3 General Relativity
Newton’s law of gravitation, Equation (1.28), runs into serious conflict with special
relativity in three different ways. First, there is no obvious way of rewriting it in terms
of invariants, since it only contains scalars. Second, it has no explicit time dependence,
so gravitational effects propagate instantaneously to every location in the Universe,
in fact, also infinitely far outside the horizon of the Universe!
Third, thegravitating mass푚Gappearing in Equation (1.28) is totally independent
of theinert mass푚appearing in Newton’s second law [Equation (2.29)], as we already
noted, yet for unknown reasons both masses appear to be equal to a precision of 10−^13
or better (10−^18 is expected soon). Clearly a theory is needed to establish a formal link
between them. Mach thought that the inert mass of a body was somehow linked to the
gravitational mass of the whole Universe. To be rigorous, he thought that axes of local
nonrotating frames, such as axes of gyroscopes, in their time-evolution precisely fol-
low some average of the motion of matter in the Universe. Einstein, who was strongly
influenced by the ideas of Mach, called thisMach’s principle.Inhisearlyworkongen-
eral relativity he considered it to be one of the basic, underlying principles, together
with the principles of equivalence and covariance, but in his later publications he no
longer referred to it. This may have been a misunderstanding of Einstein because, if
one carefully defines the wordsprecisely followandsome average, Mach’s principle is
a consequence of cosmology with Einstein Gravity.
Facing the above shortcomings of Newtonian mechanics and the limitations of
special relativity Einstein set out on a long and tedious search for a better law of
gravitation valid in the inhomogeneous gravitational field near a massive body, yet
one that would reduce to Newton’s law in some limit. Realizing that the space we live
in was not flat, except locallyequivalentto (a patchwork of flat frames describing)
a curved space-time, the law of gravitation has to be a covariant relation between
mass density and curvature. Thus Einstein proceeded to combine the principle of
equivalence which we describe in Section 3.1 and the principle of general covariance
which we meet in Section 3.2.
Introduction to Cosmology, Fourth Edition. Matts Roos
© 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.