Karl Schwarzschild: Solving GR on the Eastern Front

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Discussion Overview

The discussion centers on Karl Schwarzschild's contributions to general relativity (GR) during World War I, particularly his development of the Schwarzschild solution. Participants explore the context of his work, the physics available to him at the time, and the mathematical challenges he faced in finding solutions that aligned with physical reality.

Discussion Character

  • Exploratory
  • Technical explanation
  • Historical

Main Points Raised

  • Some participants note that any Lorentzian metric can be considered an exact solution, but this does not ensure that the solution is physically reasonable.
  • Others argue that Schwarzschild had access to Einstein's field equations and sought a solution under the conditions of vacuum and spherical symmetry, leading to the unique solution known as the Schwarzschild solution.
  • A participant mentions that Schwarzschild was not focused on the weak field limit, as Einstein had already addressed the precession of Mercury, but rather aimed for a more general solution.
  • It is noted that Schwarzschild also provided a solution for a compact star model using an incompressible fluid, which was published in 1916.
  • Some participants discuss the historical context of Schwarzschild's life, including his health issues and his son's later contributions to astrophysics.

Areas of Agreement / Disagreement

Participants generally agree on the significance of Schwarzschild's work and the conditions under which he derived his solutions. However, there are differing views on the implications of his findings and the nature of the solutions he explored, indicating that multiple perspectives remain in the discussion.

Contextual Notes

There are unresolved aspects regarding the assumptions made in deriving the solutions, such as the nature of the stress-energy tensor and the implications of Birkhoff's Theorem, which are not fully explored in the discussion.

Who May Find This Useful

Readers interested in the historical development of general relativity, the mathematical challenges in finding solutions to Einstein's equations, and the contributions of Karl Schwarzschild and his family to physics may find this discussion informative.

BWV
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Read the bio / fiction chapter on Karl Schwarzschild in Benjamin Labatut’s great Book, and curious on a little color on how he developed the solution - I had thought finding an exact solution in GR was just math chops, but actually any Lorentzian metric is an exact solution, so the difficulty was in finding a solution that reproduced the physics, but what physics would Swchwarzchild had in 1915 on the Eastern front - just the precession of Mercury, which was in the copy Einsteins GR paper he has?

FWIW, he was not directly in the trenches, he foolishly volunteered at age 40 to serve as an artillery specialist where he could employ abilities. He also was wasting away with Pemphigus, a nasty genetic skin disease that Ashkenazi Jews are susceptible to
 

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BWV said:
what physics would Swchwarzchild had in 1915 on the Eastern front
He had Einstein's field equation; Einstein had sent him a preprint of his paper giving the final correct version of the field equation. He then looked for a solution that satisfied the assumptions of vacuum (zero stress-energy) and spherical symmetry, and found the solution that now bears his name. We now know that this is the unique solution for those conditions (this result is known as Birkhoff's Theorem and was proved, IIRC, in the early 1920s).

BWV said:
just the precession of Mercury, which was in the copy Einsteins GR paper he has?
Schwarzschild wasn't interested in solving the weak field limit; EInstein had already done that and showed that the precession of Mercury came out. He was interested in the most general possible solution for the conditions given (vacuum and spherical symmetry). (He also found a solution for the case of spherical symmetry and a perfect fluid with constant density, i.e., describing a highly idealized spherical planet or star.)
 
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BWV said:
any Lorentzian metric is an exact solution
In the sense that you can compute its Einstein tensor and call that, adjusted by an appropriate constant factor, the "stress-energy tensor" of your solution, yes. But, as you note, this makes no guarantee whatever that the resulting solution will describe anything physically reasonable.

The more usual approach is to make some reasonable assumptions about things like symmetries of the spacetime (as Schwarzschild assumed spherical symmetry) and some general form for the stress-energy tensor (as Schwarzschild assumed vacuum, and then for his other solution he assumed a perfect fluid with constant density). That allows you to simplify the form of the metric using the symmetries, compute its Einstein tensor to give a set of differential equations for the metric components, and then use your assumption about the stress-energy tensor to determine the solution.
 
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Karl’s son Martin was an accomplished Astrophysicist, who fortunately fled Germany in the 30s and worked for US intelligence during the war, before landing at Princeton where he worked on stellar evolution, dying in 1997. No doubt some here knew him
 
PeterDonis said:
He had Einstein's field equation; Einstein had sent him a preprint of his paper giving the final correct version of the field equation. He then looked for a solution that satisfied the assumptions of vacuum (zero stress-energy) and spherical symmetry, and found the solution that now bears his name. We now know that this is the unique solution for those conditions (this result is known as Birkhoff's Theorem and was proved, IIRC, in the early 1920s).
Amazingly, he also gave a first solution for a "compact star" (non-vacuum solution, using the model of an incompressible fluid). Both papers appeared in the "Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin) 1916":

https://ui.adsabs.harvard.edu/abs/1916SPAW...189S/abstract
https://ui.adsabs.harvard.edu/abs/1916skpa.conf..424S/abstract