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- TL;DR Summary
- Differences in Schwarzschild r coordinate (areal radius) generally don't have any simple relation to reasonable distance definitions. This thread establishes a limited sense in which they do: Locally, near any event in the fully extended Schwarzschild geometry, distance measured along the spacelike radial geodesic orthogonal to a colocated free faller is simply radial coordinate difference.
Other ways of wording this finding about the extended SC (Schwarzschild) spacetime:
- in the local frame of a free faller, radial distance is given r coordinate difference
- in either Fermi-normal coordinates, or Riemann-Normal coordinates built from a free faller at an event, coordinate difference in the radial direction match areal radial difference very near the origin.
In particular, near the horizon, for a free faller crossing the horizon, differences in SC r coordinate are proper distances in the observers local frame.
Note, this result is quite different from the feature evident from Gullestrand-Painleive coordinates that for radial paths orthogonal to the free fall congruence, path length is given by r coordinate difference. This path is not a geodesic of the spacetime. The local result described here is specifically related to spacelike spacetime geodesics.
I will give the derivation in several posts, backwards from the final result.
It will be derived later that arclength along any radial spacelike geodesic in extended SC geometry is given by:
$$ \int_{r_0}^{r_1} (e^2+(1-R/r))^{-1/2}\,dr $$
Some definitions:
- e is a constant characterizing the geodesic, with meaning to be given soon
- R is Schwarzschild radius
References to free faller specifically mean either:
- free fall from infinity, i.e. the speed relative to colocated stationary observer goes to zero as r goes to infinity
- free rise (e.g. from past horizon) with the same limiting condition
Note that in the arclength formula, e=0 corresponds to constant SC time geodesics. Every other radial geodesic is orthogonal to some free faller, e.g. at r value ##r_f##. Then we have ##e^2=R/r_f##.
Then, if we consider a local free faller frame at any ##r_0##, small geodesic distances are given by:
$$ \int_{r_0}^{r_0+\delta} ((R/r_0)+(1-R/r))^{-1/2}\,dr $$
which is easily seen to be asymptotically ##\delta##. This establishes the claims given the claimed arclength formula and meaing for e. These latter results will be derived in succeeding posts.
- in the local frame of a free faller, radial distance is given r coordinate difference
- in either Fermi-normal coordinates, or Riemann-Normal coordinates built from a free faller at an event, coordinate difference in the radial direction match areal radial difference very near the origin.
In particular, near the horizon, for a free faller crossing the horizon, differences in SC r coordinate are proper distances in the observers local frame.
Note, this result is quite different from the feature evident from Gullestrand-Painleive coordinates that for radial paths orthogonal to the free fall congruence, path length is given by r coordinate difference. This path is not a geodesic of the spacetime. The local result described here is specifically related to spacelike spacetime geodesics.
I will give the derivation in several posts, backwards from the final result.
It will be derived later that arclength along any radial spacelike geodesic in extended SC geometry is given by:
$$ \int_{r_0}^{r_1} (e^2+(1-R/r))^{-1/2}\,dr $$
Some definitions:
- e is a constant characterizing the geodesic, with meaning to be given soon
- R is Schwarzschild radius
References to free faller specifically mean either:
- free fall from infinity, i.e. the speed relative to colocated stationary observer goes to zero as r goes to infinity
- free rise (e.g. from past horizon) with the same limiting condition
Note that in the arclength formula, e=0 corresponds to constant SC time geodesics. Every other radial geodesic is orthogonal to some free faller, e.g. at r value ##r_f##. Then we have ##e^2=R/r_f##.
Then, if we consider a local free faller frame at any ##r_0##, small geodesic distances are given by:
$$ \int_{r_0}^{r_0+\delta} ((R/r_0)+(1-R/r))^{-1/2}\,dr $$
which is easily seen to be asymptotically ##\delta##. This establishes the claims given the claimed arclength formula and meaing for e. These latter results will be derived in succeeding posts.