DOC. 9 FORMAL FOUNDATION OF RELATIVITY
53
We call
(Aa),
(A"),
V-tensors
(volume tensors).
When
multiplied by
dr,
they
represent
tensors in the
previously
defined
sense
because
\[gdr
=
/gdx1dx2dx3dx4
is
a
scalar. For
example, equation (41a)
can
be rewritten in this notation
as
=
£
a*..
1
r-\
-
öE«
^
fJTV
T/X
Of
or
(41b)
[23]
[p. 1054]
C.
The
Equations
of
Physical
Processes in
a
Given Gravitational
Field
Every equation
of
the
original theory
of
relativity
has
an equivalent, generally
covariant
equation,
in
the
sense
of
the
previous
section
(B)
which takes its
place
in
the
generalized theory
of
relativity.
For the establishment
of
these
equations,
the
fundamental tensor
of
the
guv
has to be considered
as
given.
In this
manner, one
obtains
generalizations
of those
physical
laws which
are already
known in the
original
theory
of
relativity.
The
generalized equations
show to
us
the influence
of
the
gravitational
field
upon
those
processes
to
which these
equations
relate.
Only
the
differential
relations
of
the
gravitational
field
proper
remain unknown for the time
being;
they
have to be found in
a special manner.
We want to subsume all other laws
(e.g.,
mechanical,
electromagnetical)
under the
name
"laws
of
material
processes."
§9.
Energy-Momentum
Theorem
for
"Material
Processes"
The
energy-momentum
theorem is the most
general
law
concerning
"material
processes."
In the
original theory
of
relativity
and
using
the formulation
of
Minkowski-Laue, it
can
be written
as
follows:
dPxx
dx
dx
dp*
ÖPxy
,
dP
dy
d.pw
+
dp"
U
=
dl
/,
dy
dPzy
+
ÖP
M
=/Jy
dl
dx
dy
dp
3*)
+
d(i3y)
dx
dy
dz
dK
dz
d£i)
=/Jz
dz
dl
+
d(_2?i
=
iw.
dz
dl
(42)
The time coordinate
is chosen l
=
it,
where the real time
t
is measured such that
the
velocity
of
light comes
out
equal
to
1.
[24]