Calculation of Proper Acceleration

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SUMMARY

The discussion centers on the calculation of proper acceleration, specifically the distinction between using covariant derivatives and absolute derivatives in the context of point particle motion versus fluid dynamics. Bill_K argues that the expression aμ = Uα∇αUμ is conceptually incorrect for point particles, as it implies a four-dimensional gradient applicable only to continuous fields. Instead, the absolute derivative D/Dτ should be employed for differentiating world-line functions, as it correctly incorporates Christoffel symbols. The conversation also critiques the use of covariant derivatives in the literature, emphasizing the need for clarity in distinguishing between functions of one variable and functions of four variables.

PREREQUISITES
  • Understanding of proper time and world lines in general relativity
  • Familiarity with covariant derivatives and absolute derivatives
  • Knowledge of Christoffel symbols and their role in tensor calculus
  • Conceptual grasp of vector fields versus vectors in differential geometry
NEXT STEPS
  • Study the application of absolute derivatives in general relativity
  • Learn about the role of Christoffel symbols in differentiating tensors
  • Explore the implications of using covariant derivatives in fluid dynamics
  • Investigate the mathematical distinctions between functions of one variable and functions of multiple variables
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students of general relativity who are focused on the nuances of tensor calculus and the correct application of derivatives in theoretical physics.

  • #31
Mentz114 said:
I've edited my post ! It's the symmetrized covar. diff. that is zero, isn't it ?

Yes, that is actually equivalent to the Lie derivative of the metric wrt the Killing field being zero.
 
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  • #32
George Jones said:
https://www.physicsforums.com/showthread.php?p=2625593\#post2625593
George, as academic jokes go that's pretty funny.

TrickDicky said:
Yes, that is actually equivalent to the Lie derivative of the metric wrt the Killing field being zero.
Argh ! Information overload.

Seriously, is it the case taht using the covariant derivative to test for geodesics will be OK if they are geodesic congruences - which I take to mean that the vector is extendable to a field.
 

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