Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

Examination of the Event Horizon and Internal Geometry of the Schwarzschild Metric in Kruskal-Szekeres Coordinates

Version 1 : Received: 28 March 2023 / Approved: 29 March 2023 / Online: 29 March 2023 (14:31:46 CEST)
Version 2 : Received: 12 April 2023 / Approved: 13 April 2023 / Online: 13 April 2023 (12:54:36 CEST)
Version 3 : Received: 1 May 2023 / Approved: 4 May 2023 / Online: 4 May 2023 (11:42:02 CEST)
Version 4 : Received: 10 May 2023 / Approved: 11 May 2023 / Online: 11 May 2023 (14:15:52 CEST)
Version 5 : Received: 30 May 2023 / Approved: 31 May 2023 / Online: 31 May 2023 (13:23:04 CEST)
Version 6 : Received: 4 September 2023 / Approved: 5 September 2023 / Online: 5 September 2023 (10:01:55 CEST)
Version 7 : Received: 8 November 2023 / Approved: 9 November 2023 / Online: 9 November 2023 (14:49:33 CET)
Version 8 : Received: 3 December 2023 / Approved: 4 December 2023 / Online: 5 December 2023 (09:16:27 CET)
Version 9 : Received: 24 January 2024 / Approved: 25 January 2024 / Online: 5 February 2024 (15:19:21 CET)
Version 10 : Received: 10 March 2024 / Approved: 11 March 2024 / Online: 11 March 2024 (13:23:27 CET)

How to cite: Laforet, C. Examination of the Event Horizon and Internal Geometry of the Schwarzschild Metric in Kruskal-Szekeres Coordinates. Preprints 2023, 2023030512. https://doi.org/10.20944/preprints202303.0512.v3 Laforet, C. Examination of the Event Horizon and Internal Geometry of the Schwarzschild Metric in Kruskal-Szekeres Coordinates. Preprints 2023, 2023030512. https://doi.org/10.20944/preprints202303.0512.v3

Abstract

It is demonstrated mathematically that the Schwarzschild radius is the end point of gravitational collapse using the definitions of the Kruskal-Szekeres coordinates and their relationship to the Schwarzschild coordinate basis vectors over the coordinate chart. The extrinsic nature of the Kruskal-Szekeres coordinates obscures the asymptote that separates the internal and external spacetimes at the horizon. It is proven that all observers that hypothetically reach the horizon are coincident with each other at the horizon, regardless of where or when they began falling relative to each other. It is also proven that all worldlines become null geodesics at the event horizon in Kruskal-Szekeres coordinates and intersect the point T=X=0 there. In the frame of falling observers, the event horizon relativistically contracts to zero size as the horizon is approached. The internal solution is describing a spherically symmetric vacuum surrounded by an infinitely dense shell that is infinitely far away in space and exists a finite time in the past relative to an observer in the vacuum.

Keywords

Black holes; General Relativity; Schwarzschild metric

Subject

Physical Sciences, Theoretical Physics

Comments (1)

Comment 1
Received: 4 May 2023
Commenter: Christopher Laforet
Commenter's Conflict of Interests: Author
Comment: Added discussion and depiction of the Lorentz boosted frames of the frefalling particle as it falls on page 6/7
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