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Metric for non-inertial coordinate system

  1. Nov 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Hey guys.

    So here's the problem:

    Consider an ordinary 2D flat spacetime in Cartesian coordinates with the line element
    [itex]ds^{2}=-dt^{2}+dx^{2}[/itex]

    Now consider a non-inertial coordinate system [itex](t',x')[/itex], given by

    [itex]t'=t, x'=x-vt-\frac{1}{2}at^{2}[/itex]

    (1) What is the metric in these coordinates?

    There are some more questions apart from this but I think I can do those if I know how to do this part.

    2. Relevant equations

    None


    3. The attempt at a solution

    Okay so here's why I'm confused. How do I get the line element in these coordinates? Here are the two options in my mind...which one is correct?

    OPTION 1
    The line element they are looking for is [itex]ds^{2}=-dt^{2}+dx'^{2}[/itex] where [itex]dx'=dx-(v+at)dt[/itex]

    OPTION 2
    The line element they are looking for is [itex]ds^{2}=-dt^{2}+dx^{2}[/itex], where [itex]dx=dx'+(v+at)dt[/itex]

    Both of these options give different metrics...so which one (if any) is the way to go?

    Thanks guys!
     
  2. jcsd
  3. Nov 21, 2013 #2

    WannabeNewton

    User Avatar
    Science Advisor

    All you have to do is calculate ##dx## and ##dt## in terms of the ##(t',x')## coordinates and plug them into ##ds^2 = -dt^2 + dx^2## to get the metric in ##(t',x')## coordinates.
     
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