Cartan's Understanding of Einstein Field Equation

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SUMMARY

Cartan's geometric interpretation of the Einstein Field Equation, represented as Gij + Λgij = κTij, articulates that the sum of moments of rotation for a 3-cube equals 8π times the energy-momentum within that cube. This perspective is primarily discussed in John Wheeler's texts, particularly "Gravitation" (MTW), which highlights Cartan's contributions that are overlooked in other literature, such as Wald's works. The complexity of Cartan's coordinate-free differential geometry may contribute to its limited adoption in mainstream discussions.

PREREQUISITES
  • Understanding of the Einstein Field Equation and its components
  • Familiarity with geometric interpretations in physics
  • Knowledge of John Wheeler's contributions to general relativity
  • Basic concepts of differential geometry, particularly coordinate-free approaches
NEXT STEPS
  • Read John Wheeler's "Gravitation" (MTW) for insights on Cartan's interpretation
  • Explore Cartan's coordinate-free differential geometry techniques
  • Investigate the implications of Cartan's ideas on modern physics literature
  • Study the relationship between Cartan's geometric interpretation and the Poisson equation
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Physicists, mathematicians, and students of general relativity seeking to deepen their understanding of geometric interpretations of the Einstein Field Equation and Cartan's contributions to the field.

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Cartan and general relativity
About a week ago I was reading about Cartan's geometric interpretation of the Einstein Field Equation

Gij + Λgij = κTij

According to Cartan, this equation expresses the idea

(sum of moments of rotation for the faces of a little 3-cube) = 8π * (amount of energy-momentum within that 3-cube)

As far as I can tell, it is only in John Wheeler's various books (MTW, but also his other books) where this idea of Cartan is explained. None of the other popular books like Wald discuss this. Apparently, it was Wheeler who dug it out of Cartan's papers and made it widely known. If anyone on this forum has a good understanding of this, I would appreciate it if you can share/explain this. Also, why don't more people and books adopt this viewpoint? Is it because Cartan's coordinate free differential geometry is too sophisticated?
 
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Frankly I don't know why...Btw MTW has also a section for the Cartan's geometric interpretation of Newton spacetime (basically the Poisson equation).
 

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