DOC. 9 FORMAL FOUNDATION OF RELATIVITY 59
one finally
gets5
^
=
E
K,?r
i,
-
(50b)
dxA
~
dxA
J(».*
This is the
equation
of
motion
of
a
material
point
if the fourth coordinate
("time
coordinate")
is chosen
as an
independent
variable. The
components
of
(Ia)
have the
physical meaning
of
being
the
negative components
of momentum and
energy, resp.;
this follows from the scheme
(43).
In
the
special
case
of the
original theory
of
relativity,
i.e.,
when the
g^v
have the
values
given
in
(18),
one
gets
5At
this
point
I
might
mention
why,
in
my opinion,
not
equation
(39)
but rather
equation
(41)
was
used for the formulation
of
the
energy-momentum
theorem.
According
to
(39),
the
energy
tensor has to be
seen as a
contravariant V-tensor and the
quantities
{rvu}
as
the
components
of
the
gravitational
field.
In
this
manner we
would have been
led,
in
§11,
dx
to
the
interpretation
that the
components
of
the
contravariant
four-vector
(Ia
=
m
dxa/ds)
represent
the
components
of
momentum and
energy
of
the material
point.
We will show
here in
a very special case
that such
interpretation runs
counter to
our understanding
of
the
essence
of
momentum.
Into
a
space
without
gravitational field,
we
introduce
a
coordinate
system
that deviates
from
a
"normal
system" only
inasfar
as
the
x1-axis
and the
x2-axis (judged
from
a
normal
TT
system)
enclose
an angle
(j
different from
n/2.
Then
we
have
.
2 2 2 2
ds2
=
-dxx
-dx2
-2dx1dx2cos4
-
dc3 +
dxA.
2
dx2
And in this
case one gets, e.g.,
-
I2
=
-m
dx2/ds.
This
quantity
vanishes
if
the
point moves
ds
in the direction
of
the
x1-axis.
On the other
hand,
it is clear that in the
case
under
consideration
an x2-component
of
momentum
actually
exists,
and it deviates from the
x1
component only by a
factor
of
cos
(f.
However,
if
one
bases the momentum theorem
on (41)
and
uses,
therefore, according
to
(51)
the
covariant
four-vector for the calculation
of
momentum and
energy,
then
one
gets,
in
our
case,
-I2
= -g12mdx1/ds =
m(dx1/xs)cos (f) =
(-I1)
cos
(f),
as
it
must
be
demanded.
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