Why is the Torsion Tensor Overlooked in General Relativity?

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SUMMARY

The discussion centers on the oversight of the torsion tensor in General Relativity (GR), specifically the preference for a torsion-free and metric-compatible connection on manifolds. This choice leads to the exclusion of the torsion term, which has been a point of contention among researchers. The Einstein-Cartan theory emerges as a framework that attempts to incorporate spin by considering torsion, although it is not regarded as fundamental. Notably, this theory finds relevance in supergravity and M-Theory, indicating its significance in advanced theoretical physics.

PREREQUISITES
  • Understanding of General Relativity and its foundational principles
  • Familiarity with differential geometry concepts
  • Knowledge of the Einstein-Cartan theory and its implications
  • Basic grasp of supergravity and M-Theory frameworks
NEXT STEPS
  • Research the implications of the torsion tensor in differential geometry
  • Study the Einstein-Cartan theory in detail
  • Explore the relationship between torsion and spin in theoretical physics
  • Investigate the role of torsion in supergravity and its connection to M-Theory
USEFUL FOR

The discussion is beneficial for theoretical physicists, researchers in differential geometry, and anyone interested in the advanced concepts of General Relativity and its extensions.

bchui
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I ahve always wondered why only the curvature term [tex]R_{\mu,\nu}[/tex] been considered in GR. From differential Geoetry, how about the torsion tensor?
 
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The simple answer is that in standard formulation of general relativity, we require a unique connection on a manifold which is torsion-free and metric compatible, which lead to the properties of the covariant derivative that we need.
 
It always bothered me as well that the torsion term was simply ignored. Many people have examined what happens when you keep the term, which has led to the Einstein-Cartan theory as a way to try and incorporate spin. Although this theory doesn't appear to be fundamental, it does seem to embed itself into supergravity which is in turn one of the limits of M-Theory.


Here is a link to Hehl's 1976 article from Rev. Mod Phs...
http://prola.aps.org/abstract/RMP/v48/i3/p393_1


And here is a much more recent article
http://arxiv.org/abs/0711.1535
 

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