Notes on Relativity and Cosmology - Physics Department, UCSB

194 CHAPTER 8. GENERAL RELATIVITY AND CURVED SPACETIME
field does not change with time.... but that is a story that we need not go into
here 1 .
What Einstein needed was a new way of looking at things – a new language in
which to discuss gravity that would organize all of this into something relatively
simple. Another way to say this is that he needed a better conception of what
a gravitational field actually is. This next step was very hard for Albert. It
took him several years to learn the appropriate mathematics and to make that
mathematics into useful physics. Instead of going through all of the twists and
turns in the development of the subject, I’ll try to give you the rough, boiled
down version, of how all it all works out.
8.1 A return to geometry
You see, Einstein kept coming back to the idea that freely falling observers
are like inertial observers – or at least as close as we can get. Recall that, in
the presence of a general gravitational field, there really are no global inertial
frames. When we talked about our ‘error’ in thinking of a freely falling frame
as inertial, it is not the case that there is a better frame which is more inertial
than is a freely falling frame. Instead, when gravity is present there are simply
no frames of reference that act precisely in the way that global inertial frames
act.
Anyway, Einstein focussed on the fact that freely falling frames are locally the
same as inertial frames. However, he knew that things were tricky for measurements
across a finite distance. Consider, for example, the reference frame of a
freely falling person. Suppose that this person holds out a rock and releases it.
The rock is then also a freely falling object, and the rock is initially at rest with
respect to the person.
However, the rock need not remain exactly at rest with respect to the person.
Suppose, for example, that the rock is released from slightly higher up in the
gravitational field. Then, Newton would have said that the gravitational field
was weaker higher up, so that the person should accelerate toward the earth
faster than does the rock. This means that there is a relative acceleration
between the person and the rock, and that the person finds the rock to accelerate
away! A spacetime diagram in the person’s reference frame looks like this:
1 If you’re interested, you might look up the difference between ‘stationary’ and ‘static’
spacetimes in a more technical book on General Relativity.