Would an object in orbit change facing or orientation?

[sbaker8688]

TL;DR Summary

Would an object encountering warpage in space-time change its facing or orientation?

Assume a 2D XY grid, and we'll label the obvious directions N, S, E, and W. Assume an object is just going along a straight line from S to N in flat space with no forces acting on it or imparted to it, including rotational. Let's assume some kind of mark on it so we can keep track of orientation, like a vertical line running up and down the object lining up perfectly with any grid lines you would imagine running N and S. Here, I assume you'd agree that in this situation the line on the object continues to line up perfectly with north-south grid lines.

Assume that the same object then encounters a gravity well or a warpage in spacetime as it moves S to N and gets pulled into orbit. My question is whether the object would or could change facing or orientation, i.e. would the vertical line running north-south up and down the object deviate from north-south. If the answer is "it depends" then give me the scenarios, i.e. if it depends on shape of object, first assume perfectly spherical, then "other." If it depends on shape of gravity well, assume perfectly spherical. If it depends on mass distribution, first assume perfectly uniform distribution of mass and density, then "other."

Avoid mathematical explanations for the most part - thanks.

Thanks.

 

[anuttarasammyak]

The moon orbiting around the Earth by gravity shows the fixed hemisphere to the Earth. Does this help your study ?

 

[sbaker8688]

The moon orbiting around the Earth shows the same hemisphere to the Earth. Does this help your study ?

No, because I don't know whether something gave the moon some kind of rotational force at some point, I don't know how much the imperfections in the surfaces of earth and moon impact things, I don't know how much unequal mass distributions within the objects affect things, etc.

Perhaps a simpler version of this question would be to assume the object creating the space-time warpage is a perfectly spherical non-rotating black hole, and the object going into orbit is also perfectly spherical with completely uniform mass distribution. Let's say it has a massless "N" painted on it's north side facing north as it encounters the black hole's gravity well. Does the N "process" or "rotate" or do anything?

(As an aside, the reason I'm asking is to see if my expectations line up with reality. My expectation is that the N would deviate from facing north. Why? I'm thinking logically and conceptually, not mathematically. My reasoning is that if objects follow a straight line through curved spacetime, there's no reason to assume that object alignment doesn't also, in some respect, follow a straight path through curved spacetime. To the object, it thinks it continues to align perfectly with it's "straight line." it's N pointing not to some "external N" but to an N parallel to it's straight line in curved spacetime that it's following, but to the outside observer, the object doesn't continue to line up with his reference. But, this is just theorycrafting on my end, and I'm wondering what the reality is)

 

[Dale]

Yes, this happens. It is called frame dragging.

https://en.m.wikipedia.org/wiki/Frame-dragging

It has been measured, but not with very great accuracy

 

[sbaker8688]

Yes, this happens. It is called frame dragging.

https://en.m.wikipedia.org/wiki/Frame-dragging

It has been measured, but not with very great accuracy

Assume non-rotation of the thing causing the gravity well.

 

[anuttarasammyak]

Tidal force generates stress in the rotating body. That would complicate the study.
Say we have two nearby mass points A and B, e.g. along East Axis, the gravitational interaction of which is negligible. They are in BH gravitation field, are at still to each other initially and make circular motions. After a period of A motion, is B in its same initial position ? If we tune their angular velocities I think we can do it, but that would violate the condition that they are at still to each other initially.

[EDIT]Kepler’s law
$$\frac{r_A^3}{T_A^2}=\frac{r_B^3}{T_B^2}$$
says we cannot synchronize them in the way I said above. Cloud of dusts cannot keep its shape during orbiting except special uniform configurations as rings of Jupiter and Saturn.

 

[Nugatory]

My question is whether the object would or could change facing or orientation,

Here’s an easily visualized example:
Walk straight ahead for a while, then without turning (so you’ll be walking sideways like a crab) start moving to your left, at right angles to your original path. If you are moving on a flat uncurved unwarped surface you will be end up facing the same direction as when you started.
But try this on the surface of the earth, which is curved not flat: start at the equator, walk north while facing north until you reach the pole, then start the same sideways shuffle to your left. When you get back to the equator you will be facing west not north even though you started facing north and never turned.

