1.0 Introduction

The general theory of relativity is the most attractive classical
(\ie, non-quantum) theory of the gravitational field available today.
It represents a generalization of Newtonian gravitation which is
consistent with the principles of special relativity.

Recall Newtonian gravitation.  The gravitational potential is determined
from the distribution of matter by the equation
\begin{equation}
     \nabla^2\phi=4\pi G\rho
     \label{Newtonian}
\end{equation}
where $G$ is the gravitational constant and $\rho$ is the density of
matter.  This potential then produces a force $F=-\nabla\phi$ on
a particle of mass $m$.  Thus, a distribution of matter produces a
potential $\phi$ which, in turn, affects particles.  In this way,
bodies interact gravitationally.  What matter at any one instant of
time determines the potential at that time.  This action is instantaneous.
In other words, a particular ``time'' is singled out.  Such a situation
is not permitted by the principles of special relativity.

Thus, the phenomenon of gravitation certainly exists, and
special relativity is about as well-established as a physical theory can be.
One is naturally led to search for a relativistic theory of gravitation.

One can think of special relativity as the limit of general relativity
when gravitation is unimportant, and of Newtonian gravitation as the limit
when speeds are much less than that of light
(so the Newtonian picture of space and time is a good aprroximation
to the special relativistic picture).  (Thus, the statement one
occasionally hears that ``general relativity must replace special relativity
when `accelerated frames' are considered'' is false.   Special relativity
must be abandoned when gravitation is important.)

It turns out that the gneral theory of relativity (\ie, this
incorporation of gravitation into special relativity) touches
on the structure of space and time.  Why should this be true for
gravitation (and not, say, for electromagnetism)?  It is because
the gravitational phenomenon has about it an ``all-inclusiveness''
which is apparently not possessed by other forces in nature
(\eg, electromagnetic).  Thus, particles can be positively charged,
negatively charged, or uncharged; particles can react to electromagnetic
fields in different ways.  But all particles seem to react to
gravitation in the same way.  (Quantum theory is a good example of a
physical theory with a similar ``all-inclusiveness''.  The principles
of quantum mechanics are not to be applied just to certain special
systems.  Quantum theory makes a much stronger claim.  All physical
phenomena are to be governed by its principles.)

This feature of gravitation is made more explicit bbby considering
the famous elevator.  People inside an elevator falling freely in a
uniform gravitational field cannot distinguish their situation
from that of an elevator at rest in the gravitational field.
The uniform gravitational field affects everything - so by no physical
experiment inside the elevator can these two situations be
distinguished.  (Like many important physical principles, this,
the equivalence principle, becomes a tautology in the final theory.  In
general relativity, ``unfiorm gravitational field'' is a long way of saying
``no gravitational field''.  That is, the existence of the former is simply
not recognized.  More on this later.)

The remarks above at least make it reasonable that gravitational phenomena
might occupy a special place in physics.  They do not, however, make it
clear why this ``specialness'' should manifest itself in statements about
space and time.  That this should be the case is suggested by the remarks
below.

In order to do Euclidean geometry, one needs straight lines.  I draw a
line in the room.  How do you determine whether or not it is straight?
One might think of putting the edge of meter stick against it.  But,
of course, that doesn't work unless one has a way to tell which meter
sticks have straight edges.  One might stretch a string between
two points on the line, and see whether the string lies along the
line.  But a gravitational field will bend the string in a way over
which we do not exert control.  (The complete explanation here is
more subtle.  We shall return to this.)  Finally, one might
``sight'' along the line to see if it is straight.  Here, the
implicit assumption is that light travels along striaght lines.
But it is known that a gravitational field affects the path of a light
ray, for light from distant stars is observed to be bent on passing near
the surface of the Sun.  Similar attempts fail for similar reasons.
In short, since gravitation affects all physical phenomena
(all-inclusiveness), and since such phenomena must be used, in particular,
to make the measurements which determine the geometry of space (and time),
it is perhaps not surprising that gravitation should be tied in with
the structure of space and time.


