- #31
TrickyDicky
- 3,507
- 28
Mentz114 said:
Yes, that is actually equivalent to the Lie derivative of the metric wrt the Killing field being zero.
Calculation of Proper Acceleration
- Context: Graduate
- Thread starter pervect
- Start date
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- Tags
- Acceleration Calculation
Click For Summary
Discussion Overview
The discussion revolves around the calculation of proper acceleration in the context of general relativity, specifically addressing the use of covariant derivatives versus absolute derivatives when analyzing the motion of point particles versus fields. Participants explore the implications of these mathematical formulations and their conceptual validity.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants argue that the expression aμ = Uα∇αUμ is conceptually incorrect when applied to point particles, suggesting that it only makes sense in the context of continuous fluid motion where Uμ is a field.
- Others propose that if the 4-velocity can be extended to a vector field, then the covariant derivative expression can be valid, but caution that this is not the case for all scenarios, such as closed timelike curves.
- There is a contention regarding whether the use of covariant derivatives is appropriate for calculating the orbits of particles, with some stating that it is a conceptual issue and that extending the velocity vector continuously off the world line is unnecessary.
- Participants note that while the covariant derivative approach may seem "sloppy," it might not lead to incorrect answers in certain contexts, similar to other approximations in physics.
- Disagreement exists on the interpretation of the Riemann curvature tensor, with some asserting that it is defined in terms of vector fields, while others maintain that it acts on vectors, not vector fields.
- There is a discussion about the implications of context in mathematical definitions, particularly regarding the distinction between vectors and vector fields in various references.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the appropriateness of using covariant derivatives for point particles versus fields, and multiple competing views remain regarding the conceptual framework for proper acceleration calculations.
Contextual Notes
Limitations include the dependence on the extendability of the 4-velocity to a vector field and the unresolved nature of the mathematical definitions regarding vectors and vector fields in the context of the Riemann curvature tensor.
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