# Does deceleration cause linear frame dragging?

• A
• e2m2a
e2m2a
Some Machian theories propose that its the relative acceleration of the universe opposite to the acceleration of an object that causes dragging linear inertial forces on the object. But how do they explain deacceleration of the object when the relative motion of the universe is de-accelerating? How could there be "drag" forces in this case?

e2m2a said:
Some Machian theories
Which ones? Please give specific references.

e2m2a said:
the relative motion of the universe is de-accelerating?
What does this mean?

If you are in a reference system attached to a platform, accelerating in the positive y-direction, with respect to your reference frame the whole universe would be accelerating in the negative y-direction and a spring attached to the platform with a mass attached to it would compress in the negative y-direction due to linear fictitious forces. Some theorists have maintained it is the acceleration of the whole universe in the negative y-direction that "drags" the mass to compress the spring. But if your platform starts to deaccelerate as you continue to move in the positive y-direction, the spring with the attached mass will start to stretch in the positive y-direction. But the universe is not accelerating in the positive y-direction. It is still moving in the negative y-direction, decreasing in speed, yet it causes the spring to stretch. How do theorists account for this? I can't cite anyone specifically except for Einstein who maintained if there is an object in the center of a hollow, shell sphere that is accelerating in the positive y-direction, the object in the center will experience inertial drag forces in the positive y-direction. But how is de-acceleration of the shell account for inertial forces on the object in the negative y-direction in this model?

e2m2a said:
Some theorists have maintained
Which theorists? You need to give specific references. We can't discuss vague handwaving on your part about some "theory" that you can't give a reference to. Personal speculations are off limits here.

e2m2a said:
I can't cite anyone specifically except for Einstein
Where? Please give a specific reference.

Mach’s principle is just a philosophical wish about how humans think the universe should work. The universe is never obligated to work the way we think it should.

There are two big problems with Mach’s principle, in my opinion. The first and biggest problem is that it is very difficult to translate it into something that is experimentally testable. The second problem is that insofar as people like Brans and Dicke have defined it in a testable way, it seems that the actual universe doesn’t follow Mach’s principle much.

PeterDonis said:
Where? Please give a specific reference.
"The Meaning of Relativity", Princeton University Press, 1922, page 106, Albert Einstein in his fourth lecture at Princeton University stated this: "What is to be expected along the line off Mach's thought?...2. A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as the acceleration."

"Deceleration" is just an unscientific word for an acceleration that reduces the speed in your chosen frame. Whether a given acceleration increases or decreases speed depends on your choice of definition of "rest", so it's a distinction without a difference in any theory of physics that doesn't have an absolute sense of "rest". As others have noted, Mach's ideas were never really developed to the point of being a theory, but they didn't aim to add an absolute sense of rest as far as I know.

Ibix said:
"Deceleration" is just an unscientific word for an acceleration that reduces the speed in your chosen frame. Whether a given acceleration increases or decreases speed depends on your choice of definition of "rest", so it's a distinction without a difference in any theory of physics that doesn't have an absolute sense of "rest". As others have noted, Mach's ideas were never really developed to the point of being a theory, but they didn't aim to add an absolute sense of rest as far as I know.
Let me ask you this. When Einstein stated this, he had already completed and published his general theory of relativity. I know he abandoned Mach's principle later on but is it true the mathematics of general relativity predict linear frame dragging can occur? If it is true that the mathematics predict it, how does this occur without invoking a Mach effect for this?

e2m2a said:
"The Meaning of Relativity", Princeton University Press, 1922
This is not a textbook or peer-reviewed paper. It's an informal work that gives Einstein's personal opinions. It is not a valid basis for discussion for this thread.

[Edit: The above is not correct. See posts #26 and #27.]

Last edited:
e2m2a said:
is it true the mathematics of general relativity predict linear frame dragging can occur?
As far as I know it's untested but predicted.
e2m2a said:
how does this occur without invoking a Mach effect for this?
The answer to "how does it occur" is always "because the Einstein field equations say so, there is no more fundamental answer at our current level of knowledge".

As to whether you need to invoke a Machian effect, the first thing you would need to do is define what Mach's Principle actually means in mathematical terms. As @Dale noted earlier, nobody seems able to do that in a way that isn't already experimentally falsified. So the answer is, at best, "we don't know what a Mach effect is well enough to know if one has been invoked or not".

e2m2a said:
is it true the mathematics of general relativity predict linear frame dragging can occur?
Ah, now we have a question that is actually answerable, because it's asking what general relativity predicts instead of what some unspecified vague "model" or "Machian theory" predicts.

