Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

[SR] How does Einstein solve the PDE for Tau?

  1. Jan 16, 2016 #1
    This is maybe more a maths question.

    In part 3 of his 1905 SR paper, how does Einstein solves the following PDE :

    img24.gif

    to get :

    img27.gif

    ?
     
  2. jcsd
  3. Jan 16, 2016 #2

    Ibix

    User Avatar
    Science Advisor

    He can vary x' and t' independently. If he keeps t' constant then ##\partial\tau/\partial t'## does not change - but he's still free to vary x'. That tells him that ##\partial\tau/\partial x'## is constant and independent of t' - in other words ##\tau=Ax'+f (t')##. He can make the same argument the other way around to get the dependence on t'. So the only possible solution is of the form ##\tau=Ax'+Bt'##. He then just substitutes that solution in to get the ratio of A to B, and chooses that he wants the constant a to be dimensionless.
     
  4. Jan 17, 2016 #3
    Hmm, if t is kept constant, shouldn't ∂τ/∂t be 0 ?
     
  5. Jan 17, 2016 #4

    martinbn

    User Avatar
    Science Advisor

    To solve an equation of the form
    $$
    \frac{\partial\tau}{\partial x'}+b\frac{\partial\tau}{\partial t} = 0
    $$
    where ##b## is a constant, you need the change the variables to ##\xi=x'+bt## and ##\eta=x'-bt##. Then the equation becomes very simple and you can solve it. The general solutions is
    $$
    \tau = F(x'-bt)
    $$
    where ##F## is any differentialble function. If it is linear, then you get the result in the paper.
     
  6. Jan 18, 2016 #5
    Last edited: Jan 18, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook