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Featured B Inertia and Mach's Principle

  1. Jun 6, 2017 #1
    Mach, Newton and others observed that centrifugal forces appear in a object when it rotates in relation to the stars. Einstein was convinced by this and tried, unsuccessfully as far as I understand, to incorporate what he called Mach’s Principle into General Relativity.

    From Wikipedia -”Mach’s principle”

    Einstein was convinced that a valid theory of gravity would necessarily have to include the relativity of inertia:

    So strongly did Einstein believe at that time in the relativity of inertia that in 1918 he stated as being on an equal footing three principles on which a satisfactory theory of gravitation should rest:

    1. The principle of relativity as expressed by general covariance.

    2. The principle of equivalence.

    3. Mach's principle (the first time this term entered the literature): … that the gµν are completely determined by the mass of bodies, more generally by Tµν.
    In 1922, Einstein noted that others were satisfied to proceed without this [third] criterion and added, "This contentedness will appear incomprehensible to a later generation however."

    It must be said that, as far as I can see, to this day Mach's principle has not brought physics decisively farther. It must also be said that the origin of inertia is and remains the most obscure subject in the theory of particles and fields. Mach's principle may therefore have a future – but not without the quantum theory.

    Abraham Pais, in Subtle is the Lord: the Science and the Life of Albert Einstein (Oxford University Press, 2005), pp. 287–288.”

    To illustrate it from my point of view:

    A little thought experiment: A scientist is in a closed space capsule with no windows. The capsule is set spinning. The scientist has a gyroscope and can use his thrusters to stop the spacecraft from spinning. Another closed spacecraft at a distance (the other side of the solar system or the other side of the galaxy) is doing the same thing. Both scientists do their work and declare themselves not rotating, open the door, lean out and wave to each other. They notice that the capsules are not rotating in relation to each other. How did they do this? They used no external clues, all they used was the gyroscope to determine their state of rotation. They both locked onto the same thing to stop the rotation - inertia. They also notice that the stars aren’t moving.

    So what do you have: Inertial forces are locked to the stars. This means that inertial forces cannot reside within an object but must be an interaction between mass and the distant stars.

    In fact it is not possible to rotate an object in relation to the stars without developing inertial of centrifugal forces. Also a spinning gyroscope’s axis will always point to a fixed place in the sky unless there is pressure on the axis to make it precess

    Michaelson and Morley, and others proved that there is no such thing as absolute movement.

    Acceleration, however is absolute. This is why the axis of the gyroscope points to a fixed point in the firmament. Linear acceleration is also absolute. If an object develops inertial forces it will be found that it is accelerating in relation to the stars and the forces developed are not in relation to local objects or a reference frame that we have arbitrarily defined. (let’s leave out acceleration due to gravity for now)

    As was stated in the quote at the top “It must be said that, as far as I can see, to this day Mach's principle has not brought physics decisively farther.” This may be correct but it doesn’t mean that Mach’s principle isn’t true. Inertial forces are tied to the stars, hence rotation is absolute and we must figure this out. It is not enough to ignore facts because they don’t fit our favourite theory. I even saw somebody say that Mach’s principle was passè. How can a fact of physics be passè.
  2. jcsd
  3. Jun 6, 2017 #2


    Staff: Mentor

    That depends on what your definition is of Mach's Principle. Some physicists seem to think GR incorporates it just fine; for example, John Wheeler, one of the key figures in GR (the "W" in MTW, the classic GR textbook), co-authored a textbook called "Gravitation and Inertia", which is almost entirely about how GR incorporates Mach's Principle.

    You left out one key assumption: that spacetime itself is flat. In a curved spacetime this experiment might not actually produce the result you describe: two scientists in closed capsules, widely separated in space, who both use local gyroscopes to determine that their capsules are not rotating, and then open a door (I would say a window since we don't want air to leak out :wink:) and look at each other, will not necessarily find that they are not rotating relative to each other. In other words, in a curved spacetime, "not rotating relative to local gyroscopes" is not necessarily the same as "not rotating relative to the distant stars".

    In fact this kind of thought experiment is one of the reasons why Wheeler says GR does incorporate Mach's Principle--because Mach's Principle, in GR terms, says that the geometry of spacetime, which is what determines how gyroscopes behave, is not fixed but dynamically depends on the distribution of matter and energy in the universe. The distant stars contribute to that, but so do other things. For example, if your gyroscopes are orbiting the Earth, the Earth's mass and spin will affect their behavior (look up "de Sitter precession" and "Lense-Thirring precession"--the latter is what was tested for and confirmed by Gravity Probe B) so that they precess relative to the distant stars--in other words, if you are riding along with the gyroscopes in orbit about the Earth, and use the gyroscopes to make sure your spaceship is not rotating, and then look out at the distant stars, you will find that you are rotating relative to them--not just in the sense that you are orbiting the Earth, but in addition to that: when you have completed exactly one orbit around the Earth, the distant stars will not have returned to the same positions in your sky as before.

