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

A Only the geodesic distance matters for maximally symmetric spacetimes

  1. Dec 11, 2017 #1
    Any physical quantity ##K(t,x,x')## on a maximally symmetric spacetime only depends on the geodesic distance between the points ##x## and ##x'##.

    Why is this so?

    N.B.:

    This statement is different from the statement that

    The geodesic distance on any spacetime is invariant under an arbitrary coordinate transformation of that spacetime.
     
    Last edited: Dec 11, 2017
  2. jcsd
  3. Dec 11, 2017 #2

    PeterDonis

    Staff: Mentor

    Why do you think it is so? Have you found a proof of it?
     
  4. Dec 12, 2017 #3
    I haven't found a proof of it. I read this in a paper.

    This is my understanding of the problem.

    The Euclidean plane is a maximally symmetric space with ##3## translation symmetries and ##3## rotation symmetries. Any physical quantity ##K(x,y)## on the Euclidean plane, where ##x## and ##y## are two arbitrary spacetime points, is constrained by the symmetries of the spacetime to depend only on ##(x-y)^{2}##. This is because ##(x-y)## is translation invariant and ##(x-y)^{2}## is rotation invariant. Therefore, the physical quantity ##K(x,y)## depends on the Galilean-invariant geodesic distance ##(x-y)^{2}##.
     
  5. Dec 12, 2017 #4
    But I am not sure how the dependence changes if we have a Euclidean disk (that is, a plane with a boundary).

    My intuition is that the ##K(x,y)## now depends not only on the spacetime points ##x## and ##y##, but also on the 'border.' The dependence is such that ##K(x,y)## for the Euclidean disk tends to ##(x-y)^2## as the 'border' tends to infinity.

    But I am not able to carry my intuition any further and write down an explicit form for the dependence of ##K(x,y)## for the Euclidean disk.

    It would be really helpful if you share some thoughts here.
     
  6. Dec 12, 2017 #5

    PeterDonis

    Staff: Mentor

    What paper? Please give a reference.
     
  7. Dec 12, 2017 #6
  8. Dec 12, 2017 #7

    PeterDonis

    Staff: Mentor

    A plane with a boundary is not maximally symmetric.
     
  9. Dec 12, 2017 #8

    PeterDonis

    Staff: Mentor

    Ok, this mentions the proposition but doesn't give a proof. Possibly one of the references in that paper does.

    Your reasoning in post #3 seems OK to me.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...