A Question on Dirac's derivatives of the 4-velocity w.r.t. coordinates

[PeterDonis]

I prefer the extension-independence approach.

If you're talking about an extension at all, you're talking about defining a vector field in an open region around the worldline you're interested in. Which means you're using Wald's framework or something equivalent to it, including deriving extension independence in that framework.

So I'm confused about how you think your preferred approach is different. "Different" to me would be not even using an extension at all, and working solely with quantities that only need to be defined along the worldline you're interested in, not in an open region around it. But that doesn't seem to be what you're describing.

 

[JimWhoKnew]

If you're talking about an extension at all, you're talking about defining a vector field in an open region around the worldline you're interested in. Which means you're using Wald's framework or something equivalent to it, including deriving extension independence in that framework.

Of course. I already wrote in #35 that "I don't say the books are wrong".

So I'm confused about how you think your preferred approach is different. "Different" to me would be not even using an extension at all, and working solely with quantities that only need to be defined along the worldline you're interested in, not in an open region around it. But that doesn't seem to be what you're describing.

Some implementations come with a "natural" extension, like the discussion of Raychaudhuri's equation, where the "Wald form" is very useful. Some come without, and the extension can be chosen arbitrarily. Some come with a "built in" vector field that obeys "Wald form" on certain curves, like the KVF on black holes' horizons. And some don't necessitate the "Wald form" at all, and can be worked out solely with quantities defined on the curve.

It is not different from Wald. It is different from your interpretation that there always exists a unique "natural" extension that should be used.

 

[PeterDonis]

It is different from your interpretation that there always exists a unique "natural" extension that should be used.

I made no such claim. I did say, incorrectly, that there was a unique extension in one particular case, the case of a geodesic in flat Minkowski spacetime. I was incorrectly thinking of the 1+1 case instead of the 3+1 case.

 

[JimWhoKnew]

Had MTW and Wald bothered to dedicate a sentence or two to clarify, I would have said nothing.

It turns out that Dirac does find it worthwhile to acknowledge the point. On page 46 of his GR book, he writes: "... if $v^\mu$ is defined as a continuous field function instead of having a meaning only on one world line ...".