Investigating Acceleration and Orbits in Rindler Spacetime

[latentcorpse]

On page 36 of these notes:
http://arxiv.org/PS_cache/gr-qc/pdf/9707/9707012v1.pdf
we are given a proof of the claim at the bottom of p35.
However, this proof doesn't actually seem to do anything! All he does in the proof is shows what the acceleration is - he doesn't make any connection with orbits at [itex]x=a^{-1}[/itex]. Am I missing something here?

And then in the example at the bottom of p36 for RIndler spacetime, i can see how we get 2.125 but then suddenly this just becomes 1/x in 2.126. Why is that? Is he just saying here that using the proposition on the last page that [itex]|a|[/itex] must be equal to [itex]\frac{1}{x}[/itex]. I'm just confused by the way he's written it - it looks like he got it to 1/x by some sort of substitution or something?Also, in the statement of the proposition on p35, why does x=1/a correspond to an orbit of k?

Then, what has he done in (2.128)?

And again, the last bit of this subsection confuses me - he says that such an observer (i.e. one with const x) is one with const a (since x=1/a in Rindler). I don't get the bit about why they aren't special because the normalisation was arbitrary though?

How does that definition in the box under 2.130 come about?

Cheers

 

[Nate810]

!The proof on page 36 does show that the acceleration is a constant for all orbits at x = a^(-1). This is done by showing that the derivative of the acceleration is equal to zero. In the example at the bottom of page 36, we are given the acceleration as 2.125, and then this is used in the equation from the proposition on page 35 (a = 1/x) to get the result that x = 1/2.125 = 1/a.In the statement of the proposition on page 35, x = 1/a corresponds to an orbit with constant acceleration because the acceleration, a, is inversely proportional to x. That is, when x increases, a decreases, and vice versa.In (2.128), the author is finding the general form of the acceleration a at any point x. He does this by substituting x = 1/a into the equation from the proposition on page 35.The definition in the box under 2.130 comes from the fact that the normalization of the acceleration was arbitrary. That is, the acceleration can take any value and still be considered valid. Therefore, the observer with constant x is not special, since the acceleration could have taken any value and still been valid.