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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 \partial / \partial r 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 4 \pi r^{2}. The only difference is that in these coordinates, \partial / \partial r 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 4 \pi r^{2} 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 into 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 4 \pi r^{2} = 16 \pi M^{2}. 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?
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 \partial / \partial r 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 4 \pi r^{2}. The only difference is that in these coordinates, \partial / \partial r 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 4 \pi r^{2} 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 into 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 4 \pi r^{2} = 16 \pi M^{2}. 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?