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- Dirac derives Einstein's equation for a dust ##G^{\mu\nu}=-8\pi \rho v^\mu v^\nu## from the principle of stationary action using the appropriate Lagrangian for the gravitational field and a dust. First, he "infers" an important relation between an arbitrary variation of an element of the matter field and the resultant change in the momentum vector density. How to derive this rigorously?
In Dirac's "General Theory of Relativity", chapters 26-30, he builds up various action principles from which Einstein's equation ##G^{\mu\nu}=-8\pi T^{\mu\nu}## can be obtained. In chapter 27, he extends the result of chapter 26 (Einstein's vacuum equation) to the case of a dust, where $$T^{\mu\nu}=\rho v^\mu v^\nu \,\, .$$ Here, ##\rho## is the mass density in the MCRF ("rest frame") of a fluid element of the dust.
The momentum density is $$p^\mu = \rho v^\mu \sqrt{-g} \, .$$ Dirac makes arbitrary changes ##b^\mu## in the position of the dust, and uses his Eq. (27.4) $$\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu )_{,\nu} \qquad\qquad(*)$$ to find the corresponding change in ##p^\mu##.
Using ##(*)##, Dirac does indeed derive Einstein's field equation and the geodesic equation for the motion of the matter. I'm not asking about that, and won't mention the details; Dirac's derivation is straightforward and easy.
My question is about ##(*)##, his Eq. (27.4). He infers this, but does not derive it. I am attaching below my notes, in which I had hoped to derive ##(*)##. (A clearer MSWord doc is also attached.) I start by considering the dust as a collection of individual matter particles. The first result is to show that the velocity field ##v^\mu## is divergenceless, i.e., ##v^\mu_{\,\, ,\mu}=0## (my Eq. (27.5)). Using this fact and ##p^\mu_{\,\, ,\mu}=0##, I obtain the desired result.
Unfortunately, if both ##v^\mu_{\,\, ,\mu}=0## and ##p^\mu_{\,\, ,\mu}=0##, then the proportionality factor ##\rho \sqrt{-g}## must be constant. But ##\rho \sqrt{-g}## definitely is not a constant.
In fact, ##v^\mu_{\,\, ,\mu}=0## cannot be right -- this would only be true if ##\rho## were constant, which again is not the case. The only conservation law is ##p^\mu_{\,\, ,\mu}=0##.
Is my entire derivation pure rubbish? I have a feeling that it's not. I must be missing something; something connected with the fact that this is a dust, and I am working with a sum over individual particles (which can be supposed to have mass ##m##). Somewhere, I am not using the full power of the delta functions in the derivation.
I think the fact that spacetime may be curved, hence the correct conservation law is ##p^\mu_{\,\, ,\mu}= (\rho v^\mu \sqrt{-g})_{\,\, ,\mu} =0##, has nothing whatever to do with proving ##(*)##. Dirac does not even bring in the metric until after his Eq. (27.4). (In fact, he merely states that the momentum density ##p^\mu## "lies in the same direction" as the velocity vector.)
Any help, please?
The momentum density is $$p^\mu = \rho v^\mu \sqrt{-g} \, .$$ Dirac makes arbitrary changes ##b^\mu## in the position of the dust, and uses his Eq. (27.4) $$\delta p^\mu = (p^\nu b^\mu - p^\mu b^\nu )_{,\nu} \qquad\qquad(*)$$ to find the corresponding change in ##p^\mu##.
Using ##(*)##, Dirac does indeed derive Einstein's field equation and the geodesic equation for the motion of the matter. I'm not asking about that, and won't mention the details; Dirac's derivation is straightforward and easy.
My question is about ##(*)##, his Eq. (27.4). He infers this, but does not derive it. I am attaching below my notes, in which I had hoped to derive ##(*)##. (A clearer MSWord doc is also attached.) I start by considering the dust as a collection of individual matter particles. The first result is to show that the velocity field ##v^\mu## is divergenceless, i.e., ##v^\mu_{\,\, ,\mu}=0## (my Eq. (27.5)). Using this fact and ##p^\mu_{\,\, ,\mu}=0##, I obtain the desired result.
Unfortunately, if both ##v^\mu_{\,\, ,\mu}=0## and ##p^\mu_{\,\, ,\mu}=0##, then the proportionality factor ##\rho \sqrt{-g}## must be constant. But ##\rho \sqrt{-g}## definitely is not a constant.
In fact, ##v^\mu_{\,\, ,\mu}=0## cannot be right -- this would only be true if ##\rho## were constant, which again is not the case. The only conservation law is ##p^\mu_{\,\, ,\mu}=0##.
Is my entire derivation pure rubbish? I have a feeling that it's not. I must be missing something; something connected with the fact that this is a dust, and I am working with a sum over individual particles (which can be supposed to have mass ##m##). Somewhere, I am not using the full power of the delta functions in the derivation.
I think the fact that spacetime may be curved, hence the correct conservation law is ##p^\mu_{\,\, ,\mu}= (\rho v^\mu \sqrt{-g})_{\,\, ,\mu} =0##, has nothing whatever to do with proving ##(*)##. Dirac does not even bring in the metric until after his Eq. (27.4). (In fact, he merely states that the momentum density ##p^\mu## "lies in the same direction" as the velocity vector.)
Any help, please?
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