Karl Schwarzschild: Solving GR on the Eastern Front

  • Context: Undergrad 
  • Thread starter Thread starter BWV
  • Start date Start date
  • Tags Tags
    Gr Schwarzschild
Click For Summary
SUMMARY

Karl Schwarzschild developed the Schwarzschild solution to Einstein's field equations in 1915 while serving as an artillery specialist during World War I. He sought a solution that satisfied the conditions of vacuum and spherical symmetry, ultimately leading to the unique solution known as Birkhoff's Theorem. Schwarzschild's work was influenced by the precession of Mercury, which was detailed in Einstein's paper that he received prior to his findings. Additionally, he provided a solution for a compact star using the model of an incompressible fluid, with both papers published in the "Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften" in 1916.

PREREQUISITES
  • Understanding of Einstein's field equations
  • Familiarity with Lorentzian metrics
  • Knowledge of vacuum solutions in general relativity
  • Concept of spherical symmetry in physics
NEXT STEPS
  • Study Birkhoff's Theorem and its implications in general relativity
  • Explore the derivation of the Schwarzschild solution in detail
  • Research the precession of Mercury and its significance in validating general relativity
  • Investigate models of compact stars and their equations of state
USEFUL FOR

Physicists, astrophysicists, and students of general relativity seeking to deepen their understanding of the historical context and mathematical foundations of Schwarzschild's contributions to the field.

BWV
Messages
1,631
Reaction score
1,991
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
 

Attachments

  • 1660503254308.jpeg
    1660503254308.jpeg
    39.8 KB · Views: 144
Physics news on Phys.org
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.)
 
Reply
  • Informative
Likes   Reactions: BWV
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.
 
Reply
  • Informative
  • Like
Likes   Reactions: vanhees71 and BWV
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