Gullstrand–Painlevé coordinates are –Painlevéa 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 until 1933 in Lemaître's paper that these solutions were simply coordinate transformations of the usual Schwarzschild solution, 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
G=1=c
t, r, θ, φ are the Schwarzschild coordinates, M is the mass of the black hole.
GP coordinates
Define a new time coordinate by
for some arbitrary function . Substituting in the Schwarzschild metric one gets
where . If we now choose such that the term multiplying is unity, we get
and the metric becomes
The spatial metric (i.e. the restriction of the metric on the surface where 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 is given by
where . The function is clearly singular at r=2M as it must be to remove that singularity in the Schwarzschild metric.
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
The speed tends to 0 as r approaches the event horizon. The raindrop appears to have slowed as it gets nearer the event horizon and halted at the event horizon as measured by the bookkeeper. Indeed, an observer outside the event horizon would see the raindrop plunges slower and slower. Its images infinitely redshifted and never make it through the event horizon. However, the bookkeeper does not physically measure the speed directly. He translates data relayed by the shell observer into Schwarzschild values and compute the speed. The result is only an accounting entry.