Kostik's latest activity
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Kostik reacted to renormalize's post in the thread A Weinberg's proof of ##{T^{\mu\nu}}_{,\nu}=0## for a perfect fluid with
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(1) Weinberg gets his negative sign by using the identity: $$\frac{\partial}{\partial... -
Kostik posted the thread A Weinberg's proof of ##{T^{\mu\nu}}_{,\nu}=0## for a perfect fluid in Special and General Relativity.Weinberg ("Gravitation and Cosmology") defines the energy-momentum tensor ##T^{\mu\nu}## in equations (2.8.1)-(2.8.2). He proves...
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Kostik posted the thread A Dirac's "comprehensive action principle" -- independent equations in Special and General Relativity.In Dirac's "General Theory of Relativity", he develops the "comprehensive action principle" in chapter 30. Simply put, he writes a...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I just noticed that Dirac actually derives the Lie derivative in chapter 30, on page 60. Here he calculates the variation ##\delta...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.I'm afraid not, what I wrote is 100% my own, original thought. You wrote "The bottom line: I don't understand how Dirac gets (27.4)"...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Oh, that is quite right. So, my method fails -- and I am at a loss how to show that ##\delta g_{\mu\nu}## can be varied independently of...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.But ##b^\mu## is small, which is the assumption.
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.What do you mean by "functionally independent"? It seems to me to require some justification, in order to carry out the final step after...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.Well, spurred on by your hint, I am comfortable that I have derived Dirac's (27.4) as shown in the notes pasted in #34 above. Now, as to...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.@JimWhoKnew Something still bothers me here. Dirac went through all this trouble to establish (27.4), because he writes the matter...
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Kostik replied to the thread A The Lagrangian of a free particle ##L=-m \, ds/dt##.I think it's circular reasoning because ##p^k = \partial L / \partial \dot{x}^k## is derived for a non-relativistic free particle, where...
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Kostik replied to the thread I Is ##p^k = \partial L / \partial \dot{x}^k## true for all ##L##'s?.Thanks. I did not phrase the question very well. I have made a more detailed post of the question here...
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Kostik posted the thread A The Lagrangian of a free particle ##L=-m \, ds/dt## in Special and General Relativity.In Dirac's "General Theory of Relativity" (p. 52), he postulates that the action for a free particle of mass ##m## is $$I=-m \int ds$$...
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Kostik posted the thread I Is ##p^k = \partial L / \partial \dot{x}^k## true for all ##L##'s? in Classical Physics.Using the Lagrangian $$L=T-U=\frac{1}{2}mv^2-U$$ we clearly have $$ \frac{\partial L}{\partial \dot{x}^k} = m\dot{x}^k = p^k $$ i.e...
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Kostik replied to the thread A Dirac's "GTR" Eq (27.4): how momentum ##p^\mu## varies.@JimWhoKnew My derivation is shown below. It's a little verbose, but it helps me understand what's going on. I find it a little tricky...
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