DOC. 25 FOUNDATIONS OF GENERAL THEORY 283
substitutions. The
equation
of
the
freely moving
material
point
in its Hamiltonian
form
is
S{/ds}
=
0.
(2)
In
the
case
where
we
drop
the
postulate
of the
constancy
of the
velocity
of
light,
there
exist,
a
priori,
no
privileged
coordinate
systems.
The coordinates
xv can
therefore be
replaced
by
as
yet
totally arbitrary
functions of these
quantities.
Then,
if there
are
also four-dimensional
regions
in
which, given
a
suitable choice of the
coordinates
xv,
the material
point
moves according
to
(2)
and
(1),
then the latter
can
no
longer
be viewed
as
the
general
law
of
motion
of
the
point moving
in the absence
of force. What
follows,
instead of
these,
is
equation (2)
in
conjunction
with
ds2
=
Euv
guvdxudxv,
(1a)
where the
quantities
guv
are
functions of the
xv.
3.
This law
of
motion
[(2),
(1a)]
is derived
first of
all
only
for the
case
where the
point
moves
totally
force-free,
thus where
no
gravitational
field affects the
point
either
(as
judged from
an
appropriate
reference
system).
But since
we
know from
experience
that the law of motion of
a
material
point
in
a gravitational
field
does not
depend
on
the material of the
body,
and
since, in
any case,
it
ought
to
be
possible
to
reduce this law
to
the
Hamiltonian
form, it
seems
natural
to
consider
[(2),
(1a)]
in
general
as
the law of motion of
a
point
acted
upon by
forces other than
gravitational
forces. This is the
gist
of the
"equivalence hypothesis."
4.
According
to
what has
been said
above,
we
have
to
conceive of the functions
guv
as
of the
components
of the
gravitational
field with
respect
to
the
completely
arbitrarily
chosen
reference
system.
Since the Hamiltonian
equation
of
motion
must
determine the motion of
the
point wholly independently
of
the
choice of the reference
system,
guvdxudxv
is to
be
conceived
as an
invariant with
respect
to
all
/XV
substitutions. We call the
positive square
root
of this
quantity
the
(four-dimensional)
line element of the
temporo-spatial
manifold.
5.
Minkowski based
a
four-dimensional covariant
theory
on
the invariant
(1),
which
theory yields
the
equations
of
the
original theory
of
relativity.
In
an
analogous
way,
it is
possible,
with the aid of the "absolute differential
calculus,"
to
ground on
the invariant
(1a) a
covariant
theory yielding
the
corresponding equations
of the
new
theory
of
relativity.
Since the
quantities
guv
enter
into these
equations,
the latter
easily
allow
one
to find the
influence that the
gravitational
field
exerts
on
any
physical processes.
The
equations
of the
new theory
of
relativity
reduce to those
of
the
original theory
in
the
special
case
where the
guv can
be considered
constant
(given
a proper
choice of the
xv);
this
is
the
special
case
in which the
gravitational
field
can
be
neglected.
It
is essential for the
theory
that
(1a)
be invariant with
respect
[6]
[7]
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