156
DOC.
13
GENERALIZED THEORY OF RELATIVITY
exerted
on
it:
[15]
Jx =
-m
#11*11
+
#12*2
+
#
13*3
+
#14
[14]
(7)
ds
dt
=
-m
g\\dxx +
g12dx2
+
gndx
3
+
g\^dx^
ds
(8)
**
=
_
1 ÖA1
2
1
^
A
3^ ds
dt
Further,
for the
energy
E of the
point,
one
obtains
(9)
-E=
-
.
dH
x+
•+
dx
/
+
H=
-m
§A\
dxl
dx.
dx;
+
Ä2
+ Ä3 +
Ä4
ds ds ds ds
In the
case
of the
customary theory
of
relativity
only
linear
orthogonal
substitutions
are
permissible.
It will
turn out
that
we are
able
to set
up equations
for
the influence
of the
gravitational
field
on
the material
processes
that
are
covariant
with
respect
to
arbitrary
substitutions.
First,
from the role that ds
plays
in the law of motion of the material
point, we
can
draw the conclusion that ds
must
be
an
absolute invariant
(scalar);
from this
it
follows that
the
quantities guv
form
a
covariant
tensor
of
the second
rank,6
which
we
call the covariant fundamental tensor.
This
tensor
determines the
gravitational
field.
Further,
it follows from
(7)
and
(9)
that
the
momentum
and the
energy
of the material
point
form
together
a
covariant
tensor
of the first
rank,
i.e., a
covariant vector.7
§3.
The
Significance
of the
Fundamental Tensor
of the
guv
for
the
Measurment
of
Space
and
Time
From the
foregoing,
one can
already
infer that there
cannot
exist
relationships
between the
space-time
coordinates
x1,
x2,
x3, x4
and the results of
measurements
obtainable
by
means
of
measuring
rods and clocks that would be
as
simple
as
those
in
the old
relativity theory.
With
regard
to time,
this has
already
found
to
be
true in
the
case
of the static
gravitational
field.8
The
question
therefore
arises,
what
is
the
[16]
6Cf.
Part
II, §1.
7Cf.
Part
II,
§1.
8Cf., e.g.,
A.
Einstein,
Ann. d.
Phys.
35
(1911):
903
ff.