Notes on Relativity and Cosmology - Physics Department, UCSB

8.4. WHAT IS THE METRIC OF SPACETIME? 213
is called General Coordinate Invariance. Invariance is a term that captures
the idea that the physics itself does not change when we change coordinates.
This turns out to be an important principle for the mathematical formulation
of General Relativity as we will discuss further in section 8.4.
8.4 What is the metric of spacetime?
We have now come to understand that the gravitational field is encoded in the
metric. Once a metric has been given to us, we have also learned how to use
it to compute various objects of interest. In particular, we have learned how
to test a space to see if it is flat by computing the ratio of circumference to
radius for a circle. However, all of this still leaves open what is perhaps the
most important question: just which metric is it that describes the spacetime
in which we live?
First, let’s again recall that there really is no one ‘right’ metric, since the metric
will depend on the choice of coordinates and there is no one ‘right’ choice of
coordinates. But there is a certain part of the metric that is in fact independent
of the choice of coordinates. That part is called the ‘geometry’ of the spacetime.
It is mathematically very complicated to write this part down by itself. So, in
practice, physicists work with the metric and then make sure that the things
they calculate do not depend on the choice of coordinates.
OK then, what determines the right geometry? Recall that the geometry is
nothing other than the gravitational field. So, we expect that the geometry
should in some way be tied to the matter in the universe: the mass, energy,
and so on should control the geometry. Figuring out the exact form of this
relationship is a difficult task, and Einstein worked on it for a long time. We will
not reproduce his thoughts in any detail here. However, in the end he realized
that there were actually not many possible choices for how the geometry and
the mass, energy, etc. should be related.
8.4.1 The Einstein equations
It turns out that, if we make five assumptions, then there is really just one
family of possible relationships. These assumptions are:
1) Gravity is spacetime curvature, and so can be encoded in a metric.
2) General Coordinate Invariance: Real physics is independent of the choice of
coordinates used to describe it.
3) The basic equations of general relativity should give the dynamics of the
metric, telling how the metric changes in time.
4) Energy (including the energy in the gravitational field) is conserved.
5) The (local) equivalence principle.