294 DOC. 28 NORDSTRÖM'S THEORY OF GRAVITATION
(1)
Sfds =
0,
where
(2)
ds2
=
Y,
/XV
The
gravitational
field
is
then characterized
by
ten
space-time
functions
guv.
ds is
an
invariant with
respect
to
arbitrary
substitutions that
plays
the
same
role in the
general
theory
of
relativity,
which
is based
on
the
absolute differential
calculus,
as
the
Euclidean line element does in Minkowski's invariant
theory.
As the
only
scalar
referring
to two
adjacent space-time points,
ds has the
meaning
of
the
"naturally
[6]
measured" interval between these
two
space-time points.
Since
to
every
vector
quantity
or
every
vector
operation
in the Euclidean
manifold there
corresponds
a more
general
vector
quantity
or
operation
in the
manifold
that is
given by
an
arbitrary
line
element,
the laws for
physical phenomena
[7]
of the
original theory
of
relativity
can
have
correlated to
them
corresponding
laws
of
the
generalized theory
of
relativity.
The laws thus
obtained,
which
are generally
covariant,
include the
influence of the
gravitational
field
on
physical processes.
From
among
all of these laws that describe
physical processes, we
shall
now
mention
only
one
particular one,
of
the most
general
significance,
namely
the law that
corresponds
to
the law
of
momentum
and
energy
conservation in the
original theory
of
relativity.
In that
theory,
the
energetic properties
of
processes
were
expressed by
means
of
a
stress-energy
tensor
(Tuv).
To these
quantities
Tuv
there
correspond
in the
generalized theory
the
quantities
2ov,
which
constitute the
components, multipled
by
v/-g,
of
a
mixed
tensor
that
is
obtained from
a symmetric
contravariant
tensor
(0uv)
through
mixed
multiplication
--
$ov
=
g
©
!
Z-/
Ö7/A
/IV
(g
denotes the determinant formed from the
quantities guv).
If,
for
example,
the
physical system
consists of
a moving,
uniform
mass
distribution
of
rest
density
p0,
then
©"v
=
o
Po
ds
ds'
and the
physical
meaning
of the
2ov
emerges
from the
following
table:3
^-11 ^-12 •£•13
*14
-
Xx
-Xy
-
Xz
-
ix
2-21
3"
*"22
^23
^-24
-
Yx
-
Yy
-
Yz
^-31
^32 ^33
^-34
-
Zx
~Zy
-
Zz
iz
$41 $42 ^43 ^44
fx
fy fz n.
Xx
etc. denote the
components
of the surface
pressure, ix
etc.
the
components
of
3There is
a sign error
in the
table
as given
in the
Phys.
Zeitschr.
14: 1257.
[8]