6.0 The Metric

Imagine a very large book in which there are listed all possible pairs,
$p,q$, or nearby events, and for each pair, the number $SI(p,q)$.
This book contains all the information about the geometry of
space-time.  One could in principle treat the geometry of space-time
by constantly dragging out this book, but, obviously, this would not
be very convenient.  One is thus led to attempt to describe the
geometrical information in $SI$ in a more convenient form.

Let $p$ be an event of space-time, \ie, let $p$ be a point
of the manifold $M$.  Recall that a nearby point, $q$, to $p$, is
described by a contravariant vector, $\xi^a$, at $p$.  We now introduce
the assumption: there is a tensor $g_{ab}=g_{ba}$ at $p$ such that
\begin{equation}
     SI(p,q)=g_{ab}\xi^a\xi^b
     \label{def:metric}
\end{equation}
for all ``infinitesimally nearby'' $q$.  This assumption can
be considered as a further extension of the idea that ``locally,
space-time has the structure of space-time in special relativity''.
In special relativity, expression (\ref{def:metric}) holds, where $g_{ab}$
is the usual metric $dx^2+dy^2+dz^2-dt^2$ of Minkowski space.  We therefore
assume that it holds at each point of space-time.

The above is repeated at each point of space-time.  In this way,
we obtain a symmetric tensor field $g_{ab}$ on the manifold $M$.
Since $\xi^a$ connects infinitesimally nearby which are null realted
precisely when $g_{ab}\xi^a\xi^b=0$, and since the points null related
to a point $p$ are supposed to form a cone with vertex $p$, it is clear
that this $g_{ab}$ must have signature $(-,+,+,+)$.  (Or, to state matters
another way, we have special relativity locally.)  We assume that $g_{ab}$
is a smooth tensor field.  (As a rule of thumb, whenever smoothness is
meaningful for a physical quantity, there is no loss, physically, in
assuming smoothness.  Perhaps that's a reason why differential geometry--the
study of smoothness--is so useful in physics.)  From the statement of its
signature, $g_{ab}$ has an inverse, $g^{ab}$.  Thus, $g_{ab}$ is a
metric tensor on $M$.

To summarize, the information in the $SI(p,q)$'s, \ie, the
geometry of space-time, is completely described by a metric tensor
field $g_{ab}$ on $M$ of signature $(-,+,+,+)$.

For obvious reasons, a vector $\xi^a$ at a point is called timelike,
null, or spacelike according as $g_{ab}\xi^a\xi^b$ is negative,
zero, or positive, respectively.

We now have a four-dimensional manifold $M$ with metric tensor
field $g_{ab}$ of signature $(-,+,+,+)$.  All the technology of
differential geometry can now be carried over to $M$.  In particular,
we have the notion of (indexed) tensor fields on $M$.  Indices of
such fields are raised and lowered with $_{ab}$ and its inverse.  We
have the (unique) derviative operator $\nabla_a$ on such tensor fields
satisfying $\nabla_a g_{bc}=0$.  We have a Riemann tensor $R_{abcd}$
satisfying the Bianchi identities, and having the algebraic
symmetries  $R_{abcd}=R_{[cd][ab]}$, $R_{[abc]d}=0$.  We construct
from the Riemann tensor the Ricci tensor and scalar curvature.

The basic philosophy is that all physics is to be described in terms
of the space-time manifold $M$.  We can now state this philosophy
a bit more precisely: physics is to be described in terms of tensor fields
on $M$.  Among these fields, easily the most important is the metric
$G_{ab}$.  There will soon be others.

We are now in a position to make more precise the distinction between
special relativity and general relativity.  The framework we have set up
above is called spacial relativity when the Riemann tensor vanishes
(everywhere), and general relativity otherwise.  (It is easily checked that
the vanishing of the Riemann tensor is precisely the condition for the
existence of  a coordinate system in which the components of the metric
take the usual special relativity form, diag$(-1,+1,+1,+1)$.  Suppose
that one views Minkowski space (the space-time of relativity) from a
``uniformly accelerated frame'', \eg, an accelerated elevator.  No matter
how the situation is ``viewed'', the Riemann tensor vanishes.
In other words, special relativity is applicable.

It is clear, from the way the metric was introduced, that it represents
geometrical information about space-time.  As we shall see shortly,
the metric has a second interpretation: as the ``potential of the
gravitational field''.

We conclude with an obvious remark.  The metric is uniquely determined
by certain physical measurements (of $SI$).  Hence, any statement
on makes about the metri automatically has physical significance.  Of
course, some statements (about $g_{ab}$) may be easier to interpret
that others, but all are in principle intrepretable.  One can freely
do mathematics, introducing physical interpretaions where
they are appropriate and interesting.