    Not necessarily. See above. What they are "locked" to is the overall distribution of matter and energy; but, as above, the distant stars are not the only things that contribute to that.

    Not necessarily. See above. Note also that even in flat spacetime, if we take a gyroscope and put it in circular motion, applying force only at its center of mass (zero torque), the gyroscope will precess relative to the distant stars (i.e., after exactly one orbit around the circle, the stars will not be in the same direction relative to the gyroscope). This is called Thomas precession.

    More precisely, proper acceleration--what an accelerometer measures--is absolute. But proper acceleration and rotation are different things. See below.

    I'm not sure I understand your reasoning here (also, the statement as you make it is incorrect, see above). It might be helpful to draw a careful distinction between two kinds of "rotation"--the rotation of the gyroscope itself, as the thing that keeps its axis "fixed" and makes it a good reference to use for a "direction in space", and the "rotation" of an object moving in a circular path about something else. The latter might or might not be associated with any proper acceleration; in the case of Thomas precession of a gyroscope moving in a circle in flat spacetime, it is, but in the cases of de Sitter precession and Lense-Thirring precession of a gyroscope in a free-fall orbit around a planet, it isn't (the orbit is free-fall, with zero proper acceleration). But the gyroscope's axis serves as a reference just as well in both cases.

    Yes, this is true, but not for the reason you give. Linear acceleration is proper acceleration, and proper acceleration is absolute because, as above, it is a direct local observable--just use an accelerometer.

    This is not necessarily true either. A rocket ship that is "hovering" above a planet like the Earth, at a fixed altitude and fixed spatial position, might not be moving at all relative to the distant stars (that depends on how the planet itself is moving, and we can certainly imagine a planet that is at rest relative to the distant stars). But it will still have nonzero proper acceleration. In fact, by the equivalence principle, the crew has no way to tell, just from measurements made inside the rocket, whether they are in fact hovering motionless above a planet or accelerating linearly in free space with no gravity present. The only way they can tell is to look out the window at distant objects.
    This is not correct. See above.

    This is a good idea, because "acceleration due to gravity" is not proper acceleration--objects moving solely under gravity are in free fall. So it's best to ignore any kind of "acceleration" other than proper acceleration.

    This is not correct as you state it. There are a lot of issues involved here and you have overlooked some and appear to be mistaken about others. See above.
  4. Jun 6, 2017 #3
    Thank you Peter, a lot to think about here. I am really impressed by the breadth and speed of your answer.

    I'll get back with further thoughts soon.

  5. Jun 7, 2017 #4


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  6. Jun 8, 2017 #5
    Thank you Chronos for the links. Very illuminating (the bits in English I could understand) and encouraging.

  7. Jun 8, 2017 #6
    Under Barbour's precise definitions of Mach's Principle, linked below, GR is "weakly" Machian (definition 2) in that it is slightly less predictive than required by the strong version.

    The definition of Mach's Principle
    Found. Phys. 40 1263-1284 (2010)
  8. Jun 13, 2017 #7

    Thanks I understand most of your criticism and the reason why the gyroscope might not be perfectly stable. This comes from the twists and the dents in the underlying space – or spacetime if you prefer.

    Let’s just say that the preponderance of inertial forces come from distant matter. Otherwise you would have to say that there is no connection between the fact that centrifugal forces appear as objects rotate in relation to the stars. Or just say that any of the observations and effects that make up Mach’s Principle are fictitious and not observable.

    I also understand the principle of equivalence although I don’t like it very much. The fields created by linear acceleration and by a gravitational object are drastically different as you know. The fact that it may not be possible to differentiate in a small lab is not very significant – increase the sensitivity of the testing equipment and you could, in theory, always tell which kind of field you were in. I do understand that both forces – or both examples of free fall you mentioned – come from your Spacetime – in one case a linear acceleration and in the other a local dent.

    You also say that some physicists believe that Mach’s principle is included “just fine”. Again from Wikipedia on Mach’s principle:
    “There have been other attempts to formulate a theory which is more fully Machian, such as the Brans–Dicke theory and the Hoyle–Narlikar theory of gravity, but most physicists argue that none have been fully successful. At an exit poll of experts, held in Tübingen in 1993, when asked the question, 'Is general relativity perfectly Machian?', 3 respondents replied 'yes' and 22 replied 'no'. To the question, 'Is general relativity with appropriate boundary conditions of closure of some kind very Machian?' the result was 14 'yes' and 7 'no'.”

    Why after 100 years of GR is there still so much discussion and disagreement on its application?

    The most difficult thing I found was trying to use “quotes” as you did in your response to me. I managed to get your response up as a quote but for the life of me I couldn’t figure out way to separate my response from the quote.

    All the best
  9. Jun 13, 2017 #8


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    Just make sure that your response starts after the end of quote tag. I used magic moderator powers to edit your post to fix that.
  10. Jun 13, 2017 #9


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    I would not quite put it this way. First of all, Thomas precession happens in flat spacetime, which has no "twists and dents". Second, the relationship between the way the gyroscope is pointing and the direction of some distant object like a star depends on the gyroscope's path through spacetime, not just on the geometry of spacetime. But the geometry of spacetime certainly is a factor, yes.

