104
DOC.
3
STATICS OF GRAVITATIONAL FIELD
times
greater
than
at
a
point
P0,
this
means
that in order
to
measure
the
time7 at
P,
we
must
use a
clock that
runs
c/c0
times slower than the clock
to
be used
to
measure
the time
at
P0,
if the
rates
of
the
two
clocks
are
compared
with
one
another
at
the
same
location. In other words:
a
clock
runs
faster the
greater
the
c
of
the location
to
which
we bring
it.
This
dependence
of the
speed
of the
passage
of time
on
the
gravitational potential
(c)
holds for the
passages
of time in
any process
whatsoever.
[22]
This has
already
been shown in the
previous paper.
Likewise,
the tension in
a
spring
stretched in
a
certain
way,
and,
in
general,
the
force
or
the
energy
of
an
arbitrary system always depends
on
the
magnitude
of
c
at
the
location
of
the
system.
This
can
easily
be shown
by
the
following elementary
argument.
If
we
do
experiments one
after another
in
several small
parts
of
space
with
different
c,
and
always
use
the
same
clock,
the
same
measuring
rod, etc.,
then
we
find
everywhere-apart
from
possible
differences in the
intensity
of the
gravitational
field-the
same
laws
with the
same
constants.
This follows from the
equivalence
principle.
Two mirrors
1
cm
away
from each other
may serve,
for
example,
as a
clock,
where
we
count
the number of times
a
light signal
travels back and
forth;
in
[23]
that
case we
operate
with
a
sort
of local
time,
which Abraham denotes
by l.
The
[24]
relation between this time and the universal time is then
dl
=
c
dt.
If
we measure
the time
by
means
of
l,
we
will
impart,
via the
energy
of
deformation,
a
specific velocity dx/dl to
a given spring
stretched in
a
given way
and
having
a
given
mass
m, independently
of the
magnitude
of
c
at
the location
at
which this
process
takes
place.
We have
dx dx
-
=
-
=
a,
dl cdt
where
a
is
independent
of
c.
But
according
to
(8), we can
set
the kinetic
energy
that
corresponds to
this motion
equal
to
m
2
m
(dxY
rn
2 2
ma2
-
7
=
- -
=
-
arc^
=
--
*c.
2c 2c
\
dt
2c 2
Thus,
the
energy
of the
spring
is
proportional
to
c,
and the
same
is true
of
the
energy
and forces of
any system
whatsoever.
This
dependence
has
a
direct
physical meaning.
Let
us imagine,
for
example,
a
massless thread
stretched between
two
points
P1
and
P2
of
different
gravitational
potential.
Let
one
of
two
identically
constituted
springs pull
the thread
at
P1
and the
other
at
P2
in such
a
way
that
the
system
is in
equilibrium.
The extensions
l1
and
l2
experienced by
the
two
springs
will
not
be
equal;
rather,
the
equilibrium
condition
7That is
to
say,
to
measure
the time denoted in the
equations by
"t."