FrediFizzx wrote: ↑Thu Nov 23, 2023 12:01 pm
kev01 wrote: ↑Wed Nov 22, 2023 8:26 pm
Given the complete absence....
LOL! Hi Kev, yeah nobody wants to talk much on this forum anymore. Whatever. It's like a blog for me.
Happy Thanksgiving to all!
Fair enough Fred. Still, I will take this opportunity to add a bit to OP. An alternate angle on the basic argument.
Take Birkhoff's Theorem. It states that the external gravitational field of a spherically symmetric mass distribution is entirely unaffected by any symmetric variations in e.g. density or radial velocity distributions. But is it fundamental Truth or merely a postulate? Let's test it via a specific example involving elastic stress, and see.
For instance, suppose a self-supporting thin elastic shell, say a perfectly spherical 'X-mas tree bauble' is placed in breathing mode radial oscillations.
The wall material will be under ~ sinusoidal wrt time, uniform transverse elastic stress/strain. Compression to tension back to compression...etc.
Clearly spherical symmetry precludes any contribution to external field variation from gross mass motion.
Further, there is a perfectly conservative interplay between elastic PE and KE - the TE sum is invariant (notional loss-free case).
Which means no contribution - there - to any gravitational wave source moments - monopole, dipole, quadrupole etc. Birkhoff's Theorem seems safe.
However...
At the time point of maximum compressive stress, there is a net positive stress contribution that is monopole moment in nature. And equally negative when under maximum tensile stress.
That there is no possible counteracting 'stress current' somehow analogous to the KE term in earlier mentioned elastic PE/KE -> invariant TE, simply vary the Young's modulus. The stiffer the bauble wall material, obviously the smaller the vibrational amplitude for a given maximal stress excursion.
That has no effect on the amplitude of stress contribution - which is simply linearly dependent on stress level. Unlike quadratic dependence for elastic energy density.
For an engineering solid e.g. steel, it's easy to show that stress as source is typically many orders of magnitude larger than corresponding elastic energy density as source.
From that it's clear that unlike all the other terms in RHS stress-energy-momentum tensor, stress as source cannot obey any conservative continuity relation! A cube of rubber compressed to the same final dimensions as a cube of diamond will require many orders of magnitude more energy to get to the same final stress state as the diamond. Yet both yield the same stress-only source contribution. Magic? A 'peculiarity' evidently not noticed in GR community.
We have a very straightforward counterexample to validity of Birkhoff's Theorem! Supposedly impossible monopole gravitational radiation! Provided of course stress is actually a source as per GR (and other rival theories for that matter).
Owing to the seeming 'magical' nature of stress as source, for quite some time I doubted stress was a bona fide source. Until one day, did a perturbative evaluation of a gas filled container subject to a uniform g field. Whereas a quadratic dependence of added gravitational passive mass on gas pressure was expected in accordance with pneumatic energy density, actually the first order contribution was linear in gas pressure. Hence stress/pressure is indeed a source of so-called passive gravitational mass at least.
Proving the same holds for active gravitational mass is more difficult to achieve but one assumes an equality will apply.
