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History of theories of motion; the role of inertia

  1. Sep 22, 2009 #1

    Cleonis

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    In physics there has been a sequence of three theories of motion, each superseding its predecessor: newtonian dynamics, SR and GR. Insights from GR cast the predecessors in new light. With the new insights the predecessors can be reframed in such a way that the transitions between them become the smallest possible.

    'Theory of motion' and 'theory of inertia' are effectively one and the same concept. To formulate the properties of inertia is to formulate the properties of motion.
    Among the properties of inertia:
    - The law of inertial motion: objects in motion will remain in the same state of motion, moving in a straight line and covering equal distances in equal intervals of time.
    - The law of forced acceleration: to change the state of motion a force is required; The responce to impressed force is proportional: twice the force gives twice the acceleration. (Better known as Newton's second law: F=ma)

    An explanation of the law of inertial motion would require a theory of the nature of space and time that probes deeper than current theories do. (But some theorists do attempt to formulate a quantum theory of space and time itself.)
    In the case of Newton's second law there is no explanation (as far as I know), nor are there any leads. Quite a few physicists will argue that we should not seek more fundamental explanation in the first place, but that we should simply accept newton's second law as given.

    Relativistic spacetime

    As we know GR is not just a theory of motion, GR unifies the description of inertia and gravitation into a single conceptual framework. We have that physical properties are attributed to GR-spacetime.
    John Wheeler coined the following phrase to capture the essence of GR. (I'm not quoting literally.)
    "Inertial mass is telling spacetime how to curve, curvature of spacetime is telling inertial mass how to move."

    SR is subsumed in GR, and by implication SR-spacetime has the same properties as GR-spacetime, except for the property of being "deformable". SR-spacetime is "immutable" in the sense that its morphology is unchanging.

    We have that SR-spacetime is telling matter how to move. That is, in SR inertia arises from SR-spacetime. (I know that some people will argue for a more cautious attitude. Some people will argue that since we don't know what inertia is we should only acknowledge the existence of inertia, without attributing it somewhere.)

    Classical spacetime

    With the insights gained from relativistic physics classical physics can be reinterpreted. We can define a background structure of classical dynamics, and an often used name for that background structure is 'galilean spacetime' (since in galilean spactime the applicable transformations are the galilean transformations.) Then inertia as known in classical dynamics is to be attributed to galilean spacetime.

    So, before the concept of spacetime as a physical entity, giving rise to inertia, was developed, how did physicists think about inertia?
    That is difficult to say. To this day many authors write about inertia as an innate property of objects, without any reference to some outside structure. It's very common for authors to use phrasings such as: "As the wrecking ball hits the wall of the building the ball's momentum carries it through." Inertia is rarely discussed, but when authors do discuss it the suggestion is that the inertia of an object is purely an internal affair, something purely innate to an object.

    In my opinion the attitude of regarding inertia as an innate property of individual objects is untenable. That is one of the lessons of relativistic physics.

    Cleonis
     
    Last edited: Sep 22, 2009
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  3. Sep 22, 2009 #2

    On an historical note here is a passage from Torretti, Relativity and Geometry, describing Newton's attitude to inertia.

    Newton neatly describes two kinds of force. It is ""either an external principle which, when impressed in a body, generates or destroys or otherwise changes its motion, or an internal principle, by which the motion or rest imparted to a body are conserved and by which any being endevours to persevere in its state and resist hindrance"". Innate force is said to be proportional to the 'quantity' of matter----and is said to differ from the natural 'laziness' (inertia) of matter in our conception only. The innate force must therefore be regarded as an indestructible property of the matter endowed with it. Impressed force, on the other hand, is an action only, and does not remain in the body after the action is over. ""For a body preserves in every new state by its force of inertia alone""

    Matheinste.
     
  4. Sep 22, 2009 #3

    Cleonis

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    Yes, in the Principia Newton wrote about inertia as an innate property of objects. I think this point of view has had a lasting influence; I think it persists to this day.

    But it's hard to see how Newton could have proceeded otherwise. Hypothetically, what if Newton in his own mind would have anticipated the idea of attributing inertia to physical properties of space? What could he possibly have done to convey that idea to his contemporaries?

