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October 28

According to the common theory, is the mechanical equilibrium a necessary sufficient condition, for not emitting gravitational waves?

By mechanical equilibrium I mean, both equilibrium of forces and equilibrium of torques (moments). HOTmag (talk) 09:00, 28 October 2024 (UTC)[reply]

A sufficient condition is that quadrupole moment (and all higher moments) of an isolated system is constant. Ruslik_Zero 20:23, 28 October 2024 (UTC)[reply]

Is a mechanical equilibrium a necessary condition?
Do you think you can give a concrete example of a body which is in a mechanical equilibrium and which emits GWs? HOTmag (talk) 20:35, 28 October 2024 (UTC)[reply]
1: No. Take a cylindrically symmetric flywheel (i.e., solid, no spokes) and spin it on its axle. It won't emit gravitational waves, even when the rate of spin changes when you apply a torque. Its quadrupole moment is zero (or at least, the relevant components) and therefore doesn't change on rotation.

2: Yes. Take a flywheel with two masses attached to the rim, opposite to each other, and spin it. It will emit gravitational waves. Its quadrupole moment is non-zero and changes direction during rotation. The gravitational waves will carry away some angular momentum and thereby apply a torque, but with a small motor you can compensate for that and keep the flywheel in mechanical equilibrium. PiusImpavidus (talk) 09:03, 29 October 2024 (UTC)[reply]

Thx. Now, let's assume we don't add the small motor, so the flywheell won't be in mechanical equilibrium, because as you say: "The gravitational waves will carry away some angular momentum and thereby apply a torque". However, since the natural source of this torque can't be any "real force" (namely: the electromagnetic force, the strong force, and the weak force), so: is it really reasonable to conclude, that the natural source of this torque is the gravitational (fictitious) "force", even when the system is isolated, i.e. not close to any other mass? It sounds a bit strange to my ears... HOTmag (talk) 11:57, 29 October 2024 (UTC)[reply]

Yes, the source of that torque is gravity, even when spacetime were flat if the flywheel hadn't been there. It's the effect of the flywheel itself on the surrounding spacetime, not depending on any disruptions from nearby objects. Just as a rotating electric or magnetic dipole looses angular momentum to electromagnetic radiation, even without an externally applied electromagnetic field. PiusImpavidus (talk) 13:46, 29 October 2024 (UTC)[reply]

So your flywheel example - but without the motor (along with any two-body system satisfying the same principle), seems to be an extremely rare case (isn't it?), in which a system being "both isolated and neutral", i.e both - not close to any other gravitational mass - and not influeced by any external real force, is not in mechanical equilibrium...

When I was taught about mechanical equilibrium in school (not long ago), my teacher never mentioned this rare option... HOTmag (talk) 15:13, 29 October 2024 (UTC)[reply]

Well, what's isolated? The object isn't isolated from spacetime.

General relativity is hard. Most physics school teachers consider it hard for themselves (although they must have learned something about it) and don't want to go too deeply into it. Apart from some handwavy arguments, most students wouldn't understand it at all. PiusImpavidus (talk) 16:31, 30 October 2024 (UTC)[reply]

I've explained what I mean by "isolated": not close to any other gravitational mass.

I guess if Newton were asked about, what he thought about a body - not close to any other gravitational mass - and not influenced by any force other than the gravitational one, he would immediately determine: "The body is in mechanical equilibrium". This is a direct conclusion derived from the combination of his first two laws with his law of gravity.

So Relativity theory seems to contradict, not only the Galilean transformations and the like, but also the above combination.

Indeed, I knew General relativity was a bit different from the Newtonian theory of gravitation, but I didn't expect such a basic controversy bewteen Einstein and Newton over the ~~necessary~~ sufficient condition for mechanical equilibrium. Newton could define this condition as: "not close to any other gravitational mass and not influenced by any force other than the gravitational one", but Einstein would disagree. This surprises me... HOTmag (talk) 20:02, 30 October 2024 (UTC)[reply]

Newton didn't know about gravitational waves, did he? In Newton's view, gravity is a force just like the pull on a rope, whilst the centrifugal force isn't real and only appears in invalid reference frames. In Newton's view, something is in mechanical equilibrium when all forces (including gravity) and all torques (including gravity) are balanced. In Newton's view, one can be isolated from gravity by being very far from the sun, as he wasn't aware of any object farther away than Saturn. And in Newton's view, bodies act on bodies at a distance.

In Einstein's view, gravity is as real as the centrifugal force and not really a force, but a deformation of spacetime. Gravity isn't directly considered when looking at mechanical equilibrium, but this is solved by having very complex coordinate transformations that can reintroduce the acceleration resulting from gravity. You cannot be isolated from gravity, as we are in a universe full of things, and the farther away, the more massive they get: we can have a star at one AU, but not an entire galaxy. We can have a galaxy at a megaparsec. And finally, bodies don't act on bodies at a distance, but act on local fields and are acted upon by such local fields. The fields provide for the propagation. PiusImpavidus (talk) 09:45, 31 October 2024 (UTC)[reply]

Yes, I'm aware of all these differences between both theories.

As for what I suggested as a condition for mechanical equilibrium, I was wrong when I described it as a "necessary" (and sufficient) condition, because as you say: "we are in a universe full of things", so I've just struck out the word "necessary" in my last response. Additionally: indeed, I defined an "isoloted" body as "not close to any other gravitational mass", but this definition can very easily be sharpened or idealized, by simply saying that an isolated body is a body in an ideal universe that only contains this body and not any other body. That said, Newton and Einstein wouldn't agree about the following intuitive sufficient condition for mechanical equilibrium: "being - both alone in an ideal (theoretical) universe - and also uninfluenced by any force other than the gravitational one". My point was, that Newton could agree to this sufficient condition, while Einstein would not, although this condition sounds very intuitive, if we consider both Newton's two first laws and his law of gravitation (which is of course different from Einstein's field equations). HOTmag (talk) 12:17, 31 October 2024 (UTC)[reply]

(edit conflict) A steadily rotating dumbbell in a zero-gravity environment consisting of a very thin and long bar connecting two extremely massive spheres will emit gravitational waves yet has constant linear and angular momentum. One can argue that this rotating system will actually loose angular momentum due to its rotational energy being transferred to energy dissipated by the gravitational waves. But this lack of rotational mechanical equilibrium is the result of the emission of the gravitational waves and not its cause. --Lambiam 10:47, 29 October 2024 (UTC)[reply]

Thx. Apparently, the natural source - of the torque applied on this system - can't be the gravitational (fictitious) "force", because you're referring to a "zero-gravity environment". Nor can the natural source of the torque be any other natural force (namely: the electromagnetic force, the strong force, and the weak force). Isn't this a bit bizzare? HOTmag (talk) 11:57, 29 October 2024 (UTC)[reply]

Have a look at Mach's principle and Frame dragging. Zero gravity just means no nearby masses. Mach's principle hasn't been verified but there's good reason to think it or something very like it holds so you get an inertial frame when there is 'zero gravity' set by the distant stars. By the way frame dragging happens round a rotating body even if there are no gravity waves. NadVolum (talk) 18:49, 31 October 2024 (UTC)[reply]