Annu. Rev. Astron. Astrophys. 1984. 22:
157-184
Copyright © 1984 by Annual Reviews. All rights reserved |

The standard theory assumes that electromagnetic radiation ("light" for
short) travels on null geodesics
(*x*^{a}()
: *k*^{a}_{; b}*k*^{b} = 0,
*k*^{a}*k*_{a} = 0, where
*k*^{a}
*dx*^{a} /
*d*)
in the space-time of general relativity and is characterized by
the relations
*c* = *v*,
*E* = *h*, relating
the wavelength
, frequency
,
speed *c*, and energy *E* of a photon. Then the redshift
*z*
/
(where
is the change in wavelength measured from spectra for light emitted at
wavelength ) is a
direct consequence of measurable time dilation
between the source and the observer
(25,
91,
111).
There can be various sources for this time dilation.

In the cosmological context, there is assumed to be a well-defined
average motion of matter at each point in the Universe, representing the
"standard" motion of matter there
(24,
25);
galaxies moving in this
standard way, characterized by a four-velocity *u*^{a} (where
*u*_{a} *u*^{a} = - 1),
are called "fundamental particles," and an observer moving with
four-velocity *u*^{a} is denoted as a "fundamental
observer." The
*cosmological redshift* *z*_{c} is the redshift
observed when both the source
and observer move with this velocity (at their respective space-time
positions) and there are no local gravitational redshift effects.
However, there will be *local Doppler redshifts*
*z*_{Ds}, *z*_{Do} if the source or
observer is not moving at the standard velocity, and there may be local
*gravitational redshifts*
*z*_{Gs}, *z*_{Go} generated at the
source or the observer
by local gravitational potential wells. (Notice that in each case the
effect could be a blueshift rather than redshift; this corresponds to a
negative value of *z*, and we shall assume this possibility without
explicitly mentioning it.) These contributions to the total redshift
*z*
are not distinguishable from each other by any direct measurement of the
received radiation; the measurable quantity is the total redshift *z*,
given in terms of the other quantities by

(1) |

The cosmological redshift depends on the path of the light ray from the
source to the observer. One can distinguish two contributions. The first
arises from the increase in relative distance of fundamental observers
due to integrated local relative motion of matter [characterized by the
expansion tensor
_{ab} =
*h*_{a}^{c}
*h*_{b}^{a}*u*_{(c; d)}
(24,
25),
where
*h*_{ab}
*g*_{ab} + *u*_{a}*u*_{b}
projects into the rest-space orthogonal to *u*^{a}]. The second
arises from
the noninertial motion of the fundamental observers, characterized by a
nonzero acceleration vector
^{a}
*u*^{a}_{; b} *u*^{b} (indicating that
their
world-lines are nongeodesic). The change of redshift when moving a
distance *dl*, measured by a fundamental observer along a null
geodesic in direction
*e*^{a} = *h*_{b}^{a}
*k*^{b} / (*u*_{c}*k*^{c}), obeys
the equation
(24,
25)

(2a) |

Thus one can write

(2b) |

where *z*_{Ec} is the usual Hubble (expansion) term
(obtained by integrating the first term in Equation 2a), and
*z*_{Gc} is what might be termed a
"cosmological gravitational redshift" (obtained by integrating the
second term).

In the FLRW models, it is assumed that the other redshift contributions are very small compared with the basic cosmological expansion effect. Then

(3) |

where *R*(*t*) is the "radius function" of the Universe that
represents the
time behavior of the distance between any pair of fundamental particles
as the Universe evolves (by the homogeneity and isotropy of these
models, the distances between all pairs of fundamental particles scale
in the same way); and
*t*_{e}, *t*_{0} are the times of emission and
observation
of the light, respectively. The observation of systematic redshifts
indicates that the Universe is expanding at the present time
*t*_{0}; so
*H*_{0}
(*R*^{ -1} *dR* / *dt*)|_{0} > 0.