Mathematical Tools for Physics

12—Tensors 389

space-time (”events”) can be described by rectangular coordinates(ct, x, y, z), which are concisely denoted by
xi
i= (0, 1 , 2 ,3) where x^0 =ct, x^1 =x, x^2 =y, x^3 =z

The introduction of the factorcintox^0 is merely a question of scaling. It also makes the units the same on all
axes.
The basis vectors associated with this coordinate system point along the directions of the axes.
This manifold isnotEuclidean however so that these vectors are not unit vectors in the usual sense. We
have
~e 0 .~e 0 =− 1 ~e 1 .~e 1 = 1 ~e 2 .~e 2 = 1 ~e 3 .~e 3 = 1

and they are orthogonal pairwise. The reciprocal basis vectors are defined in the usual way,

~ei.~ej=δij

so that
~e^0 =−~e 0 ~e^1 =~e 1 ~e^1 =~e 1 ~e^1 =~e 1

The contravariant (also covariant) components of the metric tensor are

gij=







−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1





=gij (48)

An observer moving in the+xdirection with speedvwill have his own coordinate system with which to
describe the events of space-time. The coordinate transformation is

x′^0 =ct′=

x^0 −vcx^1

√

1 −v^2 /c^2

=

ct−vcx

√

1 −v^2 /c^2

x′^1 =x′=

x^1 −vcx^0

√

1 −v^2 /c^2

=

x−vt

√

1 −v^2 /c^2

x′^2 =y′=x^2 =y x′^3 =z′=x^3 =z

(49)