Lense-Thirring Effect and Energy Loss

[Buckethead]

TL;DR Summary

Does the deviation of an object due to Lense-Thirring cause loss of energy?

I was thinking about the Gravity-B probe and the rotation of the satellite due to the Lense-Thirring effect. I was imagining that the satellite, once in orbit, was aligned to a distant star using retro rockets effectively rendering it non-rotating relative to that star. That being said, once free to do its thing, over time the satellite would then rotate away from the star due to the Lense-Thirring effect. The question I have is: Since the satellite was initially non rotating, and since it began to rotate, this may or may not have required an acceleration and hence an energy expenditure. Was there indeed an energy expenditure? Since the satellite simply rotates due to the whim of the geodesic of spacetime, I would think that there would not be. The effect would be tiny of course since once the satellite begins to rotate, it's just inertia that keeps it rotating from then on. Thanks.

 

[PeterDonis]

Summary:: Does the deviation of an object due to Lense-Thirring cause loss of energy?

I was imagining that the satellite, once in orbit, was aligned to a distant star using retro rockets effectively rendering it non-rotating relative to that star.

Why would you imagine that?

Summary:: Does the deviation of an object due to Lense-Thirring cause loss of energy?

Since the satellite was initially non rotating, and since it began to rotate

Do you have any evidence that this actually happened?

 

[PeterDonis]

the rotation of the satellite due to the Lense-Thirring effect

You are misunderstanding what the Lense-Thirring effect is. The whole point of the Lense-Thirring effect is that the satellite is not rotating--its orientation remains the same as seen by local gyroscopes. (In technical terms, the tetrad that describes the satellite's 4-velocity and spatial orientation is parallel transported along its worldline.) The rotation of the body that the satellite is orbiting causes it to precess relative to distant objects, but this is completely unobservable locally.

 

[Buckethead]

Why would you imagine that?

The Gravity B probe was aligned to a distant star. I can't think of any other way to align it to that star except by using retro rockets once in orbit. The on board gyroscopes would then deviate (precess) due to both gravity and Lense-Thirring relative to this alignment. Correct?

You are misunderstanding what the Lense-Thirring effect is. The whole point of the Lense-Thirring effect is that the satellite is not rotating--its orientation remains the same as seen by local gyroscopes. (In technical terms, the tetrad that describes the satellite's 4-velocity and spatial orientation is parallel transported along its worldline.) The rotation of the body that the satellite is orbiting causes it to precess relative to distant objects, but this is completely unobservable locally.

OK, right. I didn't think that through. The satellite remains pointing to the distant star at all times and only the gyroscopes precess due to both the mass of the Earth and the Lense-Thirring effect (geodetic effect and Lense-Thirring effect). So that being the case, my question still is, does the change in the precession of the gyroscopes (from 0 at setup to precession once set free) absorb energy?

 

[Buckethead]

Actually, I need to completely re-phrase my question as I was only using Gravity Probe B as a stepping stone to try and figure out my overriding question which is this:

Let's say we have a mass (planet) in an otherwise flat spacetime. Along comes an asteroid originating from flat space and enters into the curved spacetime of the planet and is deflected. Now the direction of the asteroid has changed and from above it would look like the object has accelerated. However I know it hasn't because it is simply following a spacetime geodesic. But my question is, since from outside it appears as if the object has accelerated, did the (local) spacetime have to react to that change? In other words, did the spacetime "move" in response to the change in trajectory of the asteroid? In other words, was energy exchanged?

Sorry I didn't just ask this in the first place, I was trying to figure out the answer for myself using the results from Gravity Probe B.

 

[PeterDonis]

The Gravity B probe was aligned to a distant star.

Please give a specific reference for "aligned to". My understanding is that the satellite included a telescope that was sighted on a distant star, but that is not the same as forcing the satellite itself to be aligned with the star.

 

[PeterDonis]

the change in the precession of the gyroscopes (from 0 at setup to precession once set free)

You are still claiming that the gyroscopes were somehow "aligned" and then "set free". Please give a specific reference for those statements. If you are just referring to the spin-up and preparation of the gyroscopes after the satellite was placed in orbit, of course that took energy, but I'm not sure if that's what you are referring to.

 

[PeterDonis]

did the (local) spacetime have to react to that change?

If the asteroid itself has a significant mass (enough to cause detectable spacetime curvature), then of course it will "change" the local spacetime as it moves past. ("Change" is really not the right term, since spacetime is a 4-dimensional geometry that doesn't "change", it just is. A better way of stating what is going on would be to say that the spacetime geometry includes significant curvature both from the star and from the asteroid.)

did the spacetime "move" in response to the change in trajectory of the asteroid?

Spacetime doesn't "move". See above.

was energy exchanged?

There is no invariant concept of "energy" in a scenario like this. Nor is "energy exchanged" equivalent to "spacetime curvature present".

 

[PeterDonis]

If the asteroid itself has a significant mass (enough to cause detectable spacetime curvature)

Btw, this is not true of the Gravity Probe B satellite; it does not cause detectable spacetime curvature (at least not enough to detectably affect its orbit or the behavior of its gyroscopes). That was on purpose; the point was to investigate the Earth's spacetime curvature with a "test object", i.e., an object that does not itself cause any spacetime curvature.

 

[vanhees71]

Here's the Wikipedia explanation of the experimental setup of Gravity Probe B:

https://en.wikipedia.org/wiki/Gravity_Probe_B#Experimental_setup

 

[Ibix]

But my question is, since from outside it appears as if the object has accelerated, did the (local) spacetime have to react to that change? In other words, did the spacetime "move" in response to the change in trajectory of the asteroid? In other words, was energy exchanged?

