340 DOC.

71

PRINCETON LECTURES

THE

GENERAL THEORY

How

are

these

equations

connected with

Newton’s

equations

of

motion?

According to

the

special

theory

of

relativity,

the

guv

as

well

as

the

guv,

have the

values,

with

respect to

an

inertial

system

(with

real time co-ordinate

and suitable

choice

of the

sign

of

ds2),

(91)

-1

0 0 0

0

-1

0 0

0 0

-1

0

0 0 0

1

The

equations

of motion then become

d2xu

ds2

=

0.

We

shall call this

the “first

approximation”

to

the

guv-

field.

In

considering approximations

it

is

often

useful,

as

in the

special

theory

of

relativity, to

use an

imaginary

x4-co-ordinate,

as

then the

guv,

to

the

first

approximation,

assume

the values

(91a)

-1

0 0 0

0

-1

0 0

0 0

-1

0

0 0 0

-1

These values

may

be collected

in

the relation

{2} guv

=

-Svu.

To the second

approximation

we

must

then

put

(92)

guv

=

-S.+y",

where the

yuv

are

to

be

regarded

as

small

of the

first

order.

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