flux
gives incorrect results if, for two different light paths from the
source worldline to the observation event, the time delay is
smaller than or approximately equal to the coherence time. Then
interference effects give rise to frequency-dependent corrections
to the energy flux that have to be calculated with the help of wave
optics. In multiple-imaging situations, the time delay decreases
with decreasing mass of the deflector. If the deflector is a
cluster of galaxies, a galaxy, or a star, interference effects can
be ignored. Gould [145] suggested that
they could be observable if a deflector of about
Solar masses happens to be close to the line of
sight to a gamma-ray burster. In this case, the angle-separation
between the (unresolvable) images would be of the order
arcseconds (“femtolensing”). Interference effects
could make a frequency-dependent imprint on the total intensity.
Ulmer and Goodman [328]
discussed related effects for deflectors of up to
Solar masses. Femtolensing has not been observed so
far. However, it is an interesting future perspective for lensing
effects where wave optics has to be taken into account. This would
give practical relevance to the theoretical work of Herlt and
Stephani [156, 157] who calculated
gravitational lensing on the basis of wave optics in the
Schwarzschild spacetime.
We now turn to the case of an extended source,
whose surface makes up a 3-dimensional timelike submanifold of the spacetime. In this case the radiation is
characterized by the surface brightness
(= luminosity
per area) at the source and by the intensity
(= energy flux
per solid angle) at the observer. For each
past-oriented light ray from an observation event
and to an event
on
, we can relate
and
in the following way. By definition, the area distance
relates the area at the source to the solid angle at
the observer, so we get from Equation (52
)
. As area distance and luminosity
distance are related by a redshift factor, according to the general
law (48
), this gives the
relation
The law for point sources (52) and the law for
extended sources (54
) refer to bolometric
quantities, i.e., to integration over all frequencies. As every
astronomical observation is restricted to a certain frequency
range, it is actually necessary to consider frequency-specific
quantities. For a point source, one writes
and
, where the specific luminosity
is a function of the emitted frequency
and the specific
flux
is a function of the received frequency
. As
and
are related by a redshift
factor, the frequency-specific version of Equation (52
) reads
As an example for the calculation of the
brightness of images we consider the Schwarzschild spactime (see
Figure 17).