where 
is the symmetric metric tensor (depending in general on the
spacetime coordinates 
, 
, 
and 
) and summation
over repeated indices are assumed. In flat space-time using a cartesion
coordinate system, the metric is diagonal with (-1,1,1,1) on the diagonal.
This is called a galilean coordinate system. In curved 4-dimensional spacetime
the choice of a reference system is arbitrary and therefore the laws of physics
must be written in a form that is independent of the choice of reference
system. This is achieved by writing the laws of physics in tensor form.
Consider the transformation from one coordinate system , ,
, to another coordinate system 
, 
, 
,
Any collection of four quantities 
that, under this coordinate
transformation, transform according to
is called a contravariant
four-vector. Similarly any collection of four quantities 
that
transform according to
is called a covariant four vector.
These transformation laws can be generalized to tensors of arbitrary rank.
For example will a collection of 64 quantities
that transform according to
be called a mixed tensor of rank 3 with contravariant 1st and 3rd indices and covariant 2nd index. Changing between contravariant and covariant forms of vectors and tensors is accomplished using the metric tensor as follows
where 
is the contravariant metric tensor, defined by the
relation
From equations (3) and (4) it is obvious that if
a vector (or tensor) vanishes in one coordinate system, it will vanish in any
coordinate system. Thus if the relation 
holds in one coordinate system it will hold in any coordinate system.
The same is true for more complicated tensor equations and therefore if a law
of physics can be expressed as a tensor equation in a given coordinate system
it will have the same tensor form in any other coordinate system. It is said
to have been formulated in covarint form.
From the transformation law in equation (4) it follows that the differentials of a covariant vector transform as
If the transformation is not linear the second term on the right hand side is
non-zero, which means that 
is not a vector. Therefore ordinary
differentation can not be used in the covariant formulation of the laws of
physics. In curved spacetime the result of subtracting the two vectors
and 
at points 
and
is not a vector. To get a vector it is necessary to
parallel translate to and then do the
subtraction. This is done using the following definition of the
covariant derivative of covariant and contravariant tensors
where 
are the Christoffel symbols defined as
It can be shown (Landau & Lifshitz, 1975), that it is always possible to
choose a coordinate system in which is zero at the
same time as the metric is brought to galilean form at a given point. This is
called a locally inertial coordinate system. At this point the covariant
derivative reduces to the ordinary derivative and any equation between tensors
will reduce to the special relativistic form. The general relativistic forms
of the laws of physics can thus be obtained from the special relativistic ones
by exchanging the ordinary derivatives with the covariant derivatives.