So yes, moving in a straight line through a curved space can cause a change in orientation.

 

[Filip Larsen]

My question is whether the object would or could change facing or orientation, i.e. would the vertical line running north-south up and down the object deviate from north-south.

As has been hinted already, asymmetric extended objects in orbit around a mass will experience what is called gravity gradient (tidal) torques, which means that the (very) small changes in gravity in the radial direction in general will give rise to (very) small torques on an object that initially has its orientation fixed relative to the background and these torques will over time rotate the object away from its initial orientation.

Since torques generates internal stresses in the object and all physical objects to some degree gets heated up by internal stresses, the gravity gradient will over time make (passive) objects in orbit end up in a more or less stable locked orientation relative to the mass (tidal lock). We even use this mechanism to stabilize low Earth orbit satellites so they can have a stable orientation relative to Earth.

 

[FactChecker]

I interpreted the question to be: Does an orbiting object seem to be not rotating (zero rotational rate) in the geometry of curved spacetime if it remains with one side toward the center of the orbit?

 

[JimWhoKnew]

Assume non-rotation of the thing causing the gravity well.

The Gravity Probe-B experiment was designed to measure two effects. The stronger effect is the geodetic (de Sitter) precession, which is independent of Earth's rotation (to lowest order).

 

[JimWhoKnew]

Perhaps a simpler version of this question would be to assume the object creating the space-time warpage is a perfectly spherical non-rotating black hole, and the object going into orbit is also perfectly spherical with completely uniform mass distribution. Let's say it has a massless "N" painted on it's north side facing north as it encounters the black hole's gravity well. Does the N "process" or "rotate" or do anything?

Actually, the answer to your question comes from the very basic property that "carried vectors" are "tilted" (in general) by parallel transport along a closed curve in curved spacetime. Since closed timelike worldlines are hard to come by, consider the following poor man's substitute: Suppose two observers are in opposite circular geodetic orbits around a Schwarzschild black hole at the same $r~$, and $~\theta=\pi/2~$ (ignore possible collision). Each of them has a set of gyroscopes pointing in the 3 orthogonal spatial directions. The orthogonality is maintained throughout their motion. Let's assume that at one instance when they cross each other's path, their gyroscopes are mutually aligned. When they meet again on the other side of the hole, we expect from the curvature property that their gyroscopes will no longer be aligned. So at least one of them has gyroscopes that changed orientation with regard to the global coordinates (and by symmetry, we know they both have such gyroscopes).

If you replace the sets of gyroscopes by 2 perfect balls with massless N,E,U painted on them orthogonally, the same outcome is expected (and for the same reasons).

 

[A.T.]

Avoid mathematical explanations for the most part - thanks.

Here is a simple diagram:

From this article, where you can find more explanations:
https://einstein.stanford.edu/SPACETIME/spacetime4.html#geodetic_effect

 

[sbaker8688]

I interpreted the question to be: Does an orbiting object seem to be not rotating (zero rotational rate) in the geometry of curved spacetime if it remains with one side toward the center of the orbit?

I think this is a way of stating the question? It gets at an 'essence' of what I'm asking, although I also want to know if it is initially nonrotating before entering the gravity well, would it start rotating from the perspective of an 'outside observer'?

I want to ignore tidal effects - I'm not interested in that. I'm purely interested in the 'curved spacetime' effects.

 

[sbaker8688]

As has been hinted already, asymmetric extended objects in orbit around a mass will experience what is called gravity gradient (tidal) torques...

Sorry, I should have specified that I'd like to ignore tidal effects, as well as friction effects, density or unequal distribution of mass effects, shape of object effects, etc. In other words I'm purely interested in curved spacetime effects.

