Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. 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 outgoing ones are simply the time reverse of ingoing coordinates (the time is the proper time along outgoing particles that reach infinity with zero velocity). The solution was proposed independently by Paul Painlevé in 1921 and Allvar Gullstrand 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, although Einstein immediately believed that to be true.

Derivation

The derivation of GP coordinates requires defining the following coordinate systems and understanding how data measured for events in one coordinate system is interpreted in another coordinate system. Convention: The units for the variables are all geometrized. Time and mass have units in meters. The speed of light in flat spacetime has a value of 1. The gravitational constant has a value of 1. The metric is expressed in the +−−− sign convention.

Schwarzschild coordinates

A Schwarzschild observer is a far observer or a bookkeeper. He does not directly make measurements of events that occur in different places. Instead, he is far away from the black hole and the events. 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. Thus, the physicist can compare and interpret the data intelligently. He can find meaningful information from these data. The Schwarzschild form of the Schwarzschild metric using Schwarzschild coordinates is given by where

GP coordinates

Define a new time coordinate by for some arbitrary function a(r). Substituting in the Schwarzschild metric one gets where. If we now choose a(r) such that the term multiplying dr^2 is unity, we get and the metric becomes The spatial metric (i.e. the restriction of the metric on the surface where t_r is constant) is simply the flat metric expressed in spherical polar coordinates. This metric is regular along the horizon where r=2M, since, although the temporal term goes to zero, the off-diagonal term in the metric is still non-zero and ensures that the metric is still invertible (the determinant of the metric is ). The function a(r) is given by where. The function a(r) is clearly singular at r=2M as it must be to remove that singularity in the Schwarzschild metric. Other choices for a(r) lead to other coordinate charts for the Schwarzschild vacuum; a general treatment is given in Francis & Kosowsky.

Motion of raindrop

Define a raindrop as an object which plunges radially toward a black hole from rest at infinity. 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 t_r. We have I.e. along the rain drops trajectory, the elapse of time t_r is exactly the proper time along the trajectory. One could have defined the GP coordinates by this requirement, rather than by demanding that the spatial surfaces be flat. A closely related set of coordinates is the Lemaître coordinates, in which the "radial" coordinate is chosen to be constant along the paths of the raindrops. Since r changes as the raindrops fall, this metric is time dependent while the GP metric is time independent. 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. The only change in the metric is that cross term changes sign. This metric is regular for outgoing raindrops—i.e. 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. They have a zero value for at r=2M. This is an indication that the Schwarzschild black hole has two horizons, a past horizon, and a future horizon. 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). The Kruskal–Szekeres coordinates are regular across both horizons at the expense of making the metric strongly dependent on the time coordinate.

Speeds of light

Assume radial motion. For light, d\tau =0. Therefore, The result is

A rain observer's view of the universe

How does the universe look like as seen by a rain observer plunging into the black hole? 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. It's possible to simplify the metric by assuming. The impact parameter can be computed knowing the rain observer's r-coordinate r_0,! and viewing angle. Then, the actual angle \ \phi,! of the distant star, is determined by numerically integrating dr,! from r_0,! to infinity. A chart of the sample results is shown at right.

History

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. In this way, Gullstrand's work appears to have priority. 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), 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. Painlevé wrote to Einstein to introduce his solution and invited Einstein to Paris for a debate. In Einstein's reply letter (December 7), 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" 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.