Notes on Relativity and Cosmology - Physics Department, UCSB

8.3. GRAVITY AND THE METRIC 205
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But wait! Doesn’t this seem to mean that the full spacetime curvature (gravitational
field) contains a lot more information than just specifying an acceleration
g at each point? After all, acceleration is related to how thing behave in time,
but we have just realized that at least parts of the spacetime curvature are associated
only with space. How are we to deal with this? For the answer, proceed
on to the next section below.
8.3 Gravity and the Metric
Einstein: XXIII-XXVII
Let’s recall where we are. A while back we discovered the equivalence principle:
that locally a gravitational field is equivalent to an acceleration in special relativity.
Another way of stating this is to say that, locally, a freely falling frame is
equivalent to an inertial frame in special relativity. We noticed the parallel between
this principle and the underlying ideas being calculus: that locally every
curve is a straight line.
What we found in the current chapter is that this parallel with calculus is
actually very direct. A global inertial frame describes a flat spacetime – one in
which, for example, geodesics follow straight lines and do not accelerate relative
to one another. A general spacetime with a gravitational field can be thought of
as being curved. Just as a general curved line can be thought of as being made
up of tiny bits of straight lines, a general curved spacetime can be thought of
as being made of of tiny bits of flat spacetime – the local inertial frames of the
equivalence principle.
This gives a powerful geometric picture of a gravitational field. It is nothing
else than a curvature of spacetime itself