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 ^[1] and Allvar Gullstrand^[2] in 1922. It was not explicitly shown until 1933 in Lemaître's paper ^[3] 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

${\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt^{2}-{\frac {dr^{2}}{\left(1-{\frac {2M}{r}}\right)}}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}$

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

${\displaystyle t_{r}=t+a(r)}$

for some arbitrary function ${\displaystyle a(r)}$. Substituting in the Schwarzschild metric one gets

${\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt_{r}^{2}+2\left(1-{\frac {2M}{r}}\right)\,a'dt_{r}dr-\left({\frac {1}{1-{\frac {2M}{r}}}}-\left(1-{\frac {2M}{r}}\right)\,{a'}^{2}\right)dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}$

where ${\displaystyle a'(r)={\frac {da}{dr}}}$. If we now choose ${\displaystyle a(r)}$ such that the term multiplying ${\displaystyle dr^{2}}$ is unity, we get

${\displaystyle a'=-{\frac {1}{1-{\frac {2M}{r}}}}{\sqrt {\frac {2M}{r}}}}$

and the metric becomes

${\displaystyle g=\left(1-{\frac {2M}{r}}\right)\,dt_{r}^{2}-2{\sqrt {\frac {2M}{r}}}dt_{r}dr-dr^{2}-r^{2}\,d\theta ^{2}-r^{2}\sin ^{2}\theta \,d\phi ^{2}\,}$

The spatial metric (i.e. the restriction of the metric ${\displaystyle g|_{t=t_{r}}}$ on the surface where ${\displaystyle t_{r}}$ 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 ${\displaystyle -r^{4}\sin(\theta )^{2}}$).

The function ${\displaystyle a(r)}$ is given by

${\displaystyle a(r)=-\int {\frac {\sqrt {\frac {2M}{r}}}{1-{\frac {2M}{r}}}}dr=2M\left(-2y+\ln \left({\frac {y+1}{y-1}}\right)\right)}$

where ${\displaystyle y={\sqrt {\frac {r}{2M}}}}$. The function ${\displaystyle a(r)}$ is clearly singular at r=2M as it must be to remove that singularity in the Schwarzschild metric.

Other choices for ${\displaystyle a(r)}$ lead to other coordinate charts for the Schwarzschild vacuum; a general treatment is given in Francis & Kosowsky.^[4]

Motion of raindrop

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

${\displaystyle \frac{dr}{dt}=-<!--\; -\; -->\left(1-<!--\; -\; -->\frac{2M}{r}\right)\sqrt{\frac{2M}{r}}.}$Zdroj:https://en.wikipedia.org?pojem=Gullstrand–Painlevé_coordinates
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