Tests of Special Relativity 45The Tully–Fisher relation for spiral galaxies is calibrated by nearby spiral galax-
ies having Cepheid calibrations, and it can then be applied to spiral galaxies out
to 150Mpc. Elliptical galaxies do not rotate, they are found to occupy afundamen-
tal planein which an effective radius is tightly correlated with the surface brightness
inside that radius and with the central velocity dispersion of the stars. In principle,
this method could be applied out to푧≈1, but in practice stellar evolution effects and
the nonlinearity of Hubble’s law limit the method to푧≲ 0 .1, or about 400Mpc.
For more details on distance measurements the reader is referred to the excellent
treatment in the book by Peacock [5].
2.4 Tests of Special Relativity
Special relativity is an inseparable part of quantum field theory which describes the
world of elementary particles with an almost incredible precision. Particle physics
has tested special relativity in thousands of different experiments without finding a
flaw: the Lorentz invariance is locally exact. But at astronomical and cosmological
scales the local Lorentz invariance has to be replaced by General Relativity. The quib-
ble about whether special relativity is generally true and testable at cosmological dis-
tances and time scales is therefore meaningless.
Special relativity really contains only one parameter,c, the velocity of lightin vacuo,
which has the dimension of length/time. Couldcbe variable, or isceven measurable
at all? One is free to choose푐=1 locally because that only implies a rescaling of the
units of length. To explain problematic observations the possibility of avariable speed
of lighthas sometimes been invoked. Also Newton’s constant of gravitation,G,could
in principle be variable. Already Einstein admitted that the value ofGcould depend
on the local strength of the gravitational field. In order to avoid trivial rescaling of
units, one must test the simultaneous variation ofc,Gand the fine structure constant
which can be combined to a dimensionless number. Note thatcandGenter in the
combination퐺∕푐^4 in the Einstein Equation (3.29).
A measurement of the radius of Mercury has produced no time variation ofc
푐̇
푐
= 0 ± 2 × 10 −^12 yr−^1. (2.66)The white dwarf star Stein 2051B within our Galaxy also provides only a limit when
combined with the upper limit of the variability of the fine structure constant,
퐻 0 푐̇
푐= 0 ± 2. 1 × 10 −^3 yr (2.67)using퐻 0 = 1. 5 × 1010 yr.
Determinations of the time variation ofGhave also given null results. The strongest
constraint due to a lunar laser ranging experiment gives
|퐺̇
퐺
|⩽ 1. 3 × 10 −^12 yr−^1. (2.68)