D O C . 9 E N E R G Y C O N S E RVAT I O N 5 7
where the latter one can be easily concluded from a calculation H. Weyl gave in §28
of his book Raum, Zeit, Materie, soon available from J. Springer. From (18), (18a),
and (7) follow the expressions for as
(20)
where each column belongs to a -value, each row to a -value. From (17), (18),
and (20) follow, with (16), the energy components .
The conditions (9) are satisfied for all components with the exception of the
component This exception is rooted in the fact that does not vanish for
and for . But nevertheless, as is seen, the integral
vanishes because has the same value for and . And
so, indeed, the integrals
,
[13]
t
σ
ν
( )2
[14]
κ
R
---(
tν
σ
)2 =
ϑ1
2
cos sinϑ2
0 0 0
sinϑ1 cosϑ2 cosϑ1 ϑ1
2
–cos sinϑ2
0 0
0 0
ϑ1
2
–cos sinϑ2
0
0 0 0
ϑ1
2
–cos sinϑ2
ν σ
U
σ
ν
[p. 457]
U1.
1
t1
1
( )2
ϑ1 0 = ϑ1 π =
∂U1
-----------dϑ11∂ϑ1
ϑ1 0 =
ϑ1 π =
∫
ϑ1
2
cos sinϑ2 ϑ1 0 = ϑ1 π =
∂Uσ-
1
∂x1
-----------
∂Uσ-
2
∂x2
-----------
∂Uσ-⎞
3
∂x3
-----------⎟ + +
⎝ ⎠
⎜
⎛
∫
dx1dx2dx3