Lorentz or Poincare invariant?

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SUMMARY

General Relativity (GR) is characterized by local Lorentz invariance, which pertains to the action and field equations. The action exhibits local Lorentz symmetry, while the field equations are covariant, necessitating a reference frame for solutions. The Poincaré group, which encompasses translations, does not apply universally in GR since spacetime does not maintain uniformity across different points, only across local Lorentz frames. However, specific solutions, such as those in Minkowski spacetime, do demonstrate Poincaré invariance.

PREREQUISITES
  • Understanding of General Relativity (GR)
  • Familiarity with Lorentz and Poincaré symmetries
  • Knowledge of spacetime tensors and their covariant properties
  • Basic concepts of Minkowski spacetime
NEXT STEPS
  • Study the implications of local Lorentz invariance in General Relativity
  • Explore the differences between Lorentz and Poincaré invariance
  • Investigate covariant field equations in the context of GR
  • Examine the characteristics of Minkowski spacetime and its solutions
USEFUL FOR

The discussion is beneficial for physicists, particularly those specializing in theoretical physics, cosmology, and anyone interested in the foundational principles of General Relativity and its symmetries.

xiaomaclever
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Generally we say GR is local Lorentz invariant. Does it mean the action or field equation?
Why not Poincare invariant? Thanks!
 
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The action (not only in GR but in the standard model) has local Lorentz symmetry, while the field equations, with spacetime tensors for its terms, are covariant (when we write a solution to the field equations, we have to pick a reference frame to write it in, so they must not be *invariant* equations, right?). The Poincare group includes translations, but spacetime does not "look the same" at different points...just from different local Lorentz frames. So no Poincare symmetry. However, a special solution to the field equations does have Poincare invariance: Minkowski.
 

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