38

DOC. 9 FORMAL FOUNDATION OF RELATIVITY

is

also

a

covariant tensor (of rank five). The proof follows immediately from the

representability of tensors by sums of products of four-vectors:

a*

=

V

=

£

bx1]K2V,

that

is,

A

^

=

^A^A^XV,

and, therefore,

(AaBByua)

is

a

tensor

of rank five.

[p.

1039]

This

operation

is called "outer

multiplication,"

and the

result

the

"outer

product"

of

tensors.

One

sees

that with this

operation,

the character and rank

of

tensors to

be

"multiplied"

are

immaterial.

Furthermore,

the commutative and the associative laws

apply

to

the

sequence

of such

operations.

{4}

The

inner

product

of

tensors.

The

operation

shown in formula

(3b),

with the

tensors

of

rank

one

Av

and

Av,

is called "inner

multiplication,"

and the result the

"inner

product."

Because

of

the

representability

of

tensors

of

higher

rank

by means

of

four-vectors,

this

operation

can

be extended

easily

to tensors

in

general.

If,

for

example,

AvaBy...

is

a

covariant and Aßy... is

a

contravariant

tensor,

both

of

the

same

rank,

then

E

(AA

••)

=

*

aßy

is

a

scalar. The

proof

follows

directly

if

one

puts

^aßy...

~

5-/

^

a^ß^y'''

A

aßy...

=

Y,AaBßCt...,

and then

multiplies

them

taking

(3b)

into

account.

The

mixed

product of tensors.

The

most

general multiplication

of

tensors

results

when

some

indices

are

used for

outer,

others for inner

multiplication.

The

tensors A

and

B

result

in

a

tensor

C

according

to

the

following

scheme:

E/a

a'ß'y'...

A/xv...p

aßy...lmn...\

_

^

Ä/xv...Imn...

V* aßy

...par...

**

a'ß'y'

...rst...)

~

^par...rst...

'

aßy...a'ß'y'

The

proof

that

C

is

a

tensor

follows from

combining

the last

two

proofs

we

sketched above.

§6.

On

Some

Relations

Concerning

the Fundamental

Tensor

guv

The

contravariant

fundamental

tensor.

If

one

forms in the determinantal

scheme

of