- #246

Q-reeus

- 1,115

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For whatever reason that had never struck home before. Makes the Komar expression a Clayton's really - T_ii contributions yes and no at the same time. Need to chew over that.PeterDonis said:In terms of T_ii terms appearing in the formal expression for the Komar mass integral, yes. In terms of actually contributing, recall that we saw in a series of earlier posts that, if self-gravity can be neglected, the T_ii terms must always cancel in static equilibrium; and if self-gravity cannot be neglected, then whatever "residual" extra contribution remains in the T_ii terms is compensated for by the "redshift factor", which is < 1, multiplying the entire integral; the end result being, in effect, that the negative gravitational potential energy exactly compensates for the positive contribution of T_ii. So in any static equilibrium you can essentially consider the total mass to be the sum of the T_00 contributions alone, with everything else canceling out.

I cannot see how you figure axially vibrating rod is a dipole source. Conservation of momentum forbids it. It is merely the continuously distributed version of two concentrated masses with a spring in between. And according to this site, that certainly generates quadrupole GW's: http://ned.ipac.caltech.edu/level5/ESSAYS/Boughn/boughn.html - fig.1 and caption. Each rod end has mass dipole-like motion, but they must exactly oppose at any instant to give a net quadrupole source surely. Maybe you were thinking of charge dipole, where opposite motion of unlike charges is a dipole oscillator.It wouldn't be quadrupole; it would be dipole, since by hypothesis the rod only contracts/expands along one dimension. To get a quadrupole variation you would need to have the rod expand/contract along two orthogonal dimensions. Basically what you have described is a time-varying axisymmetric spacetime; I believe there is a general class of EFE solutions that describes these, but I'll have to look it up to be sure.

Yes understood proper time was to be used and was careless with symbols - I was just focusing on that there is only time derivative of momentum, not extra dynamical terms.Q-reeus: "As far as I knew F = dp/dt holds perfectly well not just in SR but GR too. Seems not."

No, that force law is still correct, except that it's dp/dtau, not dp/dt (that's true in SR as well); i.e., the derivative is with respect to proper time along the worldline of the object to which the 4-force is being applied, and whose 4-momentum is changing.

No, at this stage Komar had been left behind, and my comment was reaction to your statement implying that for the real non-stationary spacetime case d/dtau(T_0i) exactly cancels out T_ii re overall gravitating mass for rod. Since T_ii is supposed to be a periodically sign-reversing source of mass, it can only mean d/dtau(T_0i) is at any instant an equal and opposite source also. That in turn led to question over force law, since active mass should be identical to inertial mass. Given you say there is no change to F = dp/dtau, in what sense then is, or rather can, d/dtau(T_0i) cancel T_ii? As I said earlier, all I could see was d/dtau(T_0i) being part of mass quadrupole moment - with I suppose periodic quadrupolar near-fields as well as GW's the result.Q-reeus: "If acceleration of matter constitutes in itself a source of added mass..."

It doesn't. T_0i does not appear in the Komar mass integral, so even if you are trying to adopt a model where that integral should be "approximately conserved" in a spacetime that is "approximately stationary", T_0i doesn't come into it.

Maybe time to go back and pick over your #147 - I did have some questions about meaning of the last four expressions there. :zzz:In terms of the true conservation law in GR, covariant derivative of SET = 0, the T_0i terms (more precisely their derivatives) certainly do come into play, since they appear in the covariant derivative. I wrote down the components of that equation in an earlier post, which shows how the covariant derivative constraint relates derivatives of the various components.