    Cleonis
     
  5. Sep 22, 2009 #4
    Hello cleonis,

    Despite the title of this thread it seems it is going to be aimed at explaining inertia by some property of the aether.

    Matheinste.
     
  6. Sep 23, 2009 #5

    Jonathan Scott

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    For an important insight into inertia, see Dennis Sciama's paper from the 1950s: http://adsabs.harvard.edu/abs/1953MNRAS.113...34S". This paper shows that if Newtonian gravity theory (as an approximation) is extended to the scale of the universe, then inertia appears as a consequence of relatively accelerated motion, and so do the coriolis and centripetal forces from relatively rotated motion, in a way which is fully consistent with Mach's principle.

    Unfortunately, GR does not seem to be consistent or compatible with this beautifully neat model, although it shows signs of it in frame-dragging effects. I think this is just one of several signs that GR isn't completely correct on the large scale.
     
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  7. Sep 23, 2009 #6

    PeterDonis

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    The paper looks interesting; I haven't had a chance to read it in detail, but I have read the abstract. I'm not sure all relativists would agree that GR is not consistent or compatible with this type of model.

    For example, in Cuifolini and Wheeler's _Gravitation and Inertia_, they appear to be taking the opposite viewpoint: that, since the metric in our local region of spacetime is ultimately a consequence of the metric of the universe as a whole, which is a consequence of solving the Einstein Field Equation with the RHS (the stress-energy tensor) determined by the total mass-energy in the universe, the inertial effects we observe locally *are* ultimately due to all of the mass-energy in the universe.

    Frame-dragging effects, on this view, are a manifestation of the (small) local modification that a nearby rotating body (such as the Earth) makes to the overall effect of all the rest of the mass-energy in the universe. Indeed, on this view, the overall gravitational field of the Earth (and the Sun, and other massive bodies), including the part that makes objects fall, is a local modification to the global metric in a particular region of spacetime, due to the fact that locally, the mass-energy is not uniformly distributed (as the large-scale solution of the EFE that determines the global metric assumes), but lumpy.
     
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  8. Sep 23, 2009 #7

    Cleonis

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    Invoking the concept of aether is not an option, that is the point of view that I subscribe to.

    I regard it as key to the concept of aether that it is assumed that objects have a velocity to it, as illustrated by the reasons for setting up the Michelson-Morley experiment: the intention was to find the velocity of the Earth with respect to the aether.

    As we know velocity with respect to SR-spacetime does not enter SR as a matter of principle, that was the reason for formulating SR in the first place. So the aether concept is out of the question.


    I am really curious whether at any point in the future a theory will be developed that explains inertia on a deeper level. I don't expect such a theory to re-introduce an aether concept.

    Cleonis
     
  9. Sep 24, 2009 #8

    Jonathan Scott

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    Also see MTW section 21.12. I feel that in both cases, this arguments only show that GR is weakly compatible with Mach's principle. Even though it is not possible to show exact solutions on this scale in GR, it is not difficult to see that if the "sum for inertia" principle is to work correctly, then either G cannot be a constant, or the distribution of masses in the universe has to satisfy unrealistically contrived conditions. In contrast, in Sciama's model (and in other theories such as Brans-Dicke), G is directly determined by the distribution of masses in the universe.
     
  10. Sep 24, 2009 #9

    PeterDonis

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    This paper refers to a second paper ("paper II") that was supposed to re-do this type of analysis, but using a tensor theory of gravity instead of the scalar/vector potential theory used in this paper (which Sciama admitted was only a first-order approximation). I can't find any reference anywhere to that second paper. Does anyone know if it was ever published?
     
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  11. Sep 24, 2009 #10

    PeterDonis

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    As I read MTW, they are saying that the "sum for inertia" is only a heuristic to show the general way in which things work out; it's not supposed to be a precise calculation. The precise calculation is the solution of the initial value problem, which generates the entire 4-geometry (spacetime), and therefore specifies the inertial properties of all test bodies, from a specification of the 3-geometry and its "rate of change" (speaking loosely) on a spacelike hypersurface. That solution doesn't require G to vary, or masses to be arranged in any special way--obviously, changing the arrangement of masses on the initial hypersurface will change the full solution, but each such solution will still be valid. (It is true, though, that the specifications on the initial hypersurface have to satisfy constraint equations, so they're not completely "free".)

