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I Is Flamm's paraboloid a paraboloid?

  1. Jun 26, 2016 #1
    The shape of the Einstein-Rosen bridge is often visualized/modelized with the Flamm's paraboloid, and many other references have also stated clearly that it's a "surface of revolution of a parabola".
    But as far as I can see, when we rotate the parabola w^2 = 8M(r-2M) (in natural units c=G=1) around the w axis (i.e r=0), we get no paraboloid.
    The rotation process replaces r coordinate by sqrt(x^2+y^2), so that the resulting surface equation becomes
    (w^2/8M+2M)^2 = x^2+y^2
    This is even not a quadric surface, let alone a paraboloid (which looks like a bowl).
    Only the rotation around its symmetric axis (i.e r axis) gives a paraboloid of revolution!
    Instead, the shape of the Einstein-Rosen bridge rather looks like a hyperboloid of 1 sheet (although not rigorously).

    What's annoying me is that I cannot find anyone / any article on the Internet highlighting the fact that the Flamm's paraboloid is NOT a paraboloid, am I wrong??
    Thanks for any helpful comments (for this is my first post).
     
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  3. Jun 26, 2016 #2

    A.T.

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    Maybe you rotate it around the wrong axis?
     
  4. Jun 26, 2016 #3
    But if we rotate it around the ``correct'' axis to get the real paraboloid of revolution: https://en.wikipedia.org/wiki/Paraboloid
    Now it looks nothing like the ``Flamm's paraboloid'' (or, Einstein-Rosen bridge)...
     
    Last edited: Jun 26, 2016
  5. Jun 26, 2016 #4
    From the exterior a black hole pinches a hole in spacetime so it is cut of at r=2GM/c², but when you look at the interior Schwarzschild solution you get the paraboloid in the complex plane:

    Schwarzschild_interior.jpg
     
  6. Jun 26, 2016 #5
    This is wrong.

    Spacetime is not cut at r=2GM/c² and the interior part is not parabolic but spherical.
     
  7. Jun 26, 2016 #6

    Orodruin

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    The interior Schwarzschild solution describes what happens in a spherically symmetric incompressible fluid with zero pressure on the surface. Outside you will have the regular Schwartzschild vacuum solution and the transition between the two does not occur at the Schwarzschild radius, but wherever the surface of the fluid is.
     
  8. Jun 26, 2016 #7

    PeterDonis

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    No, this is not correct. What is correct for a black hole is that the spacelike surfaces of constant Schwarzschild coordinate time ##t## in the exterior of Schwarzschild spacetime (i.e., above the horizon), whose spatial geometry is described by the Flamm paraboloid, do not extend below the horizon. This is usually reflected by cutting off the paraboloid at ##r = 2M##. But strictly speaking, the maximally extended spacetime has another exterior region, so the full spacelike surface is actually two paraboloids joined at the ##r = 2M## surface. There is no "hole in spacetime" at ##r = 2M##.
     
  9. Jun 26, 2016 #8
    No, that's the picture for spherical incompressible fluid in vacuum, which has nothing to do with the Einstein-Rosen bridge model.
     
  10. Jun 26, 2016 #9
    But then the Flamm's paraboloid looks nothing like a well-defined paraboloid!?
    Can we just rotate a parabola around an arbitrary line and call that a paraboloid!?
     
  11. Jun 26, 2016 #10

    PeterDonis

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    It depends on how fussy you want to be about terminology. The way the Flamm paraboloid is generated is perfectly clear, and the math describing it is unambiguous. Whether you think it deserves the name "paraboloid" is a question of terminology, not math or physics.
     
  12. Jun 26, 2016 #11

    A.T.

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    It's called "Flamm's paraboloid" not just "paraboloid".
     
  13. Jun 26, 2016 #12
    Alright, then we do agree that this "Flamm's paraboloid" is NOT what mathematicians defined as "paraboloid", that whenever we mention it, we should call it "Flamm's paraboloid" as a whole, right?
    I think that it's like we don't rotate an ellipse around an arbitrary inclined axis and call that an ellipsoid...
     
  14. Jun 26, 2016 #13

    PeterDonis

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    It would appear that it isn't according to the definition you referenced, no.

    That's how I've always seen it referred to in GR discussions, yes.
     
  15. Jun 26, 2016 #14
    Lucky for you, cause i've seen people calling it paraboloid surface or something in several literatures.
    In another book, the author even explains that the paraboloid is a parabola rotating around its axis of symmetry (which is generally correct but not in this case), which made me feel WTF for a second... (ok maybe that's a typo)
     
  16. Jun 26, 2016 #15

    PeterDonis

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    [Moderator's note: a subthread on the spatial geometry of constant Schwarzschild time hypersurfaces in the maximally extended spacetime has been spun off to a different thread.]
     
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