5 0 D O C . 9 E N E R G Y C O N S E RVAT I O N

the are independent of the choice of coordinates for all coordinate systems that

agree outside of with one and the same Galilean system.

By integration of (3) between and one next gets for a coordi-

nate system :

. (4)

If we introduce, furthermore, a second (primed) coordinate system which coin-

cides with outside of , then we get, similarly, for the intersections

and ,

.

We now construct a third coordinate system of the same kind which coincides,

without violation of continuity, in the neighborhood of the intersection

with , and in the neighborhood of the intersection with . Integration

of (3) between these intersections yields

. (5)

It follows from these three relations that is independent of the choice of coor-

dinates inside of . The , therefore, change only with the choice of the Galilean

coordinate system outside of . Consequently, we exhaust all possibilities if we

proceed as follows: we first select a coordinate system which is Galilean outside of

B and arbitrarily inside of B, and then we use all and only those coordinate systems

that are related to the first one by Lorentz transformations. The have tensorial

character with respect to this group of transformations, and it can be shown with

methods of the special theory of relativity that is a four-vector. We, therefore,

can write just as we do in the special theory of relativity,

, (6)

Jσ

B

[p. 451]

x4 t1 = x4 t2 =

K

Jσ)1 ( Jσ)2 ( =

K′

K B x′

4

t′1 =

x′

4

t′

2

=

J′

σ

( )1 J′

σ

( )2 =

K″

x4 t1 =

K x′4 t′2 = K′

Jσ)1 ( J′σ)2 ( =

Jσ

B Jσ

B

Uσ

ν

[8]

Jσ) (

Jσ

E0--------

dxσ

ds

=