    MTW do seem to say that, in order for GR to properly realize Mach's Principle, the universe must be spatially closed. Since we're not sure that's true, that would be one possible flaw.
     
  12. Sep 24, 2009 #11

    Jonathan Scott

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    I don't think so, and the author of this paper seems quite sure about it: http://www.phil.uga.edu/faculty/balashov/papers/laws.pdf" [Broken]. He also hypothesises that this might have been because at the time Sciama's paper was written, the suggested relationship between G, the mass and the size of the universe was a good match, but subsequent observations (which eventually led to the "dark matter" hypothesis) meant that the match became less significant.

    Sciama did write several text books, which I haven't personally read, which apparently continue the discussion of Mach's principle and inertia.
     
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  13. Sep 24, 2009 #12

    Jonathan Scott

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    I don't find MTW convincing here.

    Mach's principle, as embodied in Sciama's model, says that if you rotate a test body, the centripetal and coriolis forces which it experiences are precisely caused by the sum of the frame-dragging effects of every body in the universe. Similarly for linear motion, inertia is due to the sum of the linear frame-dragging effects. We don't know exactly how to sum them, but if you then move somewhere else and do it, you must surely get a different sum (unless G varies in a way which compensates for the different mass distribution, as in Sciama's model), yet we know that the effects are exactly the same.

    I don't see how the "initial value" calculation helps here with inertia or rotation; we have accurate local solutions for how bodies move based on GR which simply assume asymptotic flatness outside that region. As far as I know, standard GR suggests the rotational and inertial effects would still be the same even if the local system of masses was in an otherwise empty universe; this is definitely not compatible with Mach's principle.
     
    Last edited: Sep 24, 2009
  14. Sep 24, 2009 #13

    PeterDonis

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    But in our actual universe, the overall mass distribution--the average that appears in the FRW solutions--doesn't change: it's the same everywhere and in all directions (the universe is homogeneous and isotropic). So the sum due to the overall mass distribution wouldn't vary; the only variations would be due to local lumpiness, which requires local solutions (see next item).

    That's where MTW bring in the condition that the universe must be closed; if it is, then the boundary condition of asymptotic flatness is really the condition that your local solution (say the Schwarzschild solution for the solar system) is on a small enough patch of the global spacetime that it looks locally flat, even though spacetime as a whole is curved (just like my local patch of the Earth's surface looks locally flat, even though the surface is globally curved).
     
  15. Sep 24, 2009 #14

    Jonathan Scott

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    It's not just "roughly" the same everywhere; as far as we know, rotational and linear inertia are not only exactly the same everywhere, but also match one another, despite the fact that they are due to different aspects of the distribution of matter and motion. Given the complexity of calculating the frame-dragging effect of relative motion and rotation in the simplest situation involving local masses, is this really plausible that for the whole universe the GR frame-dragging effects all add up exactly everywhere? With a Machian theory where G is due to that same distribution, this is trivial, but with GR, I don't believe it's possible, and I've read that even Einstein himself became convinced at an early stage that GR was not fully compatible with Mach's Principle (for example because it allowed De Sitter's solution).

    I can't rule out the possibility that GR could perhaps be combined with some other additional physical rules to describe the actual universe we inhabit in a way which might be compatible with Mach's principle, in a way similar to that described in MTW, but I'd bet against it.
     
  16. Sep 24, 2009 #15

    PeterDonis

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    Rotational and linear inertia are the same everywhere, but the metric is not--in other words, what states of motion are inertial states of motion does vary from place to place. It's only the metric that GR attributes to the effects of the matter in the universe. The difference between GR and a theory like that in Sciama's paper is not that the latter predicts a different inertial response than GR does; it's that the latter claims to *derive* that response from other principles, whereas GR just adopts it as a principle.

