1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Accelerating frame transformation

  1. Mar 21, 2016 #1

    Fek

    User Avatar

    1. The problem statement, all variables and given/known data

    In Minkowski spacetime we are considering a (series of) frame(s), S', attached to a rocket with constant proper acceleration. The rocket's speed in S is v.

    We find with boundary conditions x = 0 at t = t' = 0 the relationships between S and S' (for x' = 0, i.e. at the rocket):

    $$t = \frac{1}{a}sinh(a't') $$
    $$ x = \frac{1}{a'}[cosh(a't') - 1] $$

    Let's use these results to construct a full coordinate transformation from the lab frame x,t, to the accelerating frame x',t'. Try

    $$t = Asinh(a't') + B$$
    $$ x= Acosh(a't') + C $$

    Prove that if
    i) surface of constant t' are surfaces of constant time in a frame moving instantaneously at v
    ii) t matches with t' at early times and small x', while x agrees with x' at early times

    then A, B and C are uniquely determined and that

    $$t = (\frac{1}{a} + x') sinh(a't'/c)$$
    $$ x = (\frac{1}{a} + x') cosh(a't'/c) - \frac{1}{a} $$

    2. Relevant equations


    3. The attempt at a solution
    For x to agree with x' at early times (cosh(a't')=1) we know:

    $$A = (k + x') $$
    $$C = -k$$

    Where k is a constant.

    Then t and t' to agree at early t' and small x' we can see k= 1/a.

    However I don't know what limit condition 1 implies and cannot see a way to make the proof "watertight".
     
  2. jcsd
  3. Mar 23, 2016 #2

    Twigg

    User Avatar
    Gold Member

    Condition 1 basically says you can set up a third frame S'' at the same speed as S' relative to S, and that t'' depends only on t' (not x').
    Sorry for the brief reply, I'll elaborate when I can.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Accelerating frame transformation
Loading...