DOC. 30 FOUNDATION

OF GENERAL RELATIVITY

163

bracket vanishes. The result then

follows

by

equation (11).

This rule

applies correspondingly

to tensors

of

any

rank and

character, and

the

proof

is

analogous

in

all

cases.

The

rule

may

also be

demonstrated

in

this

form

:

If

Bu

and

Cv are

any vectors,

and

if,

for all values of

these,

the

inner

product

AuvBuCv

is

a

scalar,

then

Auv

is

a

covariant

tensor. This latter

proposition

also holds

good even

if

only

the

more

special

assertion

is

correct,

that with

any

choice of

the

four-vector

Bu

the inner

product

AuvBuBv

is

a

scalar,

if

in addition it

is

known that

Auv

satisfies

the

condition of

symmetry

Auv

=

Auv.

For

by

the method

given

above

we

prove

the

tensor

character

of

(Auv

+

Auv),

and from

this the

tensor

character

of

Auv

follows

on

account of

symmetry.

This

also

can

be

easily generalized

to

the

case

of

covariant

and contravariant tensors

of

any

rank.

Finally,

there

follows

from what has

been

proved,

this

law,

which

may

also be

generalized

for

any

tensors:

If

for

any

choice

of

the

four-vector

Bv

the

quantities

AuvBv

form

a

tensor of

the

first rank,

then

Auv

is

a

tensor of

the

second

rank.

For,

if

Cu

is

any four-vector,

then

on

account

of

the

tensor

character

of

AuvBv,

the inner

product

AuvBvCu

is

a

scalar for

any

choice of the

two four-vectors

Bv

and

Cu.

From

which the

proposition

follows.

§

8. Some

Aspects

of

the

Fundamental

Tensor

guv

The

Covariant Fundamental Tensor.-In the

invariant

expression

for

the

square

of

the linear

element,

ds2

=

guvdxudxv,

the

part played

by

the

dxu

is

that

of

a

contravariant

vector

which may

be chosen at

will. Since further,

guV =

gvu,

it

follows

from the considerations

of

the

preceding paragraph

that

guv

is

a

covariant tensor

of

the

second

rank. We

call

it

the

"fundamental

tensor." In

what

follows

we

deduce

some

properties

of

this tensor

which,

it

is true,

apply

to

any

tensor of

the

second

rank. But

as

the fundamental

tensor

plays a special

part

in

our

theory,

which has

its

physical

basis

in the

peculiar

effects of

gravitation,

it

so

happens

that

the

relations

to be

developed

are

of

importance

to

us only

in the

case

of

the fundamental

tensor.