DOC.
71
PRINCETON LECTURES 323
THE GENERAL THEORY
neighbourhood
of
an
observer,
falling freely
in
a
gravita-
tional
field,
there
exists
no
gravitational
field. We
can
therefore
always
regard
an
infinitesimally
small
region
of
the
space-time
continuum
as
Galilean. For
such
an
infinitely
small
region
there
will be
an
inertial
system
(with
the
space
co-ordinates,
X1, X2,
X3,
and the time co-ordinate
X4)
relatively to
which
we are
to regard
the
laws of
the
special
theory
of
relativity
as
valid. The
quantity
which
is directly
measurable
by
our
unit
measuring
rods
and
clocks,
dX12
+
dX22
+
dX32
-
dX42
or
its
negative,
(54) ds2
=
-dX12
-
dX22
-
dX32
+
dX42
is
therefore
a
uniquely
determinate invariant for
two
neighbouring
events (points
in
the four-dimensional
con-
tinuum), provided
that
we use
measuring
rods
that
are
equal to
each other when
brought together
and
superim-
posed,
and
clocks whose
rates
are
the
same
when
they
are
brought together.
In
this
the
physical assumption is
essential
that the relative
lengths
of
two measuring
rods
and the relative
rates
of
two
clocks
are
independent,
in
principle,
of
their
previous history.
But this
assumption
[84]
is certainly
warranted
by
experience;
if
it
did
not
hold
there could
be
no
sharp spectral lines,
since
the
single
atoms
of
the
same
element
certainly
do
not
have the
same
history,
and since-on
the
assumption
of
relative
variability
of
the
single
atoms depending
on
previous
history-it
would
be
absurd
to
suppose
that the
masses
or
proper frequencies
of
these
atoms
ever
had been
equal to
one
another.
Space-time
regions
of finite
extent
are,
in
general,
not
Galilean,
so
that
a
gravitational
field
cannot
be done
away
[63]