The answer @Ibix gave at the top of post #12 is correct.

e2m2a said:
how does this occur without invoking a Mach effect for this?
Now you're back to vagueness and unanswerable questions. What does "invoking a Mach effect" even mean?

This thread got off to a very bad start. "Some theories" turn out to be the musings of one man, reported by another in a popularization - as Peter says, hardly suitable for one discussion.

As Ibix says, "deceleration" is an unscientific word that merely means acceleration in a particular direction. It also implies an absolute state of rest, which is itself a bad start when discussing relativity.

"Linear frame dragging" sounds scinecy, but really isn't. How do you mrasure it? How do you separate it from other effects?

My advice is to start over from the beginning.

PeterDonis said:
Now you're back to vagueness and unanswerable questions. What does "invoking a Mach effect" even mean?
Ok. Sorry I have caused so much trouble here. I like IBIX's response. I should just focus on EInstein's field equations and forget Mach's principle since it is not a rigorously, defined scientific principle.

Mordred, Dale and weirdoguy
PeterDonis said:
This is not a textbook or peer-reviewed paper.
The German title of this (archaic) book is "Grundzüge der Relativitätstheorie". Google-translate to English: "Basic principles of relativity theory". Einstein wrote in it's preface "In the present elaboration of four lectures, that I held at the university of Princeton in May 1921 ...". I find the usual English translation "The Meaning of Relativity" of the book title misleading/"clickbail". Regarding Mach, Einstein wrote in it only, that Mach's principle "appears probable".

Source:
https://en.wikisource.org/wiki/The_Meaning_of_Relativity

Sagittarius A-Star said:
The German title of this (archaic) book is "Grundzüge der Relativitätstheorie". Google-translate to English: "Basic principles of relativity theory". Einstein wrote in it's preface "In the present elaboration of four lectures, that I held at the university of Princeton in May 1921 ...". I find the usual English translation "The Meaning of Relativity" of the book title misleading/"clickbail". Regarding Mach, Einstein wrote in it only, that Mach's principle "appears probable".

Source:
https://en.wikisource.org/wiki/The_Meaning_of_Relativity
None of this changes what I said.

Sagittarius A-Star
I tend to interpret "frame dragging" in the more general context of "gravitomagnetism". This, as the wiki article on frame dragging states, says, is basically about the effect of "mass currents".

https://en.wikipedia.org/wiki/Frame-dragging
wiki said:
Frame-dragging is an effect on spacetime, predicted by Albert Einstein's general theory of relativity, that is due to non-static stationary distributions of mass–energy. A stationary field is one that is in a steady state, but the masses causing that field may be non-static ⁠— rotating, for instance. More generally, the subject that deals with the effects caused by mass–energy currents is known as gravitoelectromagnetism, which is analogous to the magnetism of classical electromagnetism.

This particular version also mentions "linear frame dragging"

wiki said:
Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging (but see Einstein, 1921).[5]

See also the article on the Lense-Thiring effect to further justify that "frame dragging" is due to gravitomagnetism.

https://en.wikipedia.org/wiki/Lense–Thirring_precession
wiki said:
In general relativity, Lense–Thirring precession or the Lense–Thirring effect (Austrian German: [ˈlɛnsə ˈtɪrɪŋ]; named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum S.

So, to my way of thinking, the basic point of the Lense-Thiring effect is that the space-time inside a large hollow rotating sphere isn't flat because the mass currents. This is similar to the magnetic field inside a hollow rotating charged sphere.

The effect of the gravitomagnetic field on a gyroscope is not too hard to calculate by analogy with electromagnetism - a rotating charge has a magnetic moment, so a rotating mass has a gravitomagnetic moment, and responds to the gravitomagnetic field. What's interesting is taking the limit as the rotation of the gyroscope approaches zero. If we imagine a long rod parallel to the spin axis of a spinning gyroscope, if there are no significant non-gravitatioanl torques on each other, the axis of the rod and the spin axis of the gyroscope should remain parallel.

A few additional comment - as I recall, there is a non-obvious factor of two in the weak-field gravitomagnetic equations (hencefort GEM equations) as compared to Maxwell's equations, and unfortunately the GEM equations aren't invariant under Lorentz boosts (unlike Maxwell's equations), so unlike Maxwell's equations, which are exact in SR, the GEM equations in GR are approximations.

Ibix and Sagittarius A-Star
I think the following is correct, but it's not from a textbook or paper, just my thoughts, and as I write this I realize I'm missing a few pieces to nail my ideas down. However, I'll write this anyway.

