Notes on Relativity and Cosmology - Physics Department, UCSB

178 CHAPTER 7. RELATIVITY AND THE GRAVITATIONAL FIELD
The answer of course is the frame that acts like an inertial frame. In this
case, this is the freely falling reference frame. We have learned that, in such a
reference frame, we can ignore gravity completely.
Now, how much sense does the above picture really make? Let’s make this easy,
and suppose that the earth were really big.... it turns out that, in this case,
the earth’s gravitational field would be nearly constant, and would weaken only
very slowly as we go upward. Does this mesh with the diagram above?
Not really..... We said that the diagram above is effectively in an inertial frame.
However, in this case we know that, if the distance between the bottom and
top of the tower does not change, then the bottom must accelerate at a faster
rate than the top does! But we just said that we want to consider a constant
gravitational field! So, what’s up?
Side note: No, it does not help to point out that the real earth’s gravitational
field is not constant. The point here is that the earth’s gravitational field
changes in a way that has nothing to do with the relationship α = c 2 /l from
the accelerated rocket.
7.3.2 How Local?
Well, we do have a way out of this: We realized before that the idea of freely
falling frames being like inertial frames was not universally true. After all, freely
falling objects on opposite side of the earth do accelerate towards each other.
In contrast, any two inertial objects experience zero relative acceleration.
However, we did say that inertial and freely falling frames are the same ‘locally.’
Let’s take a minute to refine that statement.
How local is local? Well, this is much like the question of “when is a velocity
small compared to the speed of light?” What we found before was that Newtonian
physics held true in the limit of small velocities. In the same way, our
statement that inertial frames and freely-falling frames are similar is supposed
to be true in the sense of a limit. This comparison becomes more and more
valid the smaller a region of spacetime we use to compare the two.
Nevertheless, it is still meaningful to ask how accurate this comparison is. In
other words, we will need to know exactly which things agree in the above limit.