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 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. Painlev´e-Gullstrand coordinates The line element for the unique spherically symmetric, vacuum solution to the Einstein equation can be written as ds2 = dT 2−(dr + s 2M r dT) −r2dΩ2 (1) Note that a surface of constant T is a flat Euclidean space. 1. Which value of r corresponds to the event horizon?

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While I understand Doran coordinates and Doran form (Gullstrand-Painlevé form at a=0), I'm not entirely convinced with Gullstrand-Painlevé coordinates. While the Doran time coordinate ( t ¯) is expressed-. d t ¯ = d t + β 1 − β 2 d r. where. At the end of part 1, we looked at the form the metric of the Schwarzschild geometry takes in Gullstrand-Painleve coordinates: ds^2 = – \left( 1 – \frac{2M}{r} \right) dT^2 + 2 \sqrt{\frac{2M}{r}} dT dr + dr^2 + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right) It really does not have anything to do with the Gullstrand-Painleve coordinates. (Why is Gullstrand's name first since his paper was published later?). This section also needs a reference since Wikipedia is not supposed to be original research.


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Gullstrand painleve coordinates

Painlevé-Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner-Nordström. We predict this breakdown to occur in any region containing negative Misner-Sharp-Hernandez quasilocal mass because of repulsive gravity stopping the motion of PG observers, which are in radial free fall with zero initial Technically, the Gullstrand-Painlevé metric encodes not only a metric, but also a complete orthonormal tetrad, a set of four locally inertial axes at each point of the spacetime. The Gullstrand-Painlevé tetrad free-falls through the coordinates at the Newtonian escape velocity. To describe the dynamics of collapse, we use a generalized form of the Painlevé-Gullstrand coordinates in the Schwarzschild spacetime. The time coordinate of the form is the proper time of a free-falling observer so that we can describe the collapsing star not only outside but also inside the event horizon in a single coordinate patch. The continuation of the Schwarzschild metric across the event horizon is a well-understood problem discussed in most textbooks on general relativity.

. . . 140 is invariant under a rescaling of the spacetime coordinates. xµ = (1 + λ)xµ. (2.34). av L Bring · 2021 — Finally, we extend this to the case of nonzero curvature constant, again using Gullstrand-Painlevé coordinates.
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They consist in performing a change from coordinate time $t$ to the proper time $T$ of radially infalling observers coming from infinity at rest. The transformation is the following $$ dT=dt+\left(\frac{2M}{r}\right)^{-1/2} f(r)^{-1}dr $$ For spherically symmetric spacetimes, we show that a Painlevé–Gullstrand synchronization only exists in the region where (dr)2 ≤ 1, r being the curvature radius of the isometry group orbits Painlevé–Gullstrand (PG) coordinates [3,4] penetrating the horizon (see [5] for a review). A chacteristic feature of PG coordinates is that the three-dimensional spatial sections of spacetimes foliated by these coordinates are flat.

In 1921, Painleve proposed the Gullstrand Painleve coordinates for the Schwarzschild metric. The modification in flat spacetime Schwarzschild coordinates Kruskal Szekeres coordinates Lemaitre coordinates Gullstrand Painleve coordinates Vaidya metric Eddington, A.S necessary mathematical tools for general relativity Allvar Gullstrand Gullstrand Gullstrand-Painlevé coordinates: lt;p|>|Historical overview:| |Painlevé-Gullstrand (PG) coordinates| were proposed independently World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Painlevé–Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner–Nordström.
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