Regular Frames for Spherically Symmetric Black Holes Revisited

Abstract

We consider a space-time of a spherically symmetric black hole with one simple horizon. As a standard coordinate frame fails in its vicinity, this requires continuation across the horizon and constructing frames which are regular there. Up to now, several standard frames of such a kind are known. It was shown in the literature before, how some of them can be united in one picture as different limits of a general scheme. However, some types of frames (the Kruskal–Szekeres and Lemaître ones) and transformations to them from the original one remained completely disjoint. We show that the Kruskal–Szekeres and Lemaître frames stem from the same root. Overall, our approach in some sense completes the procedure and gives the most general scheme. We relate the parameter of transformation e0 to the specific energy of fiducial observers and show that in the limit e0→0, a homogeneous metric under the horizon can be obtained by a smooth limiting transition.

Beyond Schwarzschild: new pulsating coordinates for spherically symmetric metrics

“…One goes back in the diagram when taking into account the interior solution, in such a way that for r < r s on can continue the ingoing path with r * starting from −∞ when r = r s until r * = 0 when r = 0. Notice that for this patch one should consider 1 − r/r s in the argument of the logarithm in (18).…”

“…we have straight lines that vary from 45 • with angle decreasing continuously and making X = 0 when r ≈ 1.27846 r s , till one reaches the line at −45 • ( r = r s ). If one is willing to include the interior solution in the same diagram, one could associate all the lines with angle ≥ −45 • again with the values of r < r s , similar to what occurs for r * [see comments below (18)]. Instead, we represent the interior solution by choosing a sign and interchanging the hyperbolic functions in ( 23) and ( 24), which makes that the lines at constant r vary continuously from the red line in Fig.…”

“…respectively. As before, α 1 D can be set equal to one without loss of generality, and for Schwarzschild solution it turns out to be useful √ ab = (2r s ) −1 , since then from (18) one obtains e √ abr * = e r/(2rs) r/r s − 1 for the exterior solution, and where the defining surfaces now are hyperbolas at constant r, with…”

“…Moreover, from the perspective of the present work, (30) is the minimal solution that uniquely represents the interior and exterior solution to the event horizon [4]. The interior is included in the other transformations by selecting e √ abr * = e r * /(2rs) 1 − r/r s due to (18) as well as by interchanging the hyperbolic functions in (28) and (29). See Figure 2, where the maximal extension is shown, and lines at t = const.…”

Symmetry Transformations in Cosmological and Black Hole Analytical Solutions

“…Here, we have f = f (r) and dΩ 2 = dθ 2 + sin 2 θ dϕ 2 . This encompasses a broad class of spherically symmetric metrics that can be put in a static form [12][13][14]. There is a subclass of FLRW-metrics can be put in the form of (2), such as the de Sitter (dS) and AdS spaces, as well as Milne and Lanczos universes.…”