How do you find the killing vectors for Minkowski space?

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SUMMARY

The discussion focuses on finding the Killing vectors for Minkowski space, specifically referencing the equation governing these vectors: \(\nabla_\alpha X_\beta + \nabla_\beta X_\alpha = 0\), where \(\nabla_\alpha\) denotes the covariant derivative. The participant is studying General Relativity (GR) using Carroll's textbook and seeks clarification on the notation and concepts involved. The conversation highlights the importance of understanding covariant derivatives in the context of Killing vector fields.

PREREQUISITES
  • Understanding of General Relativity (GR)
  • Familiarity with covariant derivatives
  • Knowledge of Killing vector fields
  • Basic grasp of differential geometry
NEXT STEPS
  • Study the concept of Killing vector fields in detail
  • Learn about covariant derivatives in the context of GR
  • Explore online lectures or resources on Minkowski space
  • Read Carroll's "Spacetime and Geometry" for deeper insights
USEFUL FOR

Students of General Relativity, physicists interested in differential geometry, and anyone seeking to understand the mathematical framework of spacetime symmetries.

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Homework Statement


How do you find the killing vectors for Minkowski space(or from any metric as well)?

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The Attempt at a Solution


I'm new to GR and I'm going through Carroll's book. I've been alright so far but for some reason I just don't understand what's going on here. Could you guys explain in detail or send me to an online lecture or something to help?
 
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If X^\mu is a killing vector field, then it obeys the equation:

\nabla_\alpha X_\beta + \nabla_\beta X_\alpha = 0

where \nabla_\alpha is the covariant derivative.
 
I seem to have forgotten notation. -.- I wasn't even thinking of the covariant derivative. Thanks haha
 

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