5 3 4 A P P E N D I X E
where the ’s are functions of position in the four space. This is a generalization of the
expression in the special theory of relativity:
(33)
How must we construct our equations so that they will remain valid for an arbitrary
Gaussian co-ordinate system? We must first define the concept of a VECTOR. This is de-
fined by its contravariant components for a given co-ordinate system, which are trans-
formed by the equations:
(34) similar to (35)
We also have covariant four-vectors. is a covariant four-vector if is invariant.
The law of transformation is:
TENSORS are transformed like the products of several vectors. E.g. behaves like
the products ; like Similarly we form tensors of arbitrarily high orders.
The product of two tensors is also a tensor. E.g. it is easily shown that
formed from the two tensors and satisfies the proper equations of transformation.
The generalization of CONTRACTION is that of equating a contravariant index (super-
script) and a covariant index (subscript) and summing for this variable, e.g.
(36)
If is a tensor, is a tensor of the second order.
Parallels exist in Euclidean manifolds, but not in
curved ones. But in a neighborhood we can express a
parallel translation analytically. The components of the
translated vector differ from those of the vector by
, given by:
(37)
The components of a vector moving parallel to itself are continuous point functions. The
differences themselves are the components of a vector. The tensor defines the par-
allel transformation.
To have the addition of small translations commutative, i.e.
to have a small parallelogram “close up”, must be sym-
metric. Also if a and b are parallel-translations,
must be invariant, or:
(38)
gij
g11 g22 g33 g44 – 1 ;– = = = = gij 0= i j
a
a
x
x
---------a - = dx
x
x
---------dx- =
b b dx
b
x
x
b =
T
a b T a b .
T t u =
t u
T T =
[p. 7]
a
x
a
a
x
a
a
a a x –=
a
a
a
b
b g a b
g a b 0=