DOC.

9

FORMAL FOUNDATION OF

RELATIVITY

39

the

guv

the minors for

every

guv and divides them

by

the determinant

g

=

|guv|

of

those

guv,

one

obtains

certain

quantities

guv(=

gvu)

for which

we

want to

prove

that

they

form

a

contravariant

symmetrical

tensor.

From this definition it follows

next

that

=

s;,

a

where

guv

signifies

the

quantity

1

or

0

resp., depending upon u

=

v or u

#

v.2

[p.

1040]

Furthermore,

£

SaßdXcfaß

aß

is

a

scalar

which

we can

equate-according

to (10)-with

E

S^a^a^ß

aß\i

and also

equal

to

E

S^g^dxjlxp.

aß)±\

However, according

to the

previous

paragraph,

the

d^

=

£

#/A

ß

are

the

components

of

a

covariant

vector,

and

similarly

of

course

also the

dSv

=

EW*V

a

Our

scalar, therefore,

takes the form

E«MVv

fiv

It

can

be

easily proven

that

guv

is

a

contravariant

tensor;

this is based

upon

the

facts

that the

sum

above is

a

scalar,

the

dEu are

by

nature

arbitrarily

selectable

components

of

a

covariant

four-vector,

and

guv

=

gvu.

Note.

According

to

the

multiplication

theorem

of

determinants,

IE^"V|

=

18J

|gav|.

a

On the other

hand,

(10)

{5}

2According

to

the

previous paragraph,

guv

is

a

mixed tensor

("mixed

fundamental

tensor").