46 DOC.

9

FORMAL FOUNDATION OF RELATIVITY

the fact that the transformation

equations

of the

tensor

components are

linear

and

homogeneous.

This in

turn

causes

the

tensor

components

to

vanish relative

to

any

coordinate

system

if

they

vanish relative

to

just

one

such

system. Therefore,

once a

group

of

equations

of

a

physical system

has been

brought

into

a

form which shows

the

vanishing

of

all

components

of

a

tensor,

then this

equation system

is

independent

of

the

choice of

the

coordinate

system.

In order

to

establish such

equations

of

tensors,

one

has to know the laws

by

which

new

tensors

can

be formed from

given ones.

It

has

already

been discussed how this

can

be done in

an

algebraic manner.

We still

have to derive the laws

by

which

new

tensors

can

be formed from known

ones

through

differentiation. The laws of these differential

expressions

have

already

been

given by

Christoffel,

Ricci, and

Levi-Civita.

I

give

here

a

particularly simple

derivation for

this,

which

appears

to

be

new.

[p.

1047]

All differential

operations

on

tensors

can

be traced

to

the so-called "extension."

In the

case

of the

original theory

of

relativity,

i.e.,

the

case

where

only

linear

orthogonal

substitutions

are

admitted

as

"admissible,"

the law

can

be

phrased as

follows.

If

To1...o1

is

a

tensor

of

rank

l,

then

dto1...o1/dxl

is

a

tensor

of rank

l

+

1.

The

so-called

"divergence"

of

tensors follows

easily now

with

equation

(10)

of

§6

from

the

special

tensor

8vu,

which under restriction

to

linear

orthogonal

transforma-

tions-where

the distinction between covariant and contravariant is

void-is

to be

replaced by

the notation

8uv.

By means

of inner

multiplication

of the

tensor

8uv

with

the tensor

of

rank

l

+

1,

obtained

by "extension,"

we get

the

tensor

of

rank

l

- 1

r)T r)T

{7} T

=

ai-_ V

"i

"ai

a/S a,

This is the

divergence

of

the

tensor

To1...a1

formed relative to the index

al.

It

is

now

our

task

to

find the

generalization

of this

operation

in

case

substitutions

are

not

restricted

to

the

previous

conditions

(linearity-orthogonality).

Extension

of

a

covariant

tensor.

Let

(/)(x1...x4)

be

a

scalar and

S

a

given

curve

in

our

continuum.

The

"length

of

arc"

s

is measured from

a

point

P

on

S

in

a

distinct

direction,

as

has been

explained

in

§§1

and

8.

The values of the function

$

at

points

on

S in

our

continuum

can

then

also

be looked at

as a

function

of

s. Obviously,

the

[p.

1048] quantities d(f)/ds,

d2(f)/ds2, etc.,

are

then

also

scalars, i.e., quantities

that

are

defined

independently

of

the coordinate

system. However,

since

f

'

E

(25)

ds

ox"

ds

a*

and since from

every point

the

curves

S

can

be drawn in

any

direction,

the

quantities