You aren't using helpful concepts for thinking about this. Spacetime isn't a thing that changes - it's a 4d whole, and change is something you see when comparing one 3d slice ("the universe, now") to another ("the universe, a bit later"). And energy is an extremely tricky concept when you talk about gravity in GR. It's very useful, but it's not the panacea it is for Newtonian gravitational problems.

That said, you should remember that Newtonian gravity is a limit of GR. An asteroid passing a planet is square in the range of validity of Newton, and Newton clearly says that both planet and asteroid will follow hyperbolic paths in their barycentric frame. So the planet will start to move if it was initially stationary in your frame.

Your question about "does spacetime change" is ill posed, as I said. I think what you are trying to capture is the notion that, a long time before the asteroid, a distant observer would be able to use radar and test particles to establish that they were at rest with respect to the planet and that spacetime near the planet is Schwarzschild to very high precision. After the asteroid they would be able to establish that spacetime near the planet is Schwarzschild to very high precision but that they are no longer at rest with respect to the planet. They could argue, then, that what they call "all the universe, now" has a different metric before and after the interaction. But, again, this is because they are comparing two different parts of spacetime, not because "spacetime has changed".

It's not that you cannot think about spacetime as a stack of slices (that's what the ADM formalism is, basically), but you do need to keep straight when you are talking about the slices and when you are talking about the whole. And note that GR allows spacetimes that can't be described by ADM - those with causal loops, for example.

 

[pervect]

My understanding of the GP-B experiment is that the telescope was actively "steered" to point at the guide star, while great effort was made to make the gyroscopes torque-free.

See for instance http://einstein.stanford.edu/content/fact_sheet/GP-B_Nutshell-0307.pdf

Conceptually, the GP-B experiment is simple: Place a gy-roscope and a telescope in a polar-orbiting satellite, 642 km (400 mi) above the Earth. (GP-B actually uses four gyroscopes for redundancy.) At the start of the experi-ment, align both the telescope and the spin axis of the gyroscope with a distant reference point—a guide star. Keep the telescope aligned with the guide star for a year, and measure the precession change in the spin axis align-ment of the gyros over this period in both the plane of the orbit (the geodetic precession) and orthogonally in the plane of the Earth’s rotation (frame-dragging precession).

I believe the intent of this arrangement is to minimize any sort of unwanted coupling between the satellite body and the gyroscopes. But that's from memory, not something in the reference.

As far as energy goes:

The motion of the telescope manifestly do not conserve energy and momentum - there is active station-keeping that steers the telescope so it points at the guide star, compensates for various perturbing effects.

Ignoring frictional losses (which were minimized, but present), I would expect that the gyroscopes do conserve energy in the GR sense, i.e. I'd expect them to have a constant ADM energy. I don't have an explicit reference for this.

While this may not be readily understandable in terms of Newtonian physics, it doesn't have to be, as it's not Newtonian. The idea is to point out some of the subtle differences in GR vs Newtonian physics. It is a bit confusing that gyroscopes don't keep their spin axis aligned with the guide star. But that's the whole point of the experiment. As far as energy goes, it's the natural motion of the gyroscopes that maintains energy-momnetum, not the motion of the telescope. The motion of the telescope isn't force free - the motion of the gyroscopes is as nearly force-free as experiment could make it. Force-free in this context means non-gravitational forces.

 

[PeterDonis]

I would expect that the gyroscopes do conserve energy in the GR sense, i.e. I'd expect them to have a constant ADM energy.

ADM energy is not a concept that applies to test objects, which is what the gyroscopes are in this experiment. The only relevant notion of energy for the gyroscopes is energy at infinity, which is a constant of geodesic motion (assuming a stationary spacetime, which is assumed for this experiment). But that notion also applies to the satellite--the "station keeping" you refer to to keep the telescope pointed at the distant guide star is generally done using pure torques, i.e., multiple thrusters are used that are placed symmetrically about the satellite's center of mass so its orientation can be adjusted without changing its orbit. So there is no useful notion of "energy" that is not conserved by this operation.

 

[PeterDonis]

The motion of the telescope isn't force free

While this is true, it does not imply that there is any useful notion of energy that is not conserved by the satellite's motion. See my previous post.

Force-free in this context means non-gravitational forces.

It's worth noting, though, that the torques that are applied to the satellite to keep the telescope pointed at the distant guide star will cause "fictitious forces" (which are not usefully viewed as "non-gravitational") to appear in a reference frame that is fixed with respect to the telescope. (Of course in practice these "fictitious forces" will be tiny, since the precession involved is tiny.)

 

[vanhees71]

AFAIK the telescope was used to align one axis of the satellite with the observed binary (IM Pegasi). At the beginning also the axis of the gyroscope was aligned with this star. The gyroscopes are utmost precise spheres with a layer of niob on their surfaces within a bath of liquid He, so that the niob is superconducting. This provides a magnetic moment aligned with the spin axis (London moment of a super conductor). Also possible magnetic fields are very well shielded and also friction was minimized. A free gyroscope's spin is always Fermi-Walker transported. Since to the best of technical feasibility there's no external torque acting on the gyro this Fermi-Walker transport is along the geodesics of the free-falling gyro and thus in fact parallel transported, such that a precise measurement of the precession of the spin axis (via the London magnetic moment) relative to the axis determined by the telescope's observation of the binary star gives the effects predicted by the motion in the gravitational field of the rotating Earth, allows to observe both the geodetic precession (due to the spatial curvature due to the Earth's magnetic field) and also the smaller Lense-Thirring precession (frame-dragging effect due to the rotation of the Earth around its axis).