My favourite derivation of the Schwarzschild geometry

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Dated: 2 September 2015

In this small article, I give a derivation of the Schwarzschild geometry from the field equations, skipping only over the calculation of the Ricci tensor from the assumed form of the metric. (This was done using computer algebra, and I just exhibit the result.) The calculation is set up so that the Painlevé-Gullstrand metric obtains, which is a non-singular description of the Schwarzschild geometry outside the point mass, the event horizon not constituting a coordinate singularity. I then discuss advantages and disadvantages of the standard Schwarzschild form of the metric and the Painlevé-Gullstrand form as well as their mathematical connection. Moreover, I consider some ideas about the question of an observer falling into an evaporating black hole. The "classroom approach" paper given as Ref. [2] has meanwhile been published, and a paper on a test particle approaching an evaporating black hole (and falling through the horizon without any problem) has been submitted. They are both in the list of references given below.

My favourite derivation of the Schwarzschild geometry, 02.09.2015

[1] K. Kassner, Classroom reconstruction of the Schwarzschild metric, Eur. J. Phys. 36, 065031 (20pp) (2015).

[2] K. Kassner, Radially falling test particle approaching an evaporating black hole, Canad. J. Phys. 97, 267–276 (2019). https://doi.org/10.1139/cjp-2017-1001

[3] J. Piesnack, K. Kassner, The Vaidya metric: expected and unexpected traits of evaporating black holes, Am. J. Phys. 90, 37–46 (2022). https://doi.org/10.1119/10.0006367

Next: Limiting behaviour analytic functions, large real argument   Up: Introduction science education project    Previous: Fermat's principle in general relativity

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