3.0 The Light-Cone Structure

The next piece of structure on our space-time manifold $M$ arises
from the observation (in the real world) that no particles have
been observed to move faster than light.  Thus, the motion of light rays
is of special significance.

The word ``faster'' refers to ``speed'', which refers to ``distance
traveled per unit time''.  Hence, the notion ``faster'' violates the
spirit of space-time physics.  We reformulate the observation
in the first paragraph as follows.  Suppose the explosion of
a firecracker marks an event, so an expanding spherical pulse of light
is sent out from it.  Then any particle whose world-line passes
through this event cannot get outside of the expanding sphere of light.
(This is the statement that the particle's speed doesn't exceed that of
light.)  Let's try to reformulate this as a statement in space-time.
Let the event (firecracker) be $p$.  Then the light emerging from this
event would form a cone with vertex $p$.  (Slice the space-time with a
family of three-dimensional surfaces, and you will, indeed, see a
``sphere of light expanding from the original event''.  In the figure
one dimension is, of necessity, supressed.)  Now we consider a
particle (\ie, a curve) passing through $p$.  The observation (about
speeds not exceeding that of light) now reduces to the assertion
that the world-line of the particle is inside the cone with vertex $p$.

Thus, for each event of space-time, there is a conce with vertex at that
event.  This cone is the events which occur just a light from the original
event passes by.  The world-lines of particles which pass through the
original event lie inside the cone.  This cone is called
the (future) light-cone.

Now let $p$ and $q$ be two nearby events.  (We need ``nearby'' because
some notions, logically different, coincide for nearby events
but do not coincide for more widely separated events.  Of course, ``nearby''
means nearby in space-time.)
If $q$ lies inside the future light-cone of $p$, we say that
$q$ is timelike related to $p$ and to the future of $p$.
If $p$ lies inside the future light-cone of $q$, we say that $q$ is
timelike related to $p$ and to the past of $p$.
If $q$ lies on the future light-cone of $p$ (resp., $p$ lies on the
future light-cone of $q$), we say that $q$ is null-related to $p$,
and to the future (resp., past) of $p$.  Finally, if neither event
is either on or inside the future light-cone of the other, we say
that the events are spacelike related.  Evidently, a particle
can meet first $p$ and then $q$ if and only if these events are
timelike related, and $p$ is to the past of $q$.
A light ray can meet both events if and only if they are null
related.  Spacelike related events cannot be both met by any
particle or light ray.  (Note: A ``light ray'' corresponds to
a flashlight which is on only momentarily, sending out a pulse of light.
The light from a flashlight which is on continuously defines a
two-dimensional surface in space-time.)

It should now be clear that the three-dimensional surfaces introduced
earlier (to obtain simultaneity) had the property that the future
light-cone from any event on any surface does not elsewhere intersect
that surface.  (Physically, light emitted at 1:00 cannot get to any
other point by 1:00).

In special relativity, one can arrange matters, in an appropriate
coordinate system, so that all the light-cones look like ordinary
cones, all with the same opening angle,a dn all parallel to each
other.  There is no reason, a priori, to make such an assumption.
In general relativity, no such assumption is made.  Dropping it
amounts to admitting curvature to space-time.

To summarize, one constructs a future light-cone at each event using
a light pulse emitted from that event.  Particle world-lines which
pass through the event remain inside the light-cone.  The light-cone leads
to a distinction between timelike, spacelike, and null related events
and to a distinction between past and future related events.  (The past,
future distinction is not available for spacelike related events.)



