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In another thread, Bill_K writes:
Wald, is one of the sources that does it the "wrong" way. But it seems that while the wrong way is rather "sloppy", it seems to work in practice. I' would like to see if this "slopiness" could lead to incorrect answers.
If we imagine that parallel transport was defined by a vector field, rather than a curve, would you agree that we could use aμ = Uα∇αUμ ?
Wald's argument is that parallel transport turns out to depend only on the values of the vector field along the curve. If I'm reading this right and Wald is correct, then _any_ vector field will give the right answer, as long as its orbit generates the curve in question.
This would imply that while the formulation is sloppy, it's not so bad as to give wrong answers.
I'd like to be sure I'm not missing something though,.
Bill_K said:I think you meant to write aμ = Uα∇αUμ instead, and this is the form in which it is often quoted, but is conceptually incorrect.
The covariant derivative is a four-dimensional gradient operation and as such can only be applied to quantities that are functions of all four dimensions, such as f(x,y,z,t). This formula for aμ only makes sense if you are talking about the motion of a continuous fluid, in which the velocity is a field, Uμ(x,y,z,t), and then ∇αUμ is defined and meaningful, and denotes a rank two tensor field.
But we are talking here about the motion of a single point particle, in which Uμ(τ) is a function only of the proper time along the world line, and is undefined elsewhere. By analogy, in the Newtonian mechanics of projectile motion, it would make no sense at all to talk about the "gradient" of the projectile's velocity vector.
The correct concept is not the covariant derivative but the absolute derivative, D/Dτ, and this is what must be used to differentiate world-line functions. In a manner similar to the covariant derivative, it involves Christoffel symbols, one for each tensor index. E.g. for any vector Vμ(τ) defined along a world line with tangent vector Uμ, the absolute derivative of Vμ is
DVμ/Dτ ≡ dVμ/dτ + Γμνσ UνUσ
Wald, is one of the sources that does it the "wrong" way. But it seems that while the wrong way is rather "sloppy", it seems to work in practice. I' would like to see if this "slopiness" could lead to incorrect answers.
If we imagine that parallel transport was defined by a vector field, rather than a curve, would you agree that we could use aμ = Uα∇αUμ ?
Wald's argument is that parallel transport turns out to depend only on the values of the vector field along the curve. If I'm reading this right and Wald is correct, then _any_ vector field will give the right answer, as long as its orbit generates the curve in question.
This would imply that while the formulation is sloppy, it's not so bad as to give wrong answers.
I'd like to be sure I'm not missing something though,.