Notes on Relativity and Cosmology - Physics Department, UCSB

256 CHAPTER 9. BLACK HOLES
this line of simultaneity below, and called the point p 2 (where it intersects the
worldline of the second static observer) x 2 , t 2 .
x = 0
p = x , t
1
1 1
p
2
t = 0
= x , t
2 2
Now, since spacetime is curved, this line of simultaneity need not be perfectly
straight on our diagram. However, we also know that, in a very small region near
the first Free Faller (around whom we drew our diagram), space is approximately
flat. This means that the curvature of the line of simultaneity has to vanish near
the line x = 0. Technically, the curvature of this line (the second derivative of t
with respect to x) must itself be ‘of order (x 2 − x 1 ). This means that p 1 and p 2
are related by an equation that looks like:
t 2 − t 1
x 2 − x 1
= slope at p 1 + [curvature at p 1 ] (x 2 − x 1 ) + O([x 2 − x 1 ] 2 )
= slope at p 1 + O([x 2 − x 1 ] 2 ) + O(T 2 [x 2 − x 1 ]). (9.21)
Again, we need only a rough accounting of the errors. As a result, we can just
call the errors O(L 2 ) instead of O([x 2 − x 1 ] 2 ).
Remember that, in flat space, the slope of this line of simultaneity would be
v s1 /c 2 , where v s1 is the velocity of the first static observer. Very close to x = 0,
the spacetime can be considered to be flat. Also, as long as t 1 is small, the point
p 1 is very close to x = 0. So, the slope at p 1 is essentially v s1 /c 2 . Also, for small
t 1 we have v s1 = a s1 t. Substituting this into the above equation and including
the error terms yields
t 2 = t 1 ( + (a s1 t 1 /c 2 )(x 2 − x 1 ) + O(L 3 )
= t 1 1 + a )
s1
c 2 (x 2 − x 1 ) + O(L 3 ) + O(T 2 L), (9.22)
OK, we’re making progress here. We’ve already got two useful equations written
down (9.20 and 9.23), and we know that a third will be the condition that the
proper distance between p 1 and p 2 will be the same as the initial separation L
between the two free fallers:
L 2 = (x 2 − x 1 ) 2 − c 2 (t 2 − t 1 ) 2 . (9.23)