    Only over a large enough extent of spacetime. Over a small extent of spacetime they are only a little different--and the difference gets smaller as the extent of spacetime over which you are looking gets smaller. So it's not always "drastic".

    There isn't any disagreement on the "application" of GR. Everyone agrees on how to use the theory to make predictions that can be compared with experiment. For example, everyone agrees that the distribution of matter and energy in a spacetime determines its geometry via the Einstein Field Equation, and that in turn determines what "inertial forces" are present at a given point in spacetime. If you hand a bunch of relativity physicists with different opinions on Mach's Principle the same distribution of matter and energy and ask them to predict what the inertial forces will be in the resulting spacetime, they will all give the same answer.

    The question about Mach's Principle is not an "application", it's more of a "philosophical" question about whether the theory has a certain property, the definition of which not everyone agrees on. For example, not everyone agrees on whether the fact I described just now, that the distribution of matter and energy in a spacetime determines its geometry and hence the inertial forces, means that GR fully incorporates Mach's Principle--even though, as above, they all give the same answer about the actual experimental prediction, what the inertial forces are.
  11. Jun 13, 2017 #10
    Got it, Thanks
  12. Jun 13, 2017 #11

    I found this definition of Mach's Principle very difficult to follow. Nowhere near as clear in writing as the paper linked by Chronos (http://articles.adsabs.harvard.edu/...=2&data_type=GIF&type=SCREEN_VIEW&classic=YES) by Sciama.

    Thanks all the same
  13. Jun 13, 2017 #12


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    Interesting discussion. Perhaps this is of interest:

    Max Jammer in "Concepts of Mass", 2000, page 150:

    It could be shown that a particle in an otherwise empty universe can possess inertia or that the first Machian effect is not at all a truly physical effect but can be eliminated by an appropriate choice of a coordinate system. Einstein's confidence in the principle gradually waned, so much that eventually, a year before his death, he declared that "one should no longer speak at all of Mach's principle."
  14. Jun 18, 2017 #13


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    What you cited was not disagreement on GR, it was disagreement on what "Machian" means. Mach's principle is very appealing in broad terms, but very difficult to express in a testable form. The best attempt to date, Brans Dickie gravity, seems to indicate that the universe is no more Machian than GR.
  15. Jun 19, 2017 #14


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    @Peter, the fact that topology and even global geometry may depend on boundary conditions is a limitation on how Machian GR is, true? As an extreme, eternal BH solutions and Minkowski space all have no matter or energy at all in them, but they have completely different inertial structures.
  16. Jun 19, 2017 #15


    Staff: Mentor

    For some (perhaps most) meanings of "Machian", yes.

    Yes, this is a good example of how the Einstein Field Equation alone is not sufficient to determine a global geometry, mathematically speaking.

    In terms of physically realistic solutions, though, this issue doesn't really arise. Any physically realistic solution for, e.g., a black hole will have a region of nonzero stress-energy somewhere, because some object with stress-energy in it will have collapsed to form the hole. Even the asymptotically flat boundary condition can be replaced by a smooth merging of the black hole solution into something like an FRW background spacetime describing the universe in which the hole exists. And the FRW spacetime describing the universe as a whole doesn't need a boundary condition--unless you call the fact that our current best-fit solution is spatially infinite a "boundary condition". (Which might be a tenable position--IIRC Einstein said that for a GR solution describing the universe to be truly Machian, the universe would have to be spatially closed.)
  17. Jun 19, 2017 #16


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    What definition of Mach's principle makes GR Machian? I thought that Mach's principle was the claim that acceleration is relative to other matter, so there could be no difference between (1) a bucket of water rotating clockwise and (2) the rest of the universe rotating counterclockwise. But in GR, even in an empty universe, there is a notion of geodesics, so acceleration doesn't have to be relative to anything.
  18. Jun 19, 2017 #17


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    The one that, roughly speaking, says that the spacetime geometry, and therefore the inertial properties of worldlines (which ones are geodesics, and what proper acceleration is associated with non-geodesic worldlines), are dynamical, i.e., determined by the distribution of matter and energy in the universe. (There are some possible caveats even for this definition, which PAllen and I have had an exchange about in this thread.)
  19. Jun 20, 2017 #18
    So, if Wheeler is right, the result of another thought experiment: an unfolding Newton's bucket in a completely flat universe, is that water in it does not rise. Rotation, even rotation relatively to the unfolding rope, should be equivalent to rest. Inertia would be completely absent. Am I right?
  20. Jun 20, 2017 #19


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    I don't think so. As far as I can tell today most physicists would agree that proper acceleration is a local phenomenon, thus the water in the bucket would rise. See also the statement of Max Jammer in post #12.
  21. Jun 20, 2017 #20
    Since Max Jammer and late Einstein I get the idea that absolute rotation in an empty space (that is relative to nothing) does make sense. Am I still wrong?
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