Gullstrand–Painlevé coordinates
The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial slices are flat.
There is no coordinate singularity at the Schwarzschild radius (event horizon).
The solution was proposed independently by Paul Painlevé in 1921 [1] and Allvar Gullstrand[2] in 1922.
It was not explicitly shown that these solutions were simply coordinate transformations of the usual Schwarzschild solution until 1933 in Lemaître's paper,[3] although Einstein immediately believed that to be true.
He does not directly make measurements of events that occur in different places.
Observers local to the events are enlisted to make measurements and send the results to him.
The bookkeeper gathers and combines the reports from various places.
The numbers in the reports are translated into data in Schwarzschild coordinates, which provide a systematic means of evaluating and describing the events globally.
is constant) is simply the flat metric expressed in spherical polar coordinates.
lead to other coordinate charts for the Schwarzschild vacuum; a general treatment is given in Francis & Kosowsky.
[4] Define a raindrop as an object which plunges radially toward a black hole from rest at infinity.
[5] In Schwarzschild coordinates, the velocity of a raindrop is given by In GP coordinates, the velocity is given by Integrate the equation of motion: Using this result for the speed of the raindrop we can find the proper time along the trajectory of the raindrop in terms of the time
We have I.e. along the rain drops trajectory, the elapse of time
One could have defined the GP coordinates by this requirement, rather than by demanding that the spatial surfaces be flat.
The metric obtained if, in the above, we take the function f(r) to be the negative of what we choose above is also called the GP coordinate system.
particles which leave the black hole travelling outward with just escape velocity so that their speed at infinity is zero.
In the usual GP coordinates, such particles cannot be described for r<2M.
The Original form of the GP coordinates is regular across the future horizon (where particles fall into when they fall into a black hole) while the alternative negative version is regular across the past horizon (from which particles come out of the black hole if they do so).
How does the universe look like as seen by a rain observer plunging into the black hole?
[6] The view can be described by the following equations: where Because of spherical symmetry, the trajectory of light always lies in a plane passing through the center of sphere.
of the distant star, is determined by numerically integrating
Although the publication of Gullstrand's paper came after Painlevé's, Gullstrand's paper was dated 25 May 1921, whereas Painlevé's publication was a writeup of his presentation before the Academie des Sciences in Paris on 24 October 1921.
[7] Both Painlevé and Gullstrand used this solution to argue that Einstein's theory was incomplete in that it gave multiple solutions for the gravitational field of a spherical body, and moreover gave different physics (they argued that the lengths of rods could sometimes be longer and sometimes shorter in the radial than the tangential directions).
The "trick" of the Painlevé proposal was that he no longer stuck to a full quadratic (static) form but instead, allowed a cross time-space product making the metric form no longer static but stationary and no longer direction symmetric but preferentially oriented.
In a second, longer paper (November 14, 1921),[8] Painlevé explains how he derived his solution by directly solving Einstein's equations for a generic spherically symmetric form of the metric.
The result, equation (4) of his paper, depended on two arbitrary functions of the r coordinate yielding a double infinity of solutions.
We now know that these simply represent a variety of choices of both the time and radial coordinates.
In Einstein's reply letter (December 7),[9] he apologized for not being in a position to visit soon and explained why he was not pleased with Painlevé's arguments, emphasising that the coordinates themselves have no meaning.
Finally, Einstein came to Paris in early April.
On the 5th of April 1922, in a debate at the "Collège de France" [10][11] with Painlevé, Becquerel, Brillouin, Cartan, De Donder, Hadamard, Langevin and Nordmann on "the infinite potentials", Einstein, baffled by the non quadratic cross term in the line element, rejected the Painlevé solution.
