Correspondence between areal radius differences and proper distances

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.

Next we derive radial spacelike geodesics in Gullestrand-Painleive coordinates (hereafter, GP coordinates). This is similarly easy in any coordinates which manifest the extra killing vector field (in coordinate terms, the metric does not depend on one of the non-angular coordinates, typically labeled t). I avoid SC coordinates due to difficulty treating horizon crossing geodesics. The feature that GP coords exclude the past horizon and two of the 4 Kruskal regions does not matter, because symmetry arguments generalize the derived results to the other cases.

Suppressing angular coordinates, the GP metric form is:
$$ds^2 = dr^2 + 2\sqrt{R/r}~drdt - (1-R/r)dt^2$$

Note, I write this such that the spacelike interval squared is positive, since we are interested in spacelike geodesics. For any spacelike curve, geodesic or not, if parameterized by arclength (s), we must then have:

$$1 = {\dot r}^2 + 2\sqrt{R/r}~\dot r \dot t - (1-R/r){\dot t}^2=L^2$$

where dots are derivatives by s. To avoid confusion, we will use primes for derivatives by t.

Next, note that the Euler-Lagrange equation for t, using the squared Langrangian (which is valid for geodesics parameterized by an affine parameter, which, of course s is) tells us that:
##\partial {(L^2)} / \partial {\dot t}## is constant since ##\partial {(L^2)} /\partial t## is 0. So we can say:
$$\dot r \sqrt{R/r}-(1-R/r)\dot t=e$$
Thus:
$$\dot t = (\dot r \sqrt{R/r}-e)/(1-R/r)$$
Substituting this in the equation for ##L^2## and simplifying gives:
$$\dot r = \pm \sqrt{(1-R/r)+e^2}$$
Then, using these last two equations and rearranging we can write:
$$r' = \dot r / \dot t = \frac{\pm (1-R/r)} {(1+(1-R/r)/e^2)^{-1/2} \pm \sqrt{R/r}} $$
This defines the desired geodesics. While this can be solved in closed form for t as a function of r, most features of the geodesics can be deduced from this r' formula directly.

Using the minus signs chooses geodesics where r' is negative for r>R. These are the horizon crossing geodesics. The others 'appear' to asymptote the horizon as t goes to ##-\infty##, but this is a coordinate artifact of the GP coordinates not including the past (white hole) horizon. If we switched to the 'outgoing' GP cooridinates, we would see that these geodesics simply cross the WH horizon, with the same geometry. Thus, by reflections and symmetry we only need to consider these r' negative 'ingoing' geodesics.

The next post will discuss the e=0 case a bit, and derive the geometric meaning of e (for spacelike geodesics). The final post in this series will derive the arclength formula.

The case of e=0 can easily be treated via limit of the r' expression of the last post. However, better is simply to go back to ##\dot r## and ##\dot t## from earlier in the post and consider e=0. Then you simply get
$$r' = (1-R/r)\sqrt{r/R}$$.
To see that this is constant Scwarzschild time curve (known to be goedesic if for no other reason than the prior post derivation), see the article https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates. There, from the first few lines of the coordinate derivation, one can see that constant SC time condition is given by ##r' = 1/a'##. This matches the expression just quoted except for sign. The sign difference is from our taking the metric form to have positive spacelike interval squared instead of negative.

To get a geometric meaning for the e, the constant of motion from the Euler-Lagrange equation we start with the observation that in GP coordinates, lines of constant t are orthogonal to the congruence of free fallers. While those knowledgeable of GP coordinates take this for granted, it will be derived here for benefit of others. First, note that the path of a free faller in GP coordinates is given by:
$$r'=-\sqrt{R/r}$$
as noted in https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates, the second equation under "motion of a raindrop". Then a vector tangent to this is simply a scaling of ##(1,-\sqrt{R/r)}##. A vector in the direction of constant GP t coordinate is simply a scaling of (0,1) - components given in (t,r) order. Taking the inner product using the metric as in post #2 you get zero, thus they are orthogonal. This implies that if ##dt/dr## is zero at some point on a curve, it is orthogonal to the free faller at that event.

Then, from post #2 we also have (just taking minus sign as discussed):
$$\dot t / \dot r = \frac {(1+(1-R/r)/e^2)^{-1/2} - \sqrt{R/r}} {- (1-R/r)} $$
and solving for zero gives ##R/r=e^2##, i.e. when r is given this value, the curve is orthogonal to the free faller there. Thus, as desired, a spacelike geodesic is characterized by ##e^2=R/r_f##, where ##r_f## is the r value at which the geodesic is orthogonal to a free faller. In addition, there is the e=0 case.

Finally, to get the arc-length formula, originally I laborious derived it from plugging r' for a geodesic into the metric and simplifying. However, noting the expression for ##\dot r## in post #2, the arc-length formula immediately follows (given that ##\dot r = dr/ds##).

Thus all claims have been derived. The upshot is a simple geometric meaning for small differences in areal radii in the radial direction (proper distance in the local frame of a colocated free faller), as well as a remarkably simple general formula for arclength along any radial spacelike geodesic. Note that it trivially reduces to what is obvious for constant SC time geodesics (##e^2=0##).

With that I am done, and hope some people found this interesting.