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PWiz
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I'm trying to understand *quote unquote thread title* by performing some simple (heuristic) analysis on my own. Before beginning, I'd like to present what I've been given to understand here at PF:
-r is not the distance from the center of a spherical shell to an arbitrary spatial coordinate on the shell surface (I don't know if one could even properly define the center of a sphere in curved spacetime, but I'm not too sure on this one)
-if two spherical masses have radii ##r_1## and ##r_2## respectively, the distance between the surfaces of these two masses is not equivalent to ##|r_1 - r_2|## (again this seems pretty obvious because space[time] may be curved, but I'd like to derive this fact nonetheless)
So let's look at a feature of the Schwarzschild metric. Since the metric is diagonalized (it describes non-rotating masses, and hence describes static spacetime geometries), it means that there is no need to discriminate between hypersurfaces. (I guess I could do without this statement, but spacetime geometries may not be static/invariant under a time reversal, which I think is the case for the Kerr metric, so I'm putting this in just to be sure) If we investigate hypersurfaces with a constant r coordinate, the metric incidentally reduces to ##ds^2 = r^2 (d \theta^2 + sin^2 \theta d\phi^2)##, which exactly describes the surface of a 2-sphere in standard spherical coordinates. The only reasonable deduction that I'm able to make with this is that the set of all 2-spheres surfaces in Schwarzschild coordinates represent the family of hypersurfaces with a fixed r coordinate (or is it the other way around?). I am refraining from changing '2-sphere surfaces' to simply '2-spheres' in the sentence above because I don't think the volumes in a flat and non-flat manifold commute (if memory serves right, I think the amount of deviation is what the contracted Riemann tensor really measures).
Is my reasoning correct? If so, then how do I define r? In terms of a geometric property that associates it with a 2-sphere surface, or maybe even a boundary of a circle?
And now about the difference in radii not being a trivial subtraction:
if I look at the proper distance between two r coordinates ##r_1## and ##r_2## (on an arbitrary submanifold of constant t), the difference takes the form $$s= \int_{r_1}^{r_2} \frac {1} {\sqrt{1- \frac {2GM}{rc^2}}} dr$$. Now I can go ahead and try to show that this does not, in general, equal to ##|r_1 - r_2|##, but before I go through the pains of evaluating this integral, I want to know if this is the correct approach to begin with.
Thanks for reading.
P.S. I'm a little tired right now, so please excuse any flagrant mistakes I might've accidently made in this post.
-r is not the distance from the center of a spherical shell to an arbitrary spatial coordinate on the shell surface (I don't know if one could even properly define the center of a sphere in curved spacetime, but I'm not too sure on this one)
-if two spherical masses have radii ##r_1## and ##r_2## respectively, the distance between the surfaces of these two masses is not equivalent to ##|r_1 - r_2|## (again this seems pretty obvious because space[time] may be curved, but I'd like to derive this fact nonetheless)
So let's look at a feature of the Schwarzschild metric. Since the metric is diagonalized (it describes non-rotating masses, and hence describes static spacetime geometries), it means that there is no need to discriminate between hypersurfaces. (I guess I could do without this statement, but spacetime geometries may not be static/invariant under a time reversal, which I think is the case for the Kerr metric, so I'm putting this in just to be sure) If we investigate hypersurfaces with a constant r coordinate, the metric incidentally reduces to ##ds^2 = r^2 (d \theta^2 + sin^2 \theta d\phi^2)##, which exactly describes the surface of a 2-sphere in standard spherical coordinates. The only reasonable deduction that I'm able to make with this is that the set of all 2-spheres surfaces in Schwarzschild coordinates represent the family of hypersurfaces with a fixed r coordinate (or is it the other way around?). I am refraining from changing '2-sphere surfaces' to simply '2-spheres' in the sentence above because I don't think the volumes in a flat and non-flat manifold commute (if memory serves right, I think the amount of deviation is what the contracted Riemann tensor really measures).
Is my reasoning correct? If so, then how do I define r? In terms of a geometric property that associates it with a 2-sphere surface, or maybe even a boundary of a circle?
And now about the difference in radii not being a trivial subtraction:
if I look at the proper distance between two r coordinates ##r_1## and ##r_2## (on an arbitrary submanifold of constant t), the difference takes the form $$s= \int_{r_1}^{r_2} \frac {1} {\sqrt{1- \frac {2GM}{rc^2}}} dr$$. Now I can go ahead and try to show that this does not, in general, equal to ##|r_1 - r_2|##, but before I go through the pains of evaluating this integral, I want to know if this is the correct approach to begin with.
Thanks for reading.
P.S. I'm a little tired right now, so please excuse any flagrant mistakes I might've accidently made in this post.
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