Lie Derivative in Relativity: Examples & Uses

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SUMMARY

The Lie Derivative is a crucial tool in General Relativity, particularly for identifying isometries through the calculation of Killing vectors. By setting the Lie derivative of the metric to zero, one can derive these vectors, which represent symmetries of the spacetime geometry. This concept is foundational in understanding the geometric structure of spacetime and is discussed in detail in Carroll's General Relativity notes. Its applications extend to both General and Special Relativity, emphasizing its importance in theoretical physics.

PREREQUISITES
  • Understanding of General Relativity concepts
  • Familiarity with differential geometry
  • Knowledge of Lie groups and Lie algebras
  • Basic grasp of vector fields and metrics
NEXT STEPS
  • Study the derivation and properties of Killing vectors in General Relativity
  • Explore the role of diffeomorphisms in differential geometry
  • Learn about the applications of Lie Derivatives in theoretical physics
  • Review Carroll's General Relativity notes for practical examples
USEFUL FOR

The discussion is beneficial for students and researchers in theoretical physics, particularly those focusing on General Relativity and differential geometry. It is also valuable for anyone interested in the mathematical foundations of spacetime symmetries.

kent davidge
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Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity.

I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application (and I'm not even sure whether it's easy to find out).
 
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Diffeomorphisms are generates by lie derivatives. So e.g. to find isometries one puts the lie derivative of the metric to zero and solves for the vector field, giving so called Killing vectors.
 

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