I assume that there is an isometry between the two metrics (I expect it is just a coordinate transformation). Otherwise they would not be "the same". If so, then they do indeed have the same set of geodesics.

The question could be reformulated like "is this only a coordinate transformation?". The solution is spatially symmetric with respect to the origin of the coordinate system.

https://arxiv.org/pdf/physics/9905030.pdf

Aren't the two forms of the metric just related by the definition of R that you give? Is it not just a simple case of defining the new variable R?

The two expressions are obviously identical, except for a change in metric signature, and notation:

(rs, r) ↔ (α, R)

So, it is not even a genuine coordinate transformation.

The formula relating r and R in the original work by Schwarzschild is just a red herring! It is irrelevant today, as the system has become better understood. The symbol r used by Schwarzschild should not be confused with the symbol r used in the other expression.

But if you allow r in both metrics to be only positive, one metric describes the interior of the black hole and other doesnt, how can they be identical?

Why both r's should not be confused? Both are the radial coordinate in polar coordinates

In the Schwarzschild paper, both r and R is a radial coordinate, among an infinity of possibilities (the choice of R has a simple geometric interpretation).The r only serves as a somewhat arbitrary starting point for Schwarzschild, and is best forgotten. The condition r>0 is a mistake, and does not lead to a complete coordinate system. The extension to R>0 is a better one, but not complete either. F.i. the Kruskal-Szekeres coordinates provide a complete coverage, and is moreover non-singular at the event horizon (Schwarzschild radius). This Wikipedia article has a table over various possibilities: https://en.wikipedia.org/wiki/Schwarzschild_metric

I believe R is defined by Schwarzschild as an auxiliary quantity, not a coordinate, with a lower limit: α

Why is r>0 a mistake? Isnt it the original solution to einsteins equations? It is a polar coordinate and it makes sense when translating to x,y,z,t coordinates to be always positive.

You don't have to invest beliefs in scientific papers! The Schwarzschild paper is an impressive work of lasting value, but the details of how it proceeds towards the final metric are utterly irrelevant. Contary to history and religion, in science some century-old original works are extremely bad entry points to current fields of interest. Why don't you pick up some modern text(s) to learn this subject, and similar ones? Otherwise you will forever be left some 100+ years behind.

A continuous invertible transformation from x,y,z to spherical (not polar) coordinates requires the area of a sphere around the origin to shrink to zero as r -> 0. In the Schwarzschild solution this area approaches a finite value, 4 pi alpha**2. Which shows that the solution is inconsistent with the original assumptions (all details of which were never used in the later analysis).

I just want to compare both the new metric and the old original solution's metric. Because back in the days no body thought black holes were real so their made assumptions based on that case, which reflect in the math of the solutions. Hilbert treated this particular metric in a different way than Schwarzschild. He considered time as an imaginary number by a Wick rotation (back in the day, it didnt have that name), he used l=it with l being the time coordinate in polar coordinates (he uses the word Polarkoordinaten in the original german paper) :/

Kåre Olaussen Many physicists have told me now that they are not identical. I must remember that the relationship between r and R is not linear!