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Family of nested spheres in Schwarzschild spacetime

  1. Nov 2, 2011 #1

    PeterDonis

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    On the Wikipedia page on Schwarzschild coordinates...

    http://en.wikipedia.org/wiki/Schwarzschild_coordinates

    ...it talks about a "family of nested spheres": each surface of constant t and r is a 2-sphere (i.e., setting dt = dr = 0 and r = constant in the metric results in a Euclidean 2-sphere). At the top of the page, however, it says:

    I've bolded the sentence that I'm wondering about. I understand that the chart as a whole is singular at the horizon, so the interior and exterior charts are disconnected. But it seems to me that there are some subtleties about the nested 2-spheres that are worth mentioning:

    (1) In the Schwarzschild *interior* chart, the t coordinate is spacelike and the r coordinate is timelike. But the angular part of the metric in these coordinates is the same inside the horizon as outside, so setting dt = dr = 0 and r = constant should work the same. Yes, r is a timelike coordinate, but that only means [itex]\partial / \partial r[/itex] is a timelike vector instead of a spacelike one; it doesn't affect the physical meaning of a constant value of r relative to the area of a 2-sphere at r, correct?

    (2) Another way of expressing #1 would be to point out that in Painleve coordinates, for example, the physical definition of r is the same: a 2-sphere at radial coordinate r has physical area [itex]4 \pi r^{2}[/itex]. The only difference is that in these coordinates, [itex]\partial / \partial r[/itex] is a spacelike vector all the way down to r = 0. So since a given physical 2-sphere is labeled by the same r in both coordinate charts, its physical area must be [itex]4 \pi r^{2}[/itex] regardless of which chart we are regarding r as a part of; i.e., the family of nested 2-spheres, physically, must run all the way in to r = 0.

    (3) Bringing up Painleve coordinates also raises another issue: at the horizon, Schwarzschild coordinates are singular, but physically, there is still a 2-sphere there, with physical area [itex]4 \pi r^{2} = 16 \pi M^{2}[/itex]. Painleve coordinates are not singular at r = 2M so this can be seen directly in those coordinates by setting r = 2M, dt = dr = 0.

    (4) Finally, the bit about Schwarzschild spacetime not being static inside the horizon: that means that a curve of constant r, theta, phi, which is timelike outside the horizon, is spacelike inside the horizon (and null *on* the horizon). But that doesn't affect the fact that a surface of constant t and r (or constant Painleve time T and r) is a spatial 2-sphere. It just means that, on the horizon, a curve that stays on that 2-sphere for all time is null (the path of a light ray), and inside the horizon, a curve that stays on that 2-sphere is spacelike (i.e., no object can move on it, not even light).

    Have I got all the above correct?
     
  2. jcsd
  3. Nov 2, 2011 #2

    Bill_K

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    Science Advisor

    Looks right to me.
     
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