184
DOC.
13
GENERALIZED THEORY OF RELATIVITY
\
Ta = =
E
i a
(f
gafl
)
-
dg
•
0-
+g
v/:
/XV
oyx
ß
/XV
k
fg
dx
But the last
term
of this
sum
is
equal
to
E
vk
©vk
E^
+
vcr
@
/xv'
vk
a
/XV
V
v
dx
p
Hence,
we
end
up
with
r"
=
E^WixA,-iE^'®/XV
'
/XV
ygdx
2^
dx
a
i.e.,
the left side of the
investigated equation, up
to
the factor
1
Thus,
if that
fg
equation
is divided
by
yfg,
then
its
left side
represents
the
o-component
of
a
covariant
vector,
and
is,
therefore,
in
fact,
covariant. For that
reason,
the
content
of
those four
equations can
also be
expressed
thus:
The
divergence
of the (contravariant)
stress-energy
tensor of the
material
flow
or
of the
physical process
vanishes.
2. Differential
Tensors
of
a
Manifold
Given
by
Its
Line
Element
The
problem
of
constructing
the differential
equations
of
a
gravitational
field
(Part
I, §5)
draws one's attention
to
the
differential
invariants and
differential
covariants
of the
quadratic
differential form
=
Es"vVv
/XV
In the
sense
of
our
general
vector
analysis,
the
theory
of these differential
covariants leads
to
the
differential
tensors
that
are
given
with
a
gravitational
field.
The
complete system
of these differential
tensors (with
respect
to
arbitrary
transformations)
goes
back
to
a
covariant differential tensor of fourth rank found
by
Riemann12 and,
independently
of
him,
by Christoffel,13
which
we
shall call the
Riemann
differential tensor,
and which
reads
as
follows:
(43)
Rlklm
=
(ik,lm)
=
I
d%m &gU
d%i
'd2S,nk
ydxkdx, dxidxm dxkdxm
dx,dx:/
12Riemann,
Ges.
Werke,
p.
270.
13Christoffel, l.c.,
p.
54.
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