The Principle of Covariance 55In general, tensors can have several contravariant and covariant indices running
over the dimensions of a manifold. In a푑-dimensional manifold a tensor with푟indices
is ofrank푟and has푑푟components. In particular, an푟=1 tensor is a vector, and푟= 0
corresponds to a scalar. An example of a tensor is the assembly of the푛^2 components
푋휇푌휈formed as the products (not the scalar product!) of the푛components of the
vector푋휇with the푛components of the vector푌휈. We have already met the rank 2
tensors휂휇휈with components given by Equation (2.14), and the metric tensor푔휇휈.
Any contravariant vectorAwith components퐴휇can be converted into a covariant
vector by the operation
퐴휈=푔휇휈퐴휇.
The contravariant metric tensor푔휇휈isthematrixinverseofthecovariant푔휇휈:
푔휎휇푔휇휈=훿휈휎. (3.7)The upper and lower indices of any tensor can be lowered and raised, respectively, by
operating with푔휇휈or푔휇휈and summing over repeated indices. Thus a covariant vector
Awith components퐴휇can be converted into a contravariant vector by the operation
퐴휈=푔휇휈퐴휇,For a point particle with mass푚and total energy퐸=훾푚푐^2 , (3.8)according to Einstein’s famous relation, one assigns a momentum four-vector푃with
components푝^0 =퐸∕푐,푝^1 =푝푥,푝^2 =푝푦,푝^3 =푝푧,sothat퐸and the linear momentum
p=m풗become two aspects of the same entity,푃=(퐸∕푐,p).
The scalar product푃^2 is an invariant related to the mass,
푃^2 =휂휇휈푃휇푃휈=퐸2
푐^2−푝^2 =(훾mc)^2 , (3.9)where푝^2 ≡|훾p|^2. For a massless particle like the photon, it follows that the energy
equals the three-momentum times푐.
Newton’s second law in its nonrelativistic form,
F=ma=푚풗̇=ṗ, (3.10)is replaced by the relativistic expression
퐹=d푃
d휏=훾d푃
d푡=훾
(
d퐸
푐d푡,
dp
d푡)
. (3.11)
General Covariance. Although Newton’s second law Equation (3.11) is invariant
under special relativity in any inertial frame, it is not invariant in accelerated frames.
Since this law explicitly involves acceleration, special relativity has to be general-
ized somehow, so that observers in accelerated frames can agree on the value of
acceleration. Thus the next necessary step is to search for quantities which remain
invariant under an arbitrary acceleration and to formulate the laws of physics in
terms of these. Such a formulation is calledgenerally covariant. In a curved space-time