5.0 The Interval

We can now speak of the elapsed proper time between
two events along a world-line joining those events.
This leads to the notion of the interval between two events.

Let $p$ and $q$ be two nearby events in space-time.  (More precisely,
we require that the displacement of $q$ from $p$ to be infinitesimal.
That is, what follows is good only to first order in the displacement.)
Arrange a world-line of a particle to pass through event $p$.  Now find
a light ray which first intersects this world-line, and then event $q$.
Find a second light ray which first meets $q$, and then intersects the
world-line.

Denote by $u_1$ and $u_2$ the two events at which these light rays
intersect the world-line.  Finally, let $t_1$ be the elapsed proper
time along this world-line between $u_1$ and $p$, and let $t_2$
be the elapsed proper time between $p$ and $u_2$.  (Thus, $t_1$ and $t_2$
are just a certain number of seconds.)  We define the squared interval
between $p$ and $q$ as the product of these times
$$ SI(p,q)= t_1 t_2 = \left(\frac(\delta x}{c}\right)^2 - (\delta t)^2 $$

We next require a further assumption: that the quantity $SI(p,q)$ depends
only on the (nearby) events $p$ and $q$, and not on the other
details of the construction (whether the original world-line is
passed through $p$ or $q$; the choice of that worldline).

What are the grounds of this assumption?  Let us see what the
construction above yields in special relativity.  Our space-time
is Minkowski space, with the usual $x$, $y$, $z$, $t$ coordinates.
Let $p$ be the event with coordinates $(0,0,0,0)$
and $q$ the event with coordinates $(x,y,z,t)$.
Let the world-line be the $t$-axis (the straight line $x=y=z=0$)
Then the parameter $t$ along this world-line measures proper
time (between events on this world-line).  Light rays are straight
lines making a $45^\circ$ angle to the vertical (\ie, to the
$t$-axis).  That is, the light-cones are all open $45^\circ$
from the vertical.

In this case, the event $u_1$ has coordinates $(0,0,0,t-r)$, and
$u_2$ coordinates $(0,0,0,t+r)$, where $r^2=x^2+y^2+z^2$.  Hence,
$t_1=r-t$, and $t_2=r+t$.  Thus,
$$ SI(p,q)=t_1 t_2 = r^2-t^2 =x^2+y^2+z^2- t^2 $$
This quantity will be recognized as the usual squared interval between
points in Minkowski space in special relativity.  Clearly, the construction
above yields the correct squared in terval for any two events,
and any straight world-line in special realtivity.

Of course, even in special realtivity, the quantity $SI(p,q)$ will
not be independent of the world-line if non-straight world-lines
are admitted.  It is clear intuitively, however, and easily checked, that
independence of the world-line does hold in special relativity to
first order in the displacement of $q$ from $p$.

Thus, our assumption amounts, essentially, to the assumption
that, locally, space-time has the same structure as the space-time
of special relativity.  More precisely, space-time is that of
special relativity to first order in displacements.  (Still
more precisely, and in the language of general relativity, the displacement
of $p$ from $q$ must be small compared with the radius of curvature
of space-time.)

In the discussion above, $q$ was spacelike related to $p$; and
we had $SI(p,q)$ positive.  Suppose that $q$ were null related to $p$.
Then $u_1=p$ (if $q$ is to the future of $p$), or $u_2=p$ (if $q$ is to the
past of $p$).

This is clear because ``null related'' means precisely that there is a
light ray joining $p$ and $q$.  Hence, when $q$ is null related to $p$,
either $t_1=0$ or $t_2=0$ (the cases $u_1=p$ and $u_2=p$, respectively).
In either case, $SI(p,q)=t_1 t_2=0$.

Supose, finally, that $q$ is timelike related to $p$.  Then, if $q$
is to the future of $p$, $u_1$ and $u_2$ are to the future of $p$, so
$t_1$ is negative and $t_2$ positive.

If $q$ is to the past of $p$, the $u_1$ and $u_2$ are both to the past
of $p$, so $t_1$ is positive and $t_2$ negative.  Thus, in either case,
$SI(p,q)$ is negative.

To summarize, $SI(p,q)=SI(q,p)$ is positive if $p$ and $q$ are spacelike
related, zero if $p$ and $q$ are null related, and negative if $p$
and $q$ are timelike related.

Now suppose that $p$ and $q$ are timelike related (say, with $q$
to the future of $p$).  Choose the world-line to pass through $p$ and $q$.
Then $u_1=u_2=q$ (since $q$ is already on the world-line, the ``light rays''
from $q$ to the world-line are very short).


Hence $-t_1=t_2=$(the proper elapsed time between $p$ and $q$ along this
world-line).  (To first, order, this quantity is independent of world-line.)
Hence, for $p$ and $q$ timelike related, $SI(p,q)$ is minus the square
of the proper elapsed time between $p$ and $q$ (along any
world-line joining the events).

Similarly, let $p$ and $q$ be spacelike related, and choose the
world-line so $t_1=t_2$.  AN observer on this world-line would interpret
the situation as follows.  He is bouncing a light ray off $q$,
and measuring the round trip light travel-time, $t=t_1+t_2$

He would say that the ``spatial distance'' of $q$ from $p$ is
$\frac 12 t$ (where his unit of distance is the light-second, the
distance light goes in one second).  That is, for $p$ and $q$
spacelike related, $SI(p,q)$ has the intuitive interpretation
as the square of the ``distance'' (measured in light-seconds)
of $q$ from $p$.

In this sense, then, $SI(p,q)$ is interpreted as representing
geometrical information about space-time.


