DOC. 30 FOUNDATION OF

GENERAL RELATIVITY

157

physics

shall

be

covariant in the

face of

any

substitution of

the

co-ordinates

x1

...

x4,

we

have

to

consider how

such

generally

covariant

equations

can

be found. We

now

turn

to

this

purely

mathematical

task,

and

we

shall find

that

in

its

solution

a

fundamental role is

played by

the invariant

ds

given

in

equation

(3),

which,

borrowing

from Gauss's

theory

of

surfaces, we

have

called

the

"linear

element."

The fundamental idea

of

this

general

theory

of

covariants

is

the

following

:-Let

certain

things

("tensors") be

defined

with

respect

to

any system

of

co-ordinates

by a

number

of

functions

of

the

co-ordinates,

called

the

"components"

of

the

tensor.

There

are

then certain

rules

by

which

these

components

can

be calculated for

a

new

system

of co-ordin-

ates,

if

they

are

known

for

the

original system

of

co-ordinates,

and if

the transformation

connecting

the two

systems

is

known. The

things

hereafter

called tensors

are

further

characterized

by

the fact that

the

equations

of

transformation

for

their

components

are

linear

and

homogeneous.

Accord-

ingly,

all

the

components

in the

new

system vanish,

if

they

all

vanish in

the

original system.

If,

therefore,

a

law of

nature is

expressed by

equating

all

the

components

of

a

tensor

to

zero,

it

is

generally

covariant.

By examining

the

laws

of

the formation

of tensors,

we

acquire

the

means

of formu-

lating generally

covariant

laws.

§

5.

Contravariant and

Covariant

Four-vectors

Contravariant Four-vectors.-The linear element

is de-

fined

by

the four

"components" dxv,

for

which the

law of

transformation

is

expressed by

the

equation

dx'o

vdxv

.

(5)

The

dx'o

are

expressed

as

linear and

homogeneous

functions

of

the

dxv.

Hence

we

may

look

upon

these co-ordinate differ-

entials

as

the

components

of

a

"tensor"

of the

particular

kind which

we

call

a

contravariant

four-vector.

Any

thing

which

is

defined

relatively

to

the

system of

co-ordinates

by

four

quantities

Av,

and

which

is transformed

by

the

same

law

Aio

=

. vdxv .

(5a)