4 8 D O C . 9 E N E R G Y C O N S E RVAT I O N
This differential law is equivalent to the integral law which has been abstracted
from experience; in this alone rests its meaning.
From a formal point of view, the appropriate translation of this law into the gen-
eral theory of relativity is the equation
whose left-hand side is a divergence in terms of absolute differential calculus.
is a tensor, the energy tensor of “matter.” From a physical point of view
this equation cannot be considered a full equivalent of the conservation theorems
of momentum and energy because it has no equivalent integral equations that could
be interpreted as the conservation theorems of momentum and energy. When ap-
plied to the planetary system one can, for example, never conclude that the planets
cannot move away from the sun without bounds, or that the center of gravity of the
whole system must remain at rest (or at uniform translatory motion) relative to the
fixed stars. Obviously, experience forces us to look for a differential law that is
equivalent to the integral laws of preservation of momentum and energy. This is
achieved—as I shall demonstrate in detail later on—by the equation (21 l.c.), which
I have proven, viz.,
, (1)
where is to be calculated from the total Hamiltonian function according to the
formula (19 and 20 l.c.),
. (2)
This formulation is opposed by colleagues because and are no ten-
sors, while they expect all meaningful quantities in physics to be representable as
scalars and tensor components. They emphasize,
furthermore,3
that in certain cas-
es, with a suitable choice of coordinates, one is at liberty to put all to zero or
3
See, e.g., the paper by H. Bauer, quoted above.
[3]
∂Tσ-
ν
∂xν
-----------
1gσ
2
-- - +
μν
Tμν 0, =
[p. 449]
1
–g
--------- -Tσ
ν
[3]
[4]
ν
∂Uσ
∂xν
----------- 0 =
Uσ
ν
Uσ
ν
Tσ
ν
tσ
ν
∂H*
∂gμσ
α
--------------gα -
μν
∂H*-
∂gμσ
-----------gμν⎟
+
⎝ ⎠
⎜
⎛ ⎞
– = + =
Uσ
ν
( ) tσ
ν
( )
[5]
Uσ
ν