2.0 Events and Space-Time

Recall the Newtonian picture of physics.  One has some physical
system he wishes to describe.  At any one instant of time, that system
is in one of a certain collection of possible states.  As time goes
on, the system evolves, \ie, it passes through a sequence of states.
This general approach to describing a physical system is so natural
that it is not always mentioned explicitly.  We mention it because the
description of things in relativistic physics is quite different.
Time loses its special role.  Instead, one adopts a description in
which past, present, and future of a system are incorporated into
one picture.  This idea is reflected in the following.

A fundamental notion of general relativity is that of an event.
An event is an occurrence (in the physical world) having no extension
in either space or tim.  For example, the snapping of one's fingers,
or an explosion of a firecracker, would represent an event.

The collection of all possible events - those which have happened,
are happening, or ever will happen - will be denoted by $M$ and
called space-time.  This space-time is the arena in which all the
physics takes place.  An event is sometimes called a point of $M$.

The space-time $M$ of general relativity has more structure than
just that of a set.  We introduce part of this structure now.
Suppose we consider a certain region of $M$ (\eg, the collection of
events which occur within a given room within a given interval of time).
We wish to label points of this region by numbers.  This could be done,
for example, as follows.  Let four pilots fly airplanes around
the room.  Each pilot carries with him a clock.  (These clocks do not
have to run at uniform rate, or be synchronized.)  Let an event which
takes place in our region be the explosion of a firecracker.  Each
pilot records the time (as read from his clock) when he sees the flash.
Thus, with this event, we associate the four numbers obtained by
the pilots.  These four numbers are called the coordinates of the event.
Evidently, no two distinct events have exactly the same coordinates,
and any set of coordinates (in the appropriate intervals) defines an event.

Evidently, there are many different systems of coordinates. The pilots
can fly around in arbitrary ways, using clocks that run at
arbitrary rates.  Furthermore, there are other physical ways
of obtaining systems of coordinates.  One could, for example,
fill the room from floor to ceiling with people during the time
interval.  Each person could be assigned three numbers which give
his location in the room, and each person could carry a watch.
THen, if an event takes place in the region of space-time, the
person located at the occurrence of that event could recod his watch's
reading when the event takes place, and the three numbers which give his
location.  In that way again, events are labeled by four real numbers.

Now suppose we select a region of space-time and have two systems of
coordinates $(x^1,x^2,x^3,x^4)$ and $(y^1,y^2,y^3,y^4)$, in this region.
That is to say, each event in our region is labeled by four numbers
in two different ways.  Hence (since values for the $y$-coordinates
determine an event in this region, which, in turn, determines
values for the $x$-coordinates), we obtain four functions of four
variables
$$x^1(y^1,y^2,y^3,y^4), x^2(y^1,y^2,y^3,y^4), x^3(y^1,y^2,y^3,y^4),
     x^4(y^1,y^2,y^3,y^4)$$
We assume that these functions are smooth ($C^\infty$).  This is reasonable
physically, if the coordinates are set up with sufficient care.  Thus,
the pilots would have to move the controls of their airplanes smoothly,
and their clocks would run smoothly, etc.  More generally if the $x$'s
are an admissible system of coordinates, and if the $y$'s are
smooth functions of the $x$'s with the inverse functions (the $x$'s
as functions of the $y$'s) also smooth, we permit the $y$'s as a system
of coordinates.

Thus, on the space-time $M$ we have a large number of admissible
coordinate systems, any two smoothly related to each other, and with
any system smoothly related to the admissible ones also admissible.
But this is what the mathematical object called a four-dimensional
manifold is.  Thus, the set $M$ of events has the structure of a
four-dimensional manifold.  (Such statements, which relate the physics
to its mathematical description are never proven.  All one can do is
make the mathematical description appear appropriate.  Nor is it
claimed that one can see, by abstract reasoning, that such a description
will be useful.  The foundation of a theory can be motivated, but its
testing can only occur after the structure of the theory has been
set down in detail.)

We wish to adopt the following point of view: every statement which
is made about a physical phenomenon is to be expressed in terms of the
space-time manifold $M$.  This is a somewhat different point of view
from that of Newtonian physics.  The remarks below are intended to make
the new attitude seem more natural.

How would one describe a particle in terms of space-time?  Consider
the set of all events which occur at the particle.  That is, we
consider events (snapping of fingers) such that, at the instant
the fingers are snapped, the particle is right there.   Evidently, the
set of events describes a curve in space-time.  This curve, called
the world-line of the particle, describes completely the entire history
(and future) of the particle.

Similarly, a string is described by the collection of all events which
occur on the string - a two-dimensional surface in space-time.
A bedsheet defines a three-dimensional surface in space-time.
Two particles which collide have the property that their world-lines
intersect.  The point of intersection is the event of collision
of the particles.

Note that there is no dynamics in space-time, nothing ever happens
there.  Space-time is an unchanging once-and-for-all picture
encompassing past, present, and future.

A star would be represented by a four-dimensional region of space-time.
It would be tubular-shaped with (qualitatively) time going up the
tube.  If the star suddenly exploded, then the tube would ``get wider''
beyond a certain point.  Collapse would be represented by a contracting
of the tube.

There is, as we shall see later, no natural notion of ``simultaneity''
between events in space-time.  But everyone near the Earth agrees when
it is 1:00 Greenwich mean time.  Suppose we fill the Earth (surface,
interior, atmosphere) with people, all instructed to snap their fingers
(mark the event) at 1:00 Greenwich mean time.  These events would form
a three-dimensional surface  If fingers were again snapped at 1:01, 1:02,
\etc; we would obtain a family of such surfaces, filling the space-time.
We canconsider the first surface as representing ``all of space at 1:00'',
the next as ``all of space at 1:01'', \etc; we have now recovered
simultaneity: two events are simultaneous if they occur on the same surface.
We have also recovered dynamics: the situation at 1:00 is described the
space-time on the 1:00 surface, the situation at 1:01 by the 1:01 surface,
\etc.  Things change with time, for, as one moves from one surface to
the next, he enters different regions of space-time, and so what is going
on there might be different.  A particle (\ie, a curve in space-time),
would meet the successive surfaces at different points, and thus would
be described as ``moving in space as time evolves''.

In short, we recover the Newtonian picture of physics by slicing the
space-time with a family of three-dimensional surfaces.  But, of course,
there are many differnt ways to carry out such a slicing.  There is no
natural slicing, hence, no natural simultaneity,\etc.  All
that is ``natural'' is the entire, four-dimensional space-time manifold.
(There is nothing philosophical about this.  It is just a different
- and rather attractive - way of describing what is going on.)

To summarize, an event is an occurrence having extension in neither
space nor time.  All the possible events (that have happened, are happening,
or will happen) are collected together into space-time $M$.  This $M$
has the structure of a four-dimensional manifold.  One adopts the point of
view that everything is to be described in terms of the structure on this
manifold $M$.


