Notes on Relativity and Cosmology - Physics Department, UCSB

9.4. STRETCHING AND SQUISHING 255
of the first static observer relative to the first free faller, and we know the
acceleration a s2 of the second static observer relative to the second free faller.
Note that the total acceleration of the second static observer in our coordinate
system is a FF2 + a s2 – her acceleration relative to the second free faller plus
the acceleration of the second free faller in our coordinate system. This is
represented pictorially on the diagram above.
Actually, there is something else that we know: since the two static observers
are, well, static, the proper distance between them (as measured by them) can
never change. We will use this result to figure out what a FF2 is.
The way we will proceed is to use the standard Physics/Calculus trick of looking
at small changes over small regions. Note that there are two parameters (T and
L, as shown below) that tell us how big our region is. L is the distance between
the two free fallers, and T is how long we need to watch the system. We will
assume that both L and T are very small, so that the accelerations a s1 and a s2
are not too different, and so that the speeds involved are all much slower than
the speed of light.
T
t = 0
L
Now, pick a point (p 1 ) on the worldline of the first static observer. Call the
coordinates of that point x 1 , t 1 . (We assume t 1 < T.) Since the velocity is
still small at that point, we can ignore the difference between acceleration and
proper acceleration and the Newtonian formula:
x 1 = 1 2 a s1t 2 1 + O(T 4 ) (9.20)
is a good approximation. The notation O(T 4 ) is read “terms of order T 4 .” This
represents the error we make by using only the Newtonian formula. It means
that the errors are proportional to T 4 (or possibly even smaller), and so become
much smaller than the term that we keep (t 2 1) as T → 0. Note that since this is
just a rough description of the errors, we can use T instead of t 1 .
Recall that the two static observers will remain a constant distance apart as
determined by their own measurements. To write this down mathematically,
we need to understand how these observers measure distance. Recall that any
observer will measure distance along a line of simultaneity. I have sketched in