54 General Relativity
Equation (3.1) clearly shows that light emerging from a star with mass푀is
redshifted in proportion to푀. Thus part of the redshift observed is due to this gravi-
tational effect. From this we can anticipate the existence of stars with so large a mass
that their gravitational field effectively prohibits radiation to leave. These are theblack
holesto which we shall return later.
Superluminal Photons. A cornerstone in special relativity is that no material par-
ticle can be accelerated beyond푐, no physical effect can be propagated faster than푐,
and no signal can be transmitted faster than푐. It is an experimental fact that no parti-
cle has been found travelling at superluminal speed, but a name for such particles has
been invented,tachyons. Special relativity does not forbid tachyons, but if they exist
they cannot be retarded to speeds below푐. In this sense the speed of light constitutes
a two-way barrier: an upper limit for ordinary matter and a lower limit for tachyons.
On quantum scales this may be violated, since the photon may appear to possess
a mass caused by its interaction with virtual electron–positron pairs. In sufficiently
strong curvature fields, the trajectory of a photon may then be distorted through the
interaction of gravity on this mass and on the photon’s polarization vector, so that
the photon no longer follows its usual geodesic path through curved space-time. The
consequence is that SPE may be violated at quantum scales, the photon’s lightcone
is changed, and it may propagate with superluminal velocity. This effect, calledgrav-
itational birefringence, can occur because general relativity is not constructed to obey
quantum theory. It may still modify our understanding of the origin of the Universe,
when the curvature must have been extreme, and perhaps other similar situations like
the interior of black holes. For a more detailed discussion of this effect, see Shore [2]
and references therein.
3.2 The Principle of Covariance
Tensors. In four-dimensional space-time all spatial three-vectors have to acquire
a zeroth component just like the line element four-vector d푠in Equations (2.7)
and (2.14). A vectorAwith components퐴휇in a coordinate system푥휇can be expressed
in a transformed coordinate system푥′휈as the vectorA′with components
퐴′휈=휕푥′휈
휕푥휇
퐴휇, (3.5)
where summation over the repeated index휇is implied, just as in Equation (2.14). A
vector which transforms in this way is said to becontravariant, which is indicated by
theupper indexfor the components퐴휇.
A vectorBwith components퐵휇in a coordinate system푥휇, which transforms in
such a way that
퐵휈′=휕푥휇
휕푥′휈
퐵휇, (3.6)
is calledcovariant. This is indicated by writing its components with alower index.
Examples of covariant vectors are the tangent vector to a curve, the normal to a sur-
face, and the four-gradient of a four-scalar휙,휕휙∕휕푥휇.