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- Gullstrand-Painleve coordinates and Lemaitre coordinates are based on the same foliation, but you would never know that looking at their metric expression. This thread explores that difference.

For reference, the wikipedia entries are adequate for this discussion:

https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

(henceforth, GP coordinates)

https://en.wikipedia.org/wiki/Lemaître_coordinates

(henceforth, LM coordinates)

Both of these coordinates are based on a foliation by hypersurfaces orthogonal to the congruence of radial infaller's from infinity (with asymptotically zero velocity at infinity). Both coordinate systems have the following properties:

1) The foliation hypersurfaces (surfaces of constant time coordinate in each system) are flat Euclidean 3-spaces.

2) Radial infaller's tangent 4-velocities are orthogonal to the foliation surfaces

3) For a radial infall world line, the proper time between events on it is equal to the coordinate time difference between those events.

So why do they look so different, despite using the same foliation?

GP coordinates

-----------------

The r, θ, and ϕ coordinates are the same as in both the Shwarzschild exterior and interior patches. As result, t coordinate paths (holding these coordinates constant) are the same as the integral curves of the kvf; timelike stationary world lines outside the horizon, and spacelike 'axial' paths inside the horizon. For r=2M, the t coordinate acts as a parameter of a lightlike path of the horizon.

As a result, the metric must have off diagonal elements because the t basis vector is orthogonal to the Schwarzschild foliation (being always in the direction of the kvf), and thus

Desirable results of this choice are that the metric components are independent of the t coordinate, and the flatness of spatial slices is manifest.

Undesirable result are that the coordinate expression of a radial infall world line is not trivial (you want r(t) satisfying ## dr/dt= -\sqrt {2M/}r## ). Given this expression, it is easy to then show the tangent to such a world line is orthogonal to the r basis vector. It is a good bit more work to show that property (3) holds for such a world line.

Lemaitre coordinates

-----------------------

The angular coordinates are, of course, the same as Schwarzschild case, but both the radial and timelike coordinate are different. In place of the radial coordinate defined by area of surfaces of spherical symmetry, we introduce ρ as simply an identifier of a particular radial infall trajectory. A world line of constant ρ as well as angular coordinates is a radial infall trajectory, with the LM time coordinate marking proper time along this infall trajectory.

Because this coordinate system has timelike basis orthogonal to the foliation, there are no off diagonal metric components. The metric is orthgonal, and the 3+1 coordinate distinction remains the same outside, on and inside the horizon. The changing nature of the extra kvf has no impact, because no coordinate is based on the kvf. On the other hand, because a given differences in ρ must correspond to smaller proper distances as the foliation surfaces get closer to the horizon (radial infallers get closer), the metric components must depend on the LM time coordinate.

Desirable features of Lemaitre coordinates are that the radial infall trajectories have trivial coordinate representation (all 3 spatial coordinates constant), and that proper time being coordinate time along such a trajectory is trivially obvious by inspection from the metric, and the orthogonality of such world lines to the foliation is also trivial (simply absence of off diagonal metric components). Further, the feature that all 4 coordinates maintain their spacelike/timelike character throughout the coordinate coverage is nice.

Undesirable features are that the flatness of the spatial slices is totally obscured. Further, all metric components depend on the LM time coordinate (except the tt component which is 1).

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In my next post, I will ask a specific question I would like help with.

https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

(henceforth, GP coordinates)

https://en.wikipedia.org/wiki/Lemaître_coordinates

(henceforth, LM coordinates)

Both of these coordinates are based on a foliation by hypersurfaces orthogonal to the congruence of radial infaller's from infinity (with asymptotically zero velocity at infinity). Both coordinate systems have the following properties:

1) The foliation hypersurfaces (surfaces of constant time coordinate in each system) are flat Euclidean 3-spaces.

2) Radial infaller's tangent 4-velocities are orthogonal to the foliation surfaces

3) For a radial infall world line, the proper time between events on it is equal to the coordinate time difference between those events.

So why do they look so different, despite using the same foliation?

GP coordinates

-----------------

The r, θ, and ϕ coordinates are the same as in both the Shwarzschild exterior and interior patches. As result, t coordinate paths (holding these coordinates constant) are the same as the integral curves of the kvf; timelike stationary world lines outside the horizon, and spacelike 'axial' paths inside the horizon. For r=2M, the t coordinate acts as a parameter of a lightlike path of the horizon.

*Important to note*is that the t coordinate values along these kvf paths are completely different from the Schwarzschild t coordinate, in order to achieve property (3) above.As a result, the metric must have off diagonal elements because the t basis vector is orthogonal to the Schwarzschild foliation (being always in the direction of the kvf), and thus

*not*orthogonal to the GP spatial foliation.Desirable results of this choice are that the metric components are independent of the t coordinate, and the flatness of spatial slices is manifest.

Undesirable result are that the coordinate expression of a radial infall world line is not trivial (you want r(t) satisfying ## dr/dt= -\sqrt {2M/}r## ). Given this expression, it is easy to then show the tangent to such a world line is orthogonal to the r basis vector. It is a good bit more work to show that property (3) holds for such a world line.

Lemaitre coordinates

-----------------------

The angular coordinates are, of course, the same as Schwarzschild case, but both the radial and timelike coordinate are different. In place of the radial coordinate defined by area of surfaces of spherical symmetry, we introduce ρ as simply an identifier of a particular radial infall trajectory. A world line of constant ρ as well as angular coordinates is a radial infall trajectory, with the LM time coordinate marking proper time along this infall trajectory.

Because this coordinate system has timelike basis orthogonal to the foliation, there are no off diagonal metric components. The metric is orthgonal, and the 3+1 coordinate distinction remains the same outside, on and inside the horizon. The changing nature of the extra kvf has no impact, because no coordinate is based on the kvf. On the other hand, because a given differences in ρ must correspond to smaller proper distances as the foliation surfaces get closer to the horizon (radial infallers get closer), the metric components must depend on the LM time coordinate.

Desirable features of Lemaitre coordinates are that the radial infall trajectories have trivial coordinate representation (all 3 spatial coordinates constant), and that proper time being coordinate time along such a trajectory is trivially obvious by inspection from the metric, and the orthogonality of such world lines to the foliation is also trivial (simply absence of off diagonal metric components). Further, the feature that all 4 coordinates maintain their spacelike/timelike character throughout the coordinate coverage is nice.

Undesirable features are that the flatness of the spatial slices is totally obscured. Further, all metric components depend on the LM time coordinate (except the tt component which is 1).

----

In my next post, I will ask a specific question I would like help with.

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