What we want to do is to see if there is any effect on the direction of spin of a gyroscope when there is a nearby mass current, due to a massive linear moving object flying nearby. This seems like the simplest sort of experiment which we might describe as "linear frame dragging".

Rather than do this directly, we can change the problem a bit, and make the massive object stationary and the gyroscope move by a suitable change of coordinates.

So, consider a gyroscope doing a flyby of a large mass. The gyroscope follows a geodesic, which is a curve that parallel transports itself. This curve has no proper acceleration, so there is no difference between parallel transport and Fermi-walker transport. Thus we conclude that if the gyroscope spin axis is pointed along the direction of motion initially (i.e. parallel to the direction of motion), it will remain parallel to the direction of motion. The gyroscope will parallel-transport and fermi-walker transport it's spin axis along the geodesic path of its orbit. This implies that the gyroscope spin axis rotates relative to the fixed stars since the direction of travel of the gyroscope changes during the flyby as its path is deflected. If we ignore various aberrations and other optical effects for the purpose of exposition , if we attach a telescope to the gyroscope pointed at a guide star, the telescope will not remain pointing at the guide star.

The result of this line of thought is that we conclude that the spin axis of the gyroscope rotates relative to the fixed stars. Also, we conclude that the angle of deflection depends on the orbit of the gyroscope, not how fast it spins or it's angular momentum. This does raise an interesting point, I suppose we'd really need to use the Papapetrou equations rather than the geodesic equations if the spin was high enough. I don't think it'd matter, but it might.

Now, consider a point of view of the gyroscope. Specifically, we consider the point of view of a local inertial frame attached to the gyroscope. (The idea of a point of view in GR is a bit ambiguous, so I am being more specific). The relative rotation of the spin axis to the fixed stars must be true in the gyroscope frame just as it was in the fame of the massive body. But we need a different explanation.

I am suggesting that this "something" that causes the gyroscopes spin access to change relative to the fixed stars is the gravitomagnetic field of the moving mass flying near the gyroscope, i.e. the moving mass generates a "mass current" which generates a gravitomagnetic field, which causes the spin access to change. But I haven't done a calculation to really justify this explanation by confirming that we can use it to numerically calculate the amount of deflection of the spin axis of the gyroscope relative to the fixed stars. Also, the analysis I used is rather idealized - for instance I am assuming the gyroscope is so small in extent that we can ignore effects like tidal torques which may be large in a realistic experimental flyby. And I've also assumed the spin isn't so large that it affects the travel path.

So, I haven't really done the necessary work to justify my intuitive conception, but I do think it's an interesting approach that might shed some light on the idea of "linear frame dragging".

pervect said:
we conclude that if the gyroscope spin axis is pointed along the direction of motion initially (i.e. parallel to the direction of motion), it will remain parallel to the direction of motion.
This doesn't follow.

The "spin axis" you describe is a spacelike vector which lies in the spacelike surface orthogonal to the 4-velocity at every event on the worldline; that is the condition that is preserved by Fermi-Walker transport. But since you have specified a geodesic, there is actually no local concept of "direction of motion": there is no local spacelike vector whose direction is picked out by the motion of the gyroscope, i.e., by its timelike 4-velocity vector. (Note that if the gyroscope were not in free fall, there would be such a local direction: the direction of proper acceleration. But in free fall, there is no such direction.) This should actually be easy to see if you transform to the local inertial frame in which the gyroscope is at rest: which spacelike direction is the "direction of motion"? Obviously none--the gyroscope is at rest!

It is true that, in flat spacetime, if the gyroscope is in free fall and it starts out with its axis pointing towards some fixed distant guide star, the axis will remain pointing in that direction forever. But that is not because the gyroscope has any local notion of "direction of motion" in free fall; it is because spacetime is flat. In curved spacetime, while it will in general be true that the freely falling gyroscope's axis will not remain pointing at some fixed distant guide star if it starts out that way, you cannot infer from that that the gyroscope's "direction of motion" has changed. All you can infer is that the spacetime is curved.

pervect said:
Thus we conclude that if the gyroscope spin axis is pointed along the direction of motion initially (i.e. parallel to the direction of motion), it will remain parallel to the direction of motion.
Btw, for an obvious counterexample, consider a gyroscope in a free-fall orbit about a planet. Suppose the gyroscope starts out pointing along the planet's orbit, and that there is a fixed distant star in that direction. As the gyroscope goes around the orbit, it will certainly not turn to keep pointing along the orbit; in fact it will almost keep pointing towards the fixed distant star. The only precession will be de Sitter precession and, if the planet is rotating, Lense-Thirring precession; but these are small effects that certainly don't amount to 360 degrees of precession per orbit, which is what it would take to keep the gyroscope pointing along its direction of motion.

e2m2a said:
Ok. Sorry I have caused so much trouble here. I like IBIX's response. I should just focus on EInstein's field equations and forget Mach's principle since it is not a rigorously, defined scientific principle.
First, let me say that you do not need to "forget" Mach's principle, let things inspire you as they are. Just know when talking to scientists (and especially on this forum) you need to have more than inspiration. You need a more operational understanding of terms to convey your idea into something predictable.

