According to the theory of Special Relativity, a simplified form of the theory of General Relativity, which can't deal with accelerations, the separation between two points in spacetime is:
According to the theory of General Relativity, this should be:
This is a tensor equation with the matrix of
, which is called the metric of
spacetime. It is easily seen, that (1) is a special case of (2), since
we obtain (1) from (2) if:
In the rest of this discussion, it is assumed, that the universe obeys the Robertson-Walker-metric, which is given by:
In this equation R(t) is a scaling factor, which is a function of
time. The polar coordinates are commoving coordinates: they are fixed
on a certain object, which serves as an origin. One might think, that
this is very troublesome, since there are a lot of objects in the
universe and all can be taken as origin. The argument is certainly
correct, but the conclusion is wrong, since this model assumes that
the universe is isotropic, which means, that it doesn't matter in
which direction one looks, it all looks the same and homogeneous, which
means, that it doesn't matter where you are, since the universe is
physically the same at all points. It is true, that this is incorrect,
since there are vast regions in the universe, which are almost
absolutely empty, but also regions, in which there is a lot
of mass, like galaxies, H-II-clouds, and so on. On a cosmological scale
however, this is, I think, a reasonable assumption, since on a large
scale these more or less local conditions won't be very important.
If we simplify our calculations by using only those term of (2) for
which 
and assume a Robertson-Walker-
metric, then we obtain (zeilik, Gregory and Smith, 1992):
In these equation, P(t) is the physically measurable pressure in
the universe, U(t) the internal energy of the universe, R0
is the present value of R and 
is the
cosmological constant, which represents the energy-density of the
universe. After some algebra, one can derive the following equation
from (5) and (6):
If we assume, that (Zeilik, Gregory and Smith, 1992):
we can show, using (7), that:
To obtain (8), two more assumptions were needed. Since (8) is
needed to get the formula, these conditions are also assumed when this
method is used. Are they reasonable?
As mentioned above, P denotes the physically measurable pressure in
the universe. Since there are vast regions in the universe, which are
almost totally empty, the average pressure will be very low, so I
think, it is reasonable to put this equal to zero.
The other assumption is somewhat more arbitrary, but the only thing
this assumption states, is, that the internal energy is proportional
to the density, which seems to me like a very reasonable
assumption. The constant (in this case the square of the lightspeed)
can be chosen arbitrary, since, as can be seen from (8), it drops out
whatever its value.
The main idea of the method involving redshifts is. that one selects a, for example, spherical region of the universe. Since the universe is expanding, this sphere will expand too. So the next assumption that goes into this theory is, that the expansion of the universe is uniform. This assumption is reasonable, if one takes the spherical region mentioned above large, since taking a large region flattens out the influence of local deviations.
In the line of sight, the objects on the surfaces of this spherical
region will have redshifts of z-
and z+ respectively and the projection of this sphere
on the sky will be a circle with radius
, so we can determine the ratio /z . This is the
ratio, which the formula relates to q0, 
and the redshift of the center of the
sphere, z.
As mentioned above, a Robertson-Walker-metric is assumed. The
assumption of isotropy won't cause any trouble: we can even verify it
by doing measurements in different regions of the sky. The other
assumption, homogenity, is somewhat trickier, since we need some
objects to measure the redshift (we can't measure the redshift of
empty space). In places, where there are objects to observe, the local
mass density will be larger than the average mass density of the
universe, so if one calculates a value for
, one might get a higher value for
than the actual value. One may try
to rule out those kind of local influences by taking a vast region of
the sky, because this will flatten out the local influences, but I
think, that the value of resulting
from this method, will be more often higher than lower than the actual
value. Another problem is, that when I tried to rederive the final
equation, which relates to the ratio
/z I needed
to make the assumption, that was small
enough to assume, that cos() was about
equal to unity, so the region can't be taken too vast.
Another assumption is, that the dynamical evolution of the universe can be expressed by (Alcock and Paczynski, 1979):
This result can be derived from the field equations of Einstein's theory of General Relativity, but one only obtains this result if (8) is true. The assumptions, that are needed to find (8) are already mentioned above.
Since the final goal is to determine ,
the ratio /z need to depend on . However, it isn't possible to express it
in terms of and some well determined
constant(s) alone. At least one other not to well defined cosmological
parameter is left in the equation. In this case, this is the
decelleration parameter, q0. To arrive at this result, one
has to make the following substitutions (Alcock and Paczynski,
1979):
If one traces a null-geodesic of this geometry (a null-geodesic is the path, which connects all points for which dl^2=0) and using the theory above, one finally arrives at (Alcock and Paczynski, 1979):
This expression is only valid if the curvature of the universe,
determined by the value of a constant k, follows from k=1. There are
also formulae for other curvatures (Alcock and Paczynski 1979), but
I'll only use this one, since all measurements and estimations
indicate, that <1, so k seems to be
equal to 1, so I only mention this one here.
Since we have arrived at the final result, one can now return to the main story.