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!

Spacetime Inverval Invariance using Lorentz Transformations

  1. Aug 27, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that the spacetime interval
    -(ct)[itex]^{2}[/itex] + x[itex]^{2}[/itex] + y[itex]^{2}[/itex] + z[itex]^{2}[/itex]
    is invariant.

    [/itex][itex]
    2. Relevant equations
    Lorentz transformations
    [itex]\Delta[/itex][itex]x' = \gamma(\Delta[/itex][itex]x-u\Delta[/itex][itex]t)[/itex]
    [itex]\Delta[/itex][itex]y' = \Delta[/itex][itex]y[/itex]
    [itex]\Delta[/itex][itex]z' = \Delta[/itex][itex]z[/itex]
    [itex]\Delta[/itex][itex]t' = \gamma(\Delta[/itex][itex]t-u\Delta[/itex][itex]x/c^{2})[/itex]


    3. The attempt at a solution
    I have tried to prove that [itex]\Delta S = \Delta S'[/itex]
    So first I said that [itex]\Delta S' = - \Delta (ct')^{2} + \Delta (x')^{2} + \Delta (y')^{2} + \Delta (z')^{2}[/itex]

    And inserted all the Lorentz Transformations above into the above formula.

    I end up simplyfying it to get

    [itex]\gamma^{2} (x^{2} + u^{2}t^{2} - c^{2}t^{2} - \frac{u^{2}x^{2}}{c^{2}}) + y^{2} + z^{2}[/itex]

    How does this equal [itex]S = - \Delta (ct)^{2} \Delta (x)^{2} + \Delta (y)^{2} + \Delta (z)^{2}[/itex] ? I can't see a way to get rid of the extra terms to get this simple function.

    Any help would be really really great!
     
  2. jcsd
  3. Aug 27, 2011 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    If you collect the terms, you'll see that the coefficient of x2 is [itex]\gamma^2(1-u^2/c^2)[/itex]. Use the definition of [itex]\gamma[/itex] to simplify that.
     
  4. Aug 27, 2011 #3
    Oh of course! How did I miss that?

    Thank you so much I got it out now!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Spacetime Inverval Invariance using Lorentz Transformations
  1. Lorentz invariance (Replies: 1)

  2. Lorentz invariants (Replies: 14)

Loading...