Cartan's Understanding of Einstein Field Equation

In summary, Cartan's geometric interpretation of the Einstein Field Equation proposes that the equation expresses the relationship between the sum of moments of rotation for the faces of a small 3-cube and the amount of energy-momentum within that 3-cube. This idea is only explained in John Wheeler's books and is not widely known. It is possible that this viewpoint is not adopted by more people and books due to the complexity of Cartan's coordinate-free differential geometry. However, Wheeler's book "MTW" also includes a section on Cartan's interpretation of Newton spacetime.
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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).
 

1. What is Cartan's understanding of Einstein Field Equation?

Cartan's understanding of Einstein Field Equation is a geometric interpretation of Einstein's theory of general relativity. It involves the use of differential geometry to describe the curvature of spacetime and the relationship between matter and energy.

2. How does Cartan's understanding differ from the traditional understanding of Einstein Field Equation?

Cartan's understanding differs from the traditional understanding in that it provides a more geometric and intuitive approach to understanding the equations. It also allows for a deeper understanding of the underlying principles of general relativity.

3. What are the key concepts in Cartan's understanding of Einstein Field Equation?

The key concepts in Cartan's understanding are the use of differential forms, curvature, and torsion to describe the geometry of spacetime. It also involves the concept of parallel transport and the role of matter and energy in shaping the curvature of spacetime.

4. How does Cartan's understanding of Einstein Field Equation contribute to our understanding of the universe?

Cartan's understanding of Einstein Field Equation provides a more intuitive and geometric approach to understanding the fundamental principles of general relativity. It also allows for a deeper understanding of the relationship between matter and energy and the curvature of spacetime, which is crucial in understanding the behavior of the universe on a large scale.

5. What are some potential applications of Cartan's understanding of Einstein Field Equation?

Some potential applications of Cartan's understanding include the study of black holes, gravitational waves, and the behavior of the universe on a large scale. It also has applications in cosmology and the study of the early universe.

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