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Is This Spacetime Geometry Mathematically Conceivable?

  1. May 19, 2003 #1
    Is it possible to invent a non-Riemannian geometry to justify the existence of a "metric" of the form:

    1/ds^2 = 1/dt^2 – 1/(dx^2 + dy^2 + dz^2)

    Eugene Shubert
    http://www.everythingimportant.org/relativity
     
  2. jcsd
  3. May 20, 2003 #2
    isn't that just
    ds2 = dt2- (dx2+dy2+dz2)
    ?
     
  4. May 20, 2003 #3
  5. May 20, 2003 #4
    Umm, I would agree with schwarzchildradius here. Just multiply through and suddenly you get rid of the nasty fractions.
     
  6. May 21, 2003 #5
    Yes I do remember elementary algebra, good for me. you can invert that equation.
     
  7. May 22, 2003 #6
    Would you like to expain in more detail how you think you can invert that fraction to get the required result?
     
  8. May 22, 2003 #7
    I thought flipping a fraction such as 1/3^2 would result in 3^-2.

    Doesn't it change the exponent?
     
  9. May 22, 2003 #8
    C'mon guys....

    1/ds2 = 1/dt2 – 1/(dx2 + dy2 + dz2)
    1/ds2 = (dx2 + dy2 + dz2 - dt2)/[ (dx2 + dy2 + dz2)(dt2) ]

    ds2 = [ (dx2 + dy2 + dz2)(dt2) ]/(dx2 + dy2 + dz2 - dt2)

    Which just doesn't look any cleaner.

    edit: changed to using integrated superscript.
     
    Last edited by a moderator: May 22, 2003
  10. May 22, 2003 #9
    That was my point.
     
  11. May 22, 2003 #10
    Let me suggest the physical meaning to the expression above.

    I’m thinking of ds as an invariant that represents a differential increment of proper time. That would imply that the total amount of elapsed proper time t' would equal t/sqrt (1-1/V^2) where V^2 = (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2. I would interpret V^2 > 1 to be a superluminal velocity.
     
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