Kostik's latest activity
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I might have it figured out. Will post a coherent explanation if I do. -
Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Yep, this is where the issue really lies. If we vary $$x^\mu \rightarrow \bar{x}^\mu = x^\mu + \delta x^\mu = x^\mu + b^\mu \,\, ,$$ and...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies."Gravitation and Spacetime", 3rd Ed., Ohanian & Ruffini, pp. 262-264. I like how he computes the Lie derivative, keeping careful track...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.@JimWhoKnew I think Dirac is using ##\delta## is the most basic way it is used when varying a path or dynamical coordinate, to...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.@JimWhoKnew How do you get $$\delta P^\mu = -\mathcal{L}_b \, P^\mu$$ and $$\delta g_{\mu\nu} = -\mathcal{L}_b \, g_{\mu\nu} \quad ?$$...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Sorry, brain freeze. Yes, it's obvious. Thanks, this has been really useful, and a much better way to show Dirac's relation without...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.@JimWhoKnew No, I am perfectly happy with (10.9.9) being a special case of (10.9.10). I get to Weinberg's equation above (10.9.7)...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.We know ##p^\mu = \rho v^\mu \sqrt{-g}##. Your ##P^\mu## is ##\rho v^\mu##. Your equation below "Using the conventions and results in...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I'm thinking about this. I am comfortable with ##{u^k}_{,n} = 0##, but I'm having doubts about ##\delta u^k = 0##. The displacement...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.##u^\mu## must be constant within a sufficiently small volume element of the fluid. That's what it means for all matter particles to be...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.If Dirac is not going to actually derive (27.4) (which I think I have correctly done in the original post), but rather to simple infer...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Just take ##\lambda=t## and the issue goes away. Of course, ##u^\mu## varies throughout the matter field, but the whole point of the...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I think you have misunderstood. In Ch. 27, Dirac defers introduction of the metric, so the trajectory of a matter element (fluid...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Dirac simply guesses that, for ##b^0 \neq 0##, the formula is $$\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu)_{,\nu} $$ and verifies that...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I found a reasonable proof that Dirac's (27.3) holds for ##p^k## (still in the ##b^0=0## case). Dirac should actually have done this...
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