Notes on Relativity and Cosmology - Physics Department, UCSB

9.4. STRETCHING AND SQUISHING 259
is small. What we’re going to do is to take the limit as T and L go to zero and
turn this into a differential equation. Technically, we will take T to zero faster
that L so that T 2 /L 2 → 0. Note that we are really interested in how things
change with position at t = 0, so that is is natural to take T to zero before
taking L to zero.
OK, so imagine not just two free fallers, but a whole set of them at every value
of x. Each of these starts out with zero velocity, and each of them has an
accompanying static observer. The free faller at x will have some acceleration
a FF (x), and the static observer at x will have some acceleration a s (x) relative
to the corresponding free faller. If L is very small above, notice that a s2 −a s1 =
L das
dx + O(L2 ) and that (since a FF1 = 0), a FF2 = L daFF
dx + O(L2 ).
So, we can rewrite equation (9.35) as:
L da FF
dx
= −Lda s
dx − La2 s/c 2 + O(T 2 /L) + O(L 2 ). (9.36)
We can now divide by L and take the limit as T/L and L go to zero. The result
is a lovely differential equation:
da FF
dx
= −da s
dx − a2 s /c2 (9.37)
By the way, the important point to remember about the above expression is
that the coordinate x represents proper distance. (Sound familiar?)
9.4.4 What does it all mean?
One of the best ways to use this equation is to undo part of the last step.
Say that you have two free falling observers close together that have no initial
velocity. Then, if their separation L is small enough, their relative acceleration
is L daFF
dx
or
( ) das
relative acceleration = −L
dx + a2 s /c2 . (9.38)
Let’s take a simple example of this. Suppose that you are near a black hole
and that your head and your feel are both freely falling objects. Then, this formula
tells you at what acceleration your head would separate from (or, perhaps,
accelerate toward) your feet. Of course, your head and feet are not, in reality,
separate freely falling objects. The rest of your body will pull and push on them
to keep your head and feet roughly the same distance apart at all times. However,
your head and feet will want to separate or come together, so depending
on how big the relative acceleration is, keeping your head and feet in the proper
places will cause a lot of stress on your body.
For example, suppose that the relative acceleration is 10m/s 2 (1g) away from
each other. In that case, the experience would feel much like what you feel if