If you're motivated by Mach and gravitation, I'd suggest picking up the book "Gravitation and Inertia" by John Wheeler and Ignazio Ciufolini However, it is also 30 years old and it will make you work hard. But, it is worth reading if you are truly interested in frame dragging. It is one of the few books I've seen balance experiments and theory.

Another place to look is:
https://core.ac.uk/download/pdf/30902883.pdf Which is a thesis on translational dragging, which would be the fancier way of saying linear dragging, and they discuss Mach.

https://arxiv.org/pdf/2109.14641 where they do have a paragraph on "linear dragging",

Frame dragging is fun, but it is very hard to conceptualize in my opinion. Ergospheres are one of the coolest aspects of GR IMO.

Last edited:
romsofia said:
and do cite the above thesis.
I don't see a citation to the thesis in that paper. There is a citation to a paper by Gron and Eriksen with the same title as the thesis.

romsofia said:
a thesis on translational dragging
I would be hesitant to rely on this thesis. I have not read it through in detail, but on a first skim, I found the following on p. 20:

...in Case 2 they are actually accelerated by some engine(s) that is “magically” coordinated so that they all appear to start accelerating at the same time as viewed from the spaceship. We should say something about these hypothetical engines. They obviously cannot be ordinary fuel-engines, as then the total momentum of the stars would be conserved (considering the stars plus the exhaust) and we could not speak about the total acceleration of the stars (by which we mean the whole material content of the Universe). However, we could imagine a radiation enginge that propels the stars. The radiation carries momentum but not mass, and as long as the radiation is not absorbed at a later stage the overall material momentum of the Universe would increase without violating conservation of energy or momentum.

This, of course, is simply wrong. The radiation does not have rest mass, but it does have stress-energy, and you can't leave that stress-energy out of the analysis; it gravitates, just like the "material" stars themselves do. The total momentum of stars + exhaust is still conserved; the fact that it is not all "material momentum" is irrelevant.

Again, I have not read the thesis in detail, but when I find an error like that I immediately raise my Bayesian prior for "this author's work is not reliable".

romsofia
PeterDonis said:
This is not a textbook or peer-reviewed paper. It's an informal work that gives Einstein's personal opinions. It is not a valid basis for discussion for this thread.
I don’t agree with that. The book presents a mathematical development of special and general relativity and, in later editions, is primary presentation of his last attempt at a unified field theory. I would argue that Einstein considered this to be his textbook on relativity.

PAllen said:
The book presents a mathematical development of special and general relativity and, in later editions, is primary presentation of his last attempt at a unified field theory. I would argue that Einstein considered this to be his textbook on relativity.
Oops, yes, I was confusing this with his popular book on relativity. The book actually cited is indeed a much more rigorous work.

That said, I'm still not sure the book says what the OP thinks it says.

PeterDonis said:
Oops, yes, I was confusing this with his popular book on relativity. The book actually cited is indeed a much more rigorous work.

That said, I'm still not sure the book says what the OP thinks it says.
What the book derives after the snippet quoted is a particular form of approximate equations of motion in GR that Einstein claims show a tiny effect of the form of a test body experiencing an accelerating force when neighboring masses are accelerated; and this force is in the same direction as that acceleration.

One reaction I have to this is that I would think this might also true for Newtonian gravity.

Also, I have never studied this section, so I don’t know what I think of the uncommon approximation method used, or the interpretation of its meaning.

Further, Einstein never calls this frame dragging, and neither would I.

Last edited:

• Special and General Relativity
Replies
1
Views
1K
• Special and General Relativity
Replies
24
Views
2K
• Special and General Relativity
Replies
38
Views
2K
• Special and General Relativity
Replies
9
Views
655
• Special and General Relativity
Replies
8
Views
1K
• Special and General Relativity
Replies
14
Views
1K
• Special and General Relativity
Replies
144
Views
6K
• Special and General Relativity
Replies
36
Views
3K
• Special and General Relativity
Replies
7
Views
1K
• Special and General Relativity
Replies
72
Views
4K