Recent content by Kostik

@PeterDonis What special coordinates is Dirac using? I don’t see that.

Thanks. I’m in the airport lounge in Istanbul, changing planes on the way to Tashkent. I will digest this later and probably revisit this thread when I’m relaxed to read and reply. My original post was just Dirac’s argument minus the pseudotensor — hence, whatever he says for the total...

You can ignore the pseudotensor, that’s a distraction. Just consider the ##Y## term and focus on global conservation of non-gravitational energy — the argument is the same as my opening post. Yes he’s integrating over a 3D spatial volume, but how do you integrate a tensor density over ANY...

He certainly does. See the photo in #3 above.

Thanks, I will digest later when I have the time. Note that Dirac used harmonic coordinates only in Ch 33-34 on gravitational waves. I’m still interested in understanding how Dirac can justify integrating a tensor density over a finite volume. This is contrary to his own warning against this in...

@weirdoguy Energy density ##T^{00}\sqrt{-g}## is not a scalar density … but one component of a rank-2 tensor density.

It just occurred to me — how does Dirac integrate a tensor density over a volume? A scalar density — yes; but a vector density or a tensor density — no, you cannot add vectors or tensors at different points in curved space. Dirac himself makes this point earlier in the text. So what the heck is...

Dirac’s book is easily found on the internet, see here: https://lib.undercaffeinated.xyz/get/pdf/4188 Pp. 45-46 explain the energy-momentum tensor for a dust and his comment that ##T^{\mu\nu}\sqrt{-g}## is the density and flux of energy and momentum. Schutz’s text does the same in more detail...

As I said, he gives a very clear explanation.

I don't think that is right. Dirac gives a very clear explanation why ##T^{\mu\nu}\sqrt{-g}## is the density of energy-momentum in curved spacetime.

Never mind the pseudotensor; that's a distraction. My question concerns whether the mixed tensor ##T^0_{\, 0}\sqrt{-g}## represents (in some sense) the energy density, like ##T^{00}\sqrt{-g}## does. In which case, my original derivation is valid.

You're saying that ##T^{00}\sqrt{-g}## and ##T^0_{\,0}\sqrt{-g}## both represent the energy density (in curved spacetime)? But they're different functions of ##x##. Can you please expand and clarify your comment?

Dirac seems to disagree. Here he is discussing the energy-momentum pseudotensor ##{t_\mu}^\nu##, and ##{Y_\mu}^\nu## is just the mixed tensor version of the energy-momentum tensor ##Y^{\mu\nu}##. He seems to treat the mixed tensor ##{Y_\mu}^\nu## as having the same physical significance (energy...

Please see the derivation below that, if the metric is time-independent, then the integral over a sufficiently large 4D spacetime region of ##T^0_{\,0} \sqrt{-g}## is independent of time, where ##T^{\mu\nu}## is the energy-momentum tensor of a relevant collection of matter-energy fields. Thus, I...

Of course; thanks. I think I have found another proof, but I should have seen this.