 

[sbaker8688]

Actually, the answer to your question comes from the very basic property that "carried vectors" are "tilted" (in general) by parallel transport along a closed curve in curved spacetime....

I think you are saying that in the case of the non-rotating black hole and the initially non-rotating perfect sphere with N painted on it's north side, then ignoring tidal and any other effects that are not "curved spacetime effects," the sphere with N painted on it's north side would start to rotate?

 

[sbaker8688]

Here is a simple diagram:

Thanks.

 

[sbaker8688]

Another way of stating the question might be, assuming no forces are acting on a perfectly spherical object, and assuming it's nonrotating in flat space and therefore has an 'alignment' it is maintaining N and S, which 'alignment' does it want to maintain (ignoring all forces such as tidal, frictional, this, that) when entering a gravitiy well of a nonrotating black hole? The flat space alignment (N and S)? Or is there another "warped space" alignment it wants to maintain? Would the spherical object say it it is maintaining N and S alignment, while a flat space observer would disagree and say "no, you are rotating now"?

 

[pervect]

I am unclear of the direction of the precession, but both a spinning body (gyroscope) and a non-spinning body may change their orientation (a change in spin axis for the rotating body) as they orbit. I am slightly more familiar with the gyroscopic case, so that's what I'll describe. That's also what the literature tends to discuss.

The two relevant effects that I'm aware of in curved space-time are the geodetic precession (also known as de-Sitter precession) and the much smaller frame-dragging effect. The geodetic precession happens even if the central body is not rotating, the frame-dragging effect is due to the rotation of the central body. Gravity probe B measured both the geodetic precession and the frame-dragging effect of a satellite orbiting the Earth.

In flat space-time, there is a special-relativistic effect known as Thomas precession that causes similar effects to the geodetic precession, for instance if you forced a gyroscope to orbit a central point via a force applied by a non-gravitational source such as a tether. As I recall it's proportional to the cross product of the acceleration and the velocity, so it only has an effect if the acceleration vector is not parallel to the velocity vector.

Wiki has some references on this point, see https://en.wikipedia.org/wiki/Gravity_Probe_B and https://en.wikipedia.org/wiki/Thomas_precession.

They aren't very detailed- I seem to recall a discussion in one of my texts conmparing Thomas precession to geodetic precession, but I couldn't find it. (I thought it was MTW, but the discussion on pg 175 that I reviewed didn't mention it).

 

[JimWhoKnew]

I think you are saying that in the case of the non-rotating black hole and the initially non-rotating perfect sphere with N painted on it's north side, then ignoring tidal and any other effects that are not "curved spacetime effects," the sphere with N painted on it's north side would start to rotate?

I say that a change of the orientation of the sphere ("rotation"), compatible with these conditions, is possible (provided there is a chart that allows a meaningful definition of "orientation"). Whether such a change will actually happen, depends on the details of the geodetic motion.

 

[A.T.]

I think you are saying that in the case of the non-rotating black hole and the initially non-rotating perfect sphere with N painted on it's north side, then ignoring tidal and any other effects that are not "curved spacetime effects," the sphere with N painted on it's north side would start to rotate?

Depends what "start to rotate" means. The point of the Geodetic Effect (see post #12) is that despite no locally detectable rotation, it can return to a starting position with a different orientation than it initially had there.

 

[PeterDonis]

assuming no forces are acting on a perfectly spherical object, and assuming it's nonrotating in flat space and therefore has an 'alignment' it is maintaining N and S, which 'alignment' does it want to maintain (ignoring all forces such as tidal, frictional, this, that) when entering a gravitiy well of a nonrotating black hole? The flat space alignment (N and S)? Or is there another "warped space" alignment it wants to maintain? Would the spherical object say it it is maintaining N and S alignment, while a flat space observer would disagree and say "no, you are rotating now"?

