When Painlevé–Gullstrand coordinates fail

Abstract

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 velocity. PG coordinates break down also in the static Einstein universe for completely different reasons. The more general Martel-Poisson family of charts, which normally has PG coordinates as a limit, is reported for static cosmologies (de Sitter, anti-de Sitter and the static Einstein universe).

Unsettling Physics in the Quantum-Corrected Schwarzschild Black Hole

“…Another effect induced by the negative energy density is that it will be impossible to introduce Painlevé-Gullstrand coordinates [51,52] for the line element (4). In fact, these coordinates are associated with observers starting at spatial infinity with zero velocity [53] and they cannot be introduced in regions of negative quasilocal energy [54].…”

Painlevé–Gullstrand form of the Lense–Thirring Spacetime

“…This modified metric again asymptotically approaches standard Lense-Thirring (1) at large distances, but now has the two very strong advantages that (i) for J = 0 it is an exact solution of the vacuum Einstein equations and (ii) that the azimuthal dependence is now in partial Painlevé-Gullstrand form: g φφ (dφ − v φ dt) 2 = g φφ (dφ − ωdt) 2 . See the early references [23][24][25], and more recently references [26][27][28][29][30]. We shall soon see that "completing the square" in the Lense-Thirring metric is equivalent to linearizing a suitably chosen tetrad in the Kerr spacetime (trying to explicitly calculate approximate curvature components for this approximate metric is again slightly faster, and the results are again somewhat less horrid).…”

“…We shall soon see that "completing the square" in the Lense-Thirring metric is equivalent to linearizing a suitably chosen tetrad in the Kerr spacetime (trying to explicitly calculate approximate curvature components for this approximate metric is again slightly faster, and the results are again somewhat less horrid). • Third, put the r-t plane into standard Painlevé-Gullstrand form [23][24][25][26][27][28][29][30] (we note that the shift function is v r = − √ 2m/r for a Schwarzschild black hole). We then have the modified metric…”

“…That is, the constant-t spatial 3-slices of the metric are all flat, and the lapse function is unity (g tt = −1). See the early references [23][24][25], and more recently [26][27][28][29][30] (note that the vector v i , representing the "flow" of space, is minus the shift vector in the ADM formalism). Two of the virtues of putting the metric into Painlevé-Gullstrand form is that mathematically it is particularly easy to work with, and physically it is very easy to interpret-in particular, the analogue spacetimes built from excitations in moving fluids are typically (conformally) of Painlevé-Gullstrand form [31][32][33][34][35][36][37][38][39][40][41][42], and so give a very concrete and physically intuitive visualization of such spacetimes.…”

Quasi-geodesics in relativistic gravity

“…This family of coordinates contains Painlevé–Gullstrand coordinates as a special case and has as a limit the more familiar Eddington–Finkelstein coordinates [ 4 , 5 ]. Painlevé–Gullstrand and Martel–Poisson coordinates have been generalized to arbitrary static and asymptotically flat black hole spacetimes in [ 3 ] and to de Sitter and other static universes in [ 6 , 7 ]. Other coordinates based on radial timelike geodesics in the Schwarzschild geometry are the Novikov and the Gautreau–Hoffman coordinates, corresponding to observers launched at a finite radius [ 8 , 9 ].…”