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What are Rosen Coordinates

  1. Mar 12, 2014 #1
    I see a number of gravitational wave analytic solutions with the metric given in terms of Rosen coordinates. I have no idea what these coordinates are. How do I perform a coordinate transformation from Rosen coordinates to traditional (t,x,y,z) Euclidean\Cartesian coordinates? Also, is there a difference between Rosen coordinates and Eisenstein-Rosen coordinates?

    Thanks
    CoordinatesPLZ
     
  2. jcsd
  3. Mar 13, 2014 #2

    Bill_K

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    A coordinate system for the description of gravitational plane waves:

    ds2 = 2 du dv + gij(u) dyi dyj.

    As opposed to Brinkmann coordinates for the same wave:

    ds2 = H(u) du2 + 2 du dv + dx2 + dy2.

    Here's a reference
    that discusses both, and how to convert one to the other.
     
  4. Mar 14, 2014 #3
    That doesn't directly answer my question. Also, I am not familiar with Brinkmann coordinates. If we suppose the line element in Rosen coordinates in flat space-time is:

    ds2=2 du dv+dy12+dy22

    Then we can deduce the possible coordinate mapping to Cartesian coordinates of

    t = (u-v)/√2
    x = y1
    y = y2
    z = (u+v)/√2,

    Assuming a <-1,1,1,1> metric signature

    Is this the correct mapping from Rosen coordinates to Cartesian coordinates?
     
  5. Mar 15, 2014 #4

    Bill_K

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    Sorry. What IS your question? You say you want to know what Rosen coordinates "are", but that question has no answer. Coordinates in a curved spacetime do not always have a simple interpretation.

    If you're interested in gravitational waves, you should learn about Brinkmann coordinates. Rosen coordinates have several drawbacks. One: they are not unique. Two: they can develop coordinate singularities (caustics). That's why they are not generally used.

    That's one possibility. Here's another instance of a plane wave in Rosen coordinates that is also flat:

    ds2 = 2 du dv + u2(dy12 + dy22)

    In a flat spacetime, (u, v, y1, y2) are called light-cone coordinates, and (t, x, y, z) are called Minkowski coordinates. The terms Euclidean and Cartesian do not apply!
     
  6. Mar 15, 2014 #5
    My problem is I have a formula for the metric tensor in Rosen coordinates and I would like the metric tensor in Minkowski spacetime. To go from one to the other I need the coordinate mapping, thus, how do I transform the coordinates from Rosen coordinates to Minkowski Coordinates?
     
  7. Mar 16, 2014 #6

    Bill_K

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    Can't be done. A gravitational wave is a curved spacetime, and Minkowski coordinates exist only in flat space. You can't turn a curved spacetime into a flat spacetime just by changing the coordinates. A gravitational wave is not just Minkowski space written in some weird set of coordinates.

    You can certainly do the z = (u + v)/√2, t = (u - v)/√2 thing if you like, and that will make at least part of the metric look more familiar, but there will be terms left over.
     
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