    I'm not sure I understand; in a Machian theory where G is due to the matter distribution, if that distribution isn't exactly the same everywhere, shouldn't G vary from place to place? If I'm close to a large mass, such as the Earth, shouldn't I measure G to be different than it is out in interplanetary space? More importantly, shouldn't I measure bodies' responses to inertial forces to be different out in interplanetary space than I do close to a large body?
     
  17. Sep 25, 2009 #16

    Jonathan Scott

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    Yes, that's correct.

    However, the first-order variation in G in a simple Machian theory is simply equivalent to the relative Newtonian potential, so the G which appears in Newton's theory and GR corresponds to the Machian value due to everything in the universe EXCEPT the local masses, which is effectively a constant in any system dominated by a large central mass.

    Also, since this form of G is dominated by the more distant masses, it is very nearly the same everywhere in the observable universe.

    If G varies in this Machian way, it should be theoretically possible to detect its variation with location in extremely sensitive laboratory experiments. Perhaps the well-known difficulties in establishing an exact value for G might relate to this, but at present the experimental precision is much lower than the sort of variation which Machian effects would involve. I also suspect that there may well be a MOND-like component in gravitational forces proportional to sqrt(m)/r which manifests even locally and could possibly be detected in laboratory experiments, in which case that would lead to significant variations in measurements. I'm aware of experiments to attempt to detect dependency of the gravitational force on higher powers of 1/r than the usual inverse square, but not of any attempt to rule out any 1/r component of the order of the MOND effect.

    A Machian expression for G would probably also vary with time, but it's not clear in what way, as it is not clear how the effective total energy of the universe varies with time. There are experimental limits on how the Newtonian value of G has varied in the region of the solar system, and these can be used to rule out certain cases, but even that is complicated by the difference between the meaning of the Machian and Newtonian G values.
     
  18. Sep 25, 2009 #17

    Jonathan Scott

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    Sorry, forgot to address this point. It's gravity which varies, not inertia. Inertial effects as measured locally are the same everywhere. They can be considered to be due to rotational and linear frame-dragging effects of Machian gravity which effectively add to exactly 1 when the variation of G (or more generally a scalar/tensor equivalent) is taken into account.

    Of course, inertial effects as seen from a distance vary with potential as usual (for example, a mass at rest in a low potential has less total energy and hence less inertia than one at rest in a high potential).
     
  19. Sep 25, 2009 #18
    Hi I have question. Assume the solar system and immediate local stars were duplicated outside the galaxy in unpopulated space. Would you expect that local measurements would be different than those we observe now???
    In my simplistic view of Machs principle, it is a top down metric with local values as fluctuations of a universal base. If this is the case, it could mean that against this scale, the isolated earth measurements and our own could actually be different but how would that possibly be determined if locally they appeared the same?? Doppler shift??





    Is it not possible that within this paradigm, that time itself could vary with time.
    If you consider time dilation as a function of gravity and dependant on overall distribution of matter, then the expansion could mean a universal base time would be speeding up even as local condensation might produce dilated regions relative to this base?????
    Or is this too crazy?? :-}
     
  20. Sep 25, 2009 #19

    Jonathan Scott

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    The local galaxy still has only has a tiny effect compared with the rest of the matter in the universe. The relative effect is of the order of GM/Rc2 where M is the mass of the galaxy and R is the distance from the centre.

    Of course, in a Machian theory, if you could move the solar system to an otherwise empty universe, then gravity would be very different.

    Within GR (or any other potentially viable relativistic theory) the local time rate at any point in the universe varies with the local gravitational potential, but even within clusters of galaxies the difference from other areas is still tiny. The only thing which makes a big difference to the time rate is being very close to an extremely massive object.
     
  21. Sep 25, 2009 #20

    PeterDonis

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    So basically, what GR attributes to a local variation in the metric due to a nearby massive body, a Machian theory of the type you're describing attributes to a local variation in G due to that massive body.

    Wouldn't the constant of proportionality for a term of this sort have to be extremely small to be consistent with known solar system dynamics? And, for that matter, the binary pulsar data, which is an even stronger test of GR? Since a 1/r term acts like a "radiative" term, it would seem that if it were present, it would affect the "spin-down" rate of the binary pulsar, by effectively increasing the rate of emission of gravitational radiation.
     
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