2 Special Relativity
The foundations of modern cosmology were laid during the second and third decade
of the twentieth century: on the theoretical side by Einstein’s theory of general relativ-
ity, which represented a deep revision of current concepts; and on the observational
side by Hubble’s discovery of the cosmic expansion, which ruled out a static Uni-
verse and set the primary requirement on theory. Space and time are not invariants
under Lorentz transformations, their values being different to observers in different
inertial frames. Nonrelativistic physics uses these quantities as completely adequate
approximations, but in relativistic frame-independent physics we must find invariants
to replace them. This chapter begins, in Section 2.1, with Einstein’s theory of special
relativity, which gives us such invariants.
In Section 2.2 we generalize the metrics in linear spaces to metrics in curved spaces,
in particular the Robertson–Walker metric in a four-dimensional manifold. This gives
us tools to define invariant distance measures in Section 2.3, which are the key to Hub-
ble’s parameter. To conclude we discuss briefly tests of special relativity in Section 2.4.
2.1 Lorentz Transformations
In Einstein’s theory of special relativity one studies how signals are exchanged
between inertial frames in linear motion with respect to each other with constant
velocity. Einstein made two postulates about such frames:
(i) the results of measurements in different frames must be identical;
(ii) light travels by a constant speed,푐,invacuo,inallframes.The first postulate requires that physics be expressed in frame-independent invariants.
The latter is actually a statement about the measurement of time in different frames,
as we shall see shortly.
Introduction to Cosmology, Fourth Edition. Matts Roos
© 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.