306 THE
RELATIVITY
PRINCIPLE
According
to
§17,
equation
(30)
is also
applicable to
a
coordinate
system
in
which
a
homogeneous
gravitational
field is
acting.
In that
case we
have
to
put
$
=
?£,
where $
is the gravitational
potential,
so
that
we
obtain
a
=
r
1
(30a)
We
have
defined
two
kinds
of
times for
S.
Which
of the
two
definitions
do
we
have to
use
in the various cases?
Let
us assume
that
at
two
locations
of different gravitational potentials
(y£)
there exists
one
physical
system
each, and
we
want
to
compare
their
physical
quantities.
To
do
this,
the
most
natural
procedure
might
be
as
follows: First
we
take
our
measuring
tools
to
the
first
physical
system
and
carry
out
our
measurements there;
then
we
take
our
measuring
tools
to
the
second
system
to
carry
out
the
same
measurement
here.
If
the
two sets
of measurements
give
the
same
results,
we
shall denote
the
two
physical
systems
as
"equal."
The
measuring
tools include
a
clock with
which
we
measure
local
times
a.
From
this it follows that
to
define the
physical
quantities at
some
position
of the
gravitational
field, it is natural
to
use
the time
a.
However,
if
we
deal with
a
phenomenon
in
which
objects situated
at
posi-
tions with different
gravitational
potentials
must
be
considered simultan-
eously,
we
have to
use
the time
r
in those
terms
in which
time
occurs
explicitly
(i.e.,
not
only
in the definition
of
physical
quantities), because
otherwise the
simultaneity of the events
would not
be
expressed
by
the
equal-
ity
of the
time values
of
the
two events.
Since in the definition
of
the
time
r a
clock situated
in
an
arbitrarily
chosen position
is
used,
but
not
an
arbitrarily
chosen
instant,
when using
time
r
the
laws
of
nature
can
vary
with
position
but
not
with time.
§19. The
effect
of
the gravitational
field
on
clocks
If
a
clock
showing
local time is located in
a
point
P
of gravitational
potential $,
then,
according to (30a),
its
reading
will
be
(1
+ o\2)
times
greater
than the time
r,
i.e.,
it
runs (1
+
o/c2) times faster than
an