**5.2 Redshifts and distances**

Light and particles (galaxies) move along (different) geodesics. When,
in an expanding universe, light travels along a smooth geodesic from
the source towards the observer, its frequency changes with the
changing scale factor according to *z*_{cosm} = / = *R / R*
(Sect. 3.2.1). This is
the global effect.

Redshift *z*_{cosm} is related to the distance *r* of
the object, given by the
invariable fraction of the scale length, and the present scale factor
*R*_{0}. When distance is *measured* by a distance-dependent
object property,
such as apparent brightness or angular extent, the deduced *distance
value* depends not only on the cosmological parameters: Hubble constant
*H*_{0}, deceleration parameter *q*_{0} and
curvature constant *k*, but also on
the *measuring process*. If the process is brightness measurement the
resulting distance is luminosity distance *r*_{M}; angular
diameters give
angular distances *r*_{}, parallax measurements parallax distances
*r*_{p},
etc. The differences occur because the scale factor *R*_{0} enters
differently into the measured quantities.

McCrea (1935,
following
Tolman, 1930,
Walker, 1933,
and others) gives
an extensive discussion of distance determinations which he
introduces:

``. . . . any specific astronomical measurement of `distance' . . . . carried
out in any relativity model of space-time must lead to a result which
depends on the particular operations of measurement.''

For small distances from the observer, *z* can be approximated by the
relativistic or the classical Doppler formula, and the distance *r* is
determined from *v / H*_{0}.

Contributions to the *observed* redshift *z*_{total}
result from the warping of
the smooth geodesic due to local mass concentrations:

*Local effects on the light path* contribute
*z*_{l}. *Local effects*, which
can be described as *peculiar motions of the emitting particle* (galaxy)
and the *observer* along particle geodesics, contribute
*z*_{m}. When they
are sufficiently small, *z* and *r* can again be approximated by the
classical or special relativistic Doppler formula. The superposition
*z*_{total} = z_{cosm} + z_{l} +
z_{m} makes it observationally difficult to separate the
cosmological and the local contributions.