76 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
following ones:
a
E
-
tf-88aßKß)
=
-
k($£
+
O,
aß
dx
(81)
E
a
v
a*v
($:
+
o
=
o,
(42c)
where
or
rv
-
1
O
VT
00
2V
av (81a)
t
=
.£k
4k
E
Wi'pr
8
VT
dy
ag^
ax"
a*T
3g»'
V
öjfp
a*,
_
f~g
LvrpP
ppr
-
lsv)?""pp P
s
1lial
r\
(TO
A/W1pr'
^
ApTT'
\
^
(81b)
The
equations (81) together
with
(81a)
and
(81b) are
the differential
equations
of
the
gravitational
field.
Following
the deliberations
of
§10,
the
equations (42c)
represent
the conservation laws of momentum and
energy
for matter and
gravitational
field combined. The
tvo
are
those
quantities,
related to the
gravitational field,
which
are
in
physical analogy
to the
components
Zvo
of
the
energy
tensor
(V-tensor).
It is
to be
emphasized
that the
tvo
do not have tensorial covariance
under
arbitrary
admissible transformations but
only
under
linear
transformations.
Nevertheless,
we
call
(tvo)
the
energy
tensor of the
gravitational
field. A similar
analogy applies
to
the
components
TvoB
of
the field
strength
of
the
gravitational
field.
The
system
of
equations (81)
allows for
a simple physical interpretation
in
spite
of
its
complicated
form. The left-hand side
represents
a
kind
of
divergence
of the
gravitational
field. As the
right-hand
side
shows,
this is caused
by
the
components
of
the
total
energy
tensor. A
very
important
aspect
of
this
is the
result
that
the
energy
tensor
of
the
gravitational
field itself acts
field-generatingly,
just as
does
the
energy
tensor
of
matter.
[p.
1078]
§16.
Critical Remarks
on
the Foundation
of the
Theory
It is the
essence
of
the
theory we
derived here that the
original theory
of
relativity
holds in the
infinitesimally
small. This becomes obvious
once we
have shown that
under
a
suitable choice
of
real-valued coordinates the
quantities guv assume
the
values