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Volume 37: Pages 74-79, 2024
How Schwarzschild could have discovered and fixed the problem with his metric
Lane R. Miller,1,a)Larry M. Silverberg,2,b)Jeffrey W. Eischen,2,c)and Samuel R. Miller3,d)
13328 Bentwillow Drive, Fuquay Varina, North Carolina 27526, USA
2Department of Mechanical and Aerospace Engineering, North Carolina State University, Campus Box 7910, Raleigh, North Carolina 27695-7910, USA
623 Torrey Circle Q108, San Diego, California 92130, USA
Karl Schwarzschild’s landmark paper revealed the first ever exact solution to Einstein’s gravitational field equation. This was a major scientific achievement. However, his solution for the metric of a spherically symmetric spacetime is different than the one found in textbooks today— because Schwarzschild assumed, like in Newtonian gravity, that only one singularity could exist. Given the newness of Einstein’s general theory of relativity and the existing paradigm of Newtonian gravity, his assumption was natural. While other authors have previously pointed out this shortcoming and corrected it, the contribution of this paper is to show how Schwarzschild might have discovered it himself. A simple geometric proof indicates how Schwarzschild, with a different assignment of a single constant of integration, would have arrived at the metric found in textbooks today.
L’article historique de Karl Schwarzschild révèle la toute première solution exacte de l’équation du champ gravitationnel d’Einstein.. Ce fut une réalisation scientifique majeure. Cependant, sa solution pour la métrique d’un espace-temps à symétrie sphérique est différente de celle trouvée dans les manuels scolaires aujourd’hui – parce que Schwarzschild supposait, comme dans la gravité newtonienne, qu’une seule singularité pouvait exister. Compte tenu de la nouveauté de la théorie de la relativité générale d'Einstein et du paradigme existant de la gravité newtonienne, son hypothèse était logique. Alors que d’autres auteurs ont déjà souligné cette lacune et l’ont corrigée, la contribution de cet article est de montrer comme Schwarzschild aurait pu la découvrir lui-même. Une simple preuve géométrique indique comme, avec une affectation différente d'une seule constante d'intégration, Schwarzschild serait arrivé à la métrique trouvée dans les manuels scolaires aujourd'hui.
Key words: Schwarzschild; Gravity; Black Hole; Singularity; General Relativity; Einstein’s Equation.
Received: October 3 2023; Accepted: January 1 2024; Published Online: January 23, 2024
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