Related papers: Some notes on the Kruskal - Szekeres completion
In this pedagogical note, the Kruskal - Szekeres completion of the Schwarzschild spacetime is obtained explicitly in a few steps.
Starting from the Einstein equations in Schwarzschild-de Sitter (SdS) spacetime and imposing Friedmann-Robertson-Walker coordinates at large distances, we find two coordinate systems with time-dependent metrics that are smooth across both…
The Kruskal-Szekeres coordinates construction for the Schwarzschild spacetime could be viewed geometrically as a squeezing of the $t$-line associated with the asymptotic observer into a single point, at the event horizon $r=2M$. Starting…
We introduce a coordinate system that complements the Kruskal--Szekeres extension. Like the standard construction, it covers the maximally extended Schwarzschild manifold in its entirety, while offering an additional advantage of expressing…
Static spherically symmetric spacetimes with vanishing second Ricci invariant constitute an important class of solutions to Einstein's equations and more generally as archetypes of regular black holes. When studying completeness one is most…
The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g. $r=2M$ in the Schwarzschild black hole) or to an…
The continuation of the Schwarzschild metric across the event horizon is almost always (in textbooks) carried out using the Kruskal-Szekeres coordinates, in terms of which the areal radius r is defined only implicitly. We argue that from a…
Among the coordinates used to construct a conformal compactification of the Schwarzschild spacetime, none of them simultaneously extend smoothly both through an event horizon and beyond null infinity.To construct such coordinates, instead…
We discuss how to construct the full Schwarzschild (Kruskal-Szekeres) spacetime in one swoop by using the bundle of orthonormal Lorentz frames and the Einstein equation without the use of coordinates. We never have to write down the…
A variety of historical coordinates in which the Schwarzschild metric is regular over the whole of the extended spacetime are compared and the hypersurfaces of constant coordinate are graphically presented. While the Kruscal form (one of…
Schwarzschild coordinates (r,t) fail to describe the region within the event horizon (EH), (r <= 2 M), of a Black Hole (BH) because the metric coefficients exhibit singularity at r=2 M, and the radial geodesic of a particle appears to be…
We formulate the concept of time machine structure for spacetimes exhibiting a compactely constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by A. Ori, we…
The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other…
The well-known Schwarzschild black hole was first obtained as a stationary, spherically symmetric solution of the Einstein's vacuum field equations. But until thirty years later, efforts were made for the analytic extension from the…
We present the geodesical completion of the Schwarzschild black hole in four dimensions which covers the entire space in (u,v) Kruskal-Szekeres coordinates, including the spacetime behind the black and white hole singularities. The…
We reconsider space-time singularities in classical Einsteinian general relativity: with the help of several new co-ordinate systems we show that the Schwarzschild solution can be extended beyond the curvature singularity at r=0. The…
A reformulation of the Schwarzschild solution of the linearised Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the…
Maximally extended, explicit and regular coverings of the Schwarzschild - de Sitter family of vacua are given, first in spacetime (generalizing a result due to Israel) and then for all dimensions $D$ (assuming a $D-2$ sphere). It is shown…
We find a specific coordinate system that goes from the Painlev\'e-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by…
Every general relativity textbook emphasizes that coordinates have no physical meaning. Nevertheless, a coordinate choice must be made in order to carry out real calculations, and that choice can make the difference between a calculation…