These questions all make a hidden assumption, that there is some unique definition of "alignment" or "orientation". But there isn't. There are actually several different concepts involved here:

(1) An object can be modeled as a set of worldlines (non-intersecting timelike curves in spacetime) that fill the "world tube" occupied by the object. The object is "non-rotating" if that set of worldlines has zero vorticity (which has a precise mathematical meaning and can be computed if the worldlines are known).

(2) An object can be modeled as a single worldline (which can be thought of as the worldline of its center of mass) plus a set of three orthonormal spacelike vectors that describe a set of "spatial axes" that are carried along by the object through spacetime. The object is "non-rotating" if the spacelike vectors are Fermi-Walker transported along its worldline (which again has a precise mathematical meaning and can be computed if the worldline is known).

Note that it can be proven that, under fairly general assumptions, senses #1 and #2 of "rotation" above are equivalent if we make appropriate choices about how to match them up.

(3) If we have a worldline and at least one spacelike vector carried along the worldline (which we obviously have in #2 above, and which we can have in #1 above if we match up #1 and #2 appropriately so they are equivalent as above), we can look to see in which direction the spacelike vector points globally (for example, at a particular distant star), and whether that direction changes along the worldline. The object is "non-rotating" if that direction doesn't change.

In experiments like Gravity Probe B, the "precession" being tested for was a failure of the satellite to be non-rotating in sense #3 above, even though it was non-rotating in senses #1 and #2 above. The latter was because (a) the satellite was set to be non-rotating in senses #1 and #2 above when it was launched, and (b) the satellite was small enough that tidal effects, i.e., any effects of spacetime curvature that might cause it to start rotating in senses #1 and #2 above, were negligible.

However, in a case like the Moon orbiting the Earth, the Moon is rotating in all three senses above. The Moon is large enough that tidal effects on it are not negligible, and of course the reason it is rotating in all three senses above now is that it is tidally locked to the Earth.

 

[pervect]

The issue of definition of "alignment" is an important one. There are two popular choices - alignment relative to the fixed stars, which is the one I use, and the GPB experiment uses. A second possible source of alignment is what a gyroscope does. The point of the GPB experiment is that the two aren't the same - the gyroscope doesn't keep pointing at the "fixed" guide star.

A subsidary point is whether or not the universe as a whole rotates, this sort of cosmological question would be at odds with the idea that the "fixed stars" create a (non-rotating) reference frame. The standard cosmolgical model doesn't rotate, I believe I might have glanced at some more direct measurements or expeirments, but I can't really recall, and nothing obvious comes to mind.

 

[PeterDonis]

Moderator's note: A subthread on the details of what I described in post #21 has been moved to its own separate "I" level thread here:

https://www.physicsforums.com/threads/frame-fields-and-congruences-of-worldlines.1065770/

 

[pervect]

I wanted to confirm that for circular motion in a flat-space time (with some non-gravitational central force providing the needed centripetal acceleration), the Thomas precession would be 1/3 of the geodetic precession of an orbit with the same velocity in curved space-time. This is my interpretation of https://einstein.stanford.edu/SPACETIME/spacetime4.html#geodetic_effect.

Velocity here is measured relative to a stationary co-located stationary observer, I am ignoring time dilation effects as the original source didn't really discuss them (though comments on this point would be somewhat interesting as well).

In the framework of the gravito-electromagnetic analogy, the geodetic effect arises partly as a spin-orbit interaction between the spin of the test body (the gyroscope in the case of GP-B) and the "mass current" of the central body (the earth). This is the exact analog of Thomas precession in electromagnetism, where the electron experiences an induced magnetic field (in its rest frame) due to the apparent motion of the nucleus. In the gravitomagnetic case, the orbiting gyroscope feels the massive earth whizzing around it (in its rest frame) and experiences an induced gravitomagnetic torque, causing its spin vector to precess. This spin-orbit interaction accounts for one third of the total geodetic precession; the other two thirds arise due to space curvature alone and cannot be interpreted gravito-electromagnetically. They can, however, be understood geometrically. Model flat space as a 2-dimensional sheet, as shown in the diagram below (left).