Coordinate families for the Schwarzschild geometry based on radial timelike geodesics

Abstract

We explore the connections between various coordinate systems associated with observers moving inwardly along radial geodesics in the Schwarzschild geometry. Painlevé-Gullstrand (PG) time is adapted to freely falling observers dropped from rest from infinity; Lake-Martel-Poisson (LMP) time coordinates are adapted to observers who start at infinity with non-zero initial inward velocity; Gautreau-Hoffmann time coordinates are adapted to observers dropped from rest from a finite distance from the black hole horizon. We construct from these an LMP family and a proper-time family of time coordinates, the intersection of which is PG time. We demonstrate that these coordinate families are distinct, but related, one-parameter generalizations of PG time, and show linkage to Lemaître coordinates as well.

Interior volume of (1 + D)-dimensional Schwarzschild black hole

“…One can check that for D = 3 its value matches with the earlier analysis [9], done for usual (1 + 3) dimensional case. Substituting this in (6) we obtain the maximum volume inside the horizon as…”

Slicing black hole spacetimes

“…In the literature, adapted coordinates to these situations (in the special case of a Schwarzschild spacetime) are referred to as "hail" (L 0 > 1), "rain" (L 0 = 1) and "drip" (L 0 < 1) coordinates (see, e.g., Ref. [2] and references therein). Let us consider some explicit examples.…”

“…Geodesic observer families instead characterize the rain, drip and hail coordinate systems [1,2] which include the Painlevé-Gullstrand coordinates [3,4] and their generalizations [5]. For nonrotating black holes the latter are also characterized by the intrinsic curvature properties of their associated slicing, whose induced geometry is flat.…”

When Painlevé–Gullstrand coordinates fail

“…A chacteristic feature of PG coordinates is that the three-dimensional spatial sections of spacetimes foliated by these coordinates are flat. PG coordinates constitute a very useful chart also in other problems in classical and quantum gravity where Schwarzschild-like (or "curvature") coordinates fail [1,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Therefore, from the point of view of tool building and in view of their many applications, it is important to have a complete understanding of PG coordinates.…”