4.0 Clock rates

We shall shortly obtain a geometry (more precisely, a metric tensor field)
on the space-time manifold $M$.  This geometry is determined physically by
measurements of time intervals between certain pairs of events.  Finally,
possibility of making meaningful measurements of elapsed times rests on a
certain assumption.  We now introduce this assumption.

There exist clocks.  We suppose that it is feasible to build clocks whose
workings are not significantly affected by external influences (e.g.,
electromagnetic fields, accelerations, etc.) (This is a rather vague
statement, whose meaning tends to change as our understanding of physics
changes.)  Now suppose we build two such clocks.  Suppose, furthermore,
that it is found that, when the clocks are held side by side (in
space-time language, when the world-lines of the clocks are made to
practically coincide over a certain stretch) their ticking rates coincide.
(For example, this would, presumably, be true of two cesium clocks.)
Now the clocks are separated and, for a while, travel different paths
in space-time.  Finally, at some later time, the clocks are held again
side by side.  We assume that, if the clocks initially ticked at the
same rate, then, when they are again brought side by side, they again
tick at the same rate.  Note that we do not assume that the total
elapsed times measured by the clocks while they are separated are the
same.  In fact, general relativity predicts that this will not in general
be the case.  The assumption, essentially, is only that
{\em the ticking rate of an ``isolated'' clock along a certain stretch
of its world-line} depends only on the world-line in that region,
not on what the world-line was before the region of interest.  In other
words, the ticking rate of a clock does not depend on what happened
to it in the past.

Why should we believe such an assumption?  It would imply,
for example, that two atomic clocks, one assembled in Chicago
and the other in Bombay, will ,when they are brought together,
tick at the same rate.  Physically, one might expect this to be
the case.

An ``experimental test'' of this assumption has been suggested by Wheeler.
An electron has a mass, and hence a Compton wavelength, and hence
a characteristic time (the light-travel time across a Compton
wavelength).  Hence, an electron is essentially a ``clock''.
The electrons in an atom inside the Earth have, presumably, traversed
different world-lines in spce-time to get where they are.
A ``comparison of the ticking rates'' is provided by the Pauli
exlusion principle.  It is because of this principle (i.e.,
because all electrons are identical) that, after the first few electrons
in an atom have occupied the lowest states, the remaining electrons
must occupy higher states.  Now, a typical electron, in an atom in the
Earth has gone around the nucleus $10^{30}$ times during the age of the
earth.  Hence, if mass differences between different electrons
were more than about one part in $10^{30}$, all electrons would by
now have dropped to the lowest level. (because they would have ``discovered''
their mass differences, i.e. discovered they do not have to obey
the exclusion principle).  But this collapse of atoms in the Earth
has not taken place.  Hence, our assumption has been ``tested''
to one part in $10^{30}$.

Now suppose we build one clock, and call the elapsed time between its
ticks {\em the second}.  Then any other clock can, by being brought
side by side with this one, be calibrated so that it ticks in
seconds too.  Once the second clock has been so calibrated, it
can be carried around anywhere in space-time.  We calibrate a large
number of clocks in the way.  Then, by our assumption, any two will,
when brought side by side, tick at the same rate.  In other words,
if we have a world-line of a particle, and two events $p$ and $q$
on this world-line, then we can count the number of ticks (number
of seconds) between the events $p$ and $q$ using a calibrated clock
which traverses the world-line between $p$ and $q$.

Our assumption ensures that this number (of seconds) is independent
of which calibrated clock we use.  (Although it will in general depend
on what the world-line is between $p$ and $q$.)

Thus, given a world-line between events $p$ and $q$, it is meaningful
to speak of the elpased proper time (in seconds) along that
world-line between $p$ and $q$.
