History of theories of motion; the role of inertia

[Cleonis]

if I wanted to quickly summarize the difference between galilean relativity and SR, I would simply say that galilean relativity uses galilean spacetime, whereas SR uses Minkowski spacetime. [...] Galilean spacetime, if we lived in it, would have physical effects too: after all, objects are predicted to experience inertial forces (e.g., centrifugal force) in pre-SR physics.

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

 

[matheinste]

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)

Cleonis

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.

 

[Cleonis]

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.

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

 

[matheinste]

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.

 

[Jonathan Scott]

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.

 

[PeterDonis]

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.

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.

 

[Cleonis]

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.

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

 

[Jonathan Scott]

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.

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.

 

[PeterDonis]

For an important insight into inertia, see Dennis Sciama's paper from the 1950s: http://adsabs.harvard.edu/abs/1953MNRAS.113...34S".

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?

 

[PeterDonis]

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.

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.

 

[Jonathan Scott]

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?

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" . 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 textbooks, which I haven't personally read, which apparently continue the discussion of Mach's principle and inertia.

 

[Jonathan Scott]

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.

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.

 

[PeterDonis]

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.

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).

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.

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).

 

[Jonathan Scott]

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).

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.

 

[PeterDonis]

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?

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.

With a Machian theory where G is due to that same distribution, this is trivial.

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?

 

[Jonathan Scott]

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?

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.

 

[Jonathan Scott]

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?

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).

 

[Austin0]

=Jonathan Scott;2364517]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.

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??

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

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?? :-}

 

[Jonathan Scott]

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?

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.

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?? :-}

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.

 

[PeterDonis]

...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.

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.

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.

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.

 

[Jonathan Scott]

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.

A MOND-like gravitational force would appear to be nearly ruled out for the solar system, as effects of the same order as MOND have been suggested as explanations for the Pioneer anomaly, but do not appear to fit. However, I don't think that the models assume it applies everywhere, but only in the MOND low-acceleration zone, so this might not be conclusive.

It should also be possible to rule it out in the laboratory, but it appears that every lab experiment to try to measure G does so by maximizing the mass and minimizing the distance (of course) which means that the ordinary Newtonian force is orders of magnitude stronger than this MOND-like sqrt(m)/r force.

Anyway, this MOND-like force I mentioned is just my personal speculative idea anyway, and is not central to the topic of this thread. It arises from considerations of "solid angle deficit" if you investigate the idea that mass is what closes the universe. The actual result is an acceleration of c²/r sqrt(2m/M) where m is the local mass and M is a measure of the total mass of the universe, so it happens to match MOND if c²/sqrt(M) is equal to the MOND acceleration parameter.

 

[Cleonis]

Anyway, this MOND-like force I mentioned is just my personal speculative idea anyway, and is not central to the topic of this thread.

In fact, the original topic of this thread did not involve Mach's principle.

The subject of what I posted was the peculiar attitude towards inertia, particularly in classical mechanics. The suggestion of inertia as a purely innate property of objects is a very weird concept.

I would like to divide questions about inertia into two categories:
- How inertia is distributed in the Universe
- The nature of inertia as local manifestation.

As to the second question, about the origin of F=m*a: there are as far as I know not even clues where a deeper theory is to be looked for, if it exists at all.

Inertia is stunningly linear. If you double the force the acceleration doubles, there doesn't seem to be any saturation effect. Just how can a physical effect be absolutely linear?

Cleonis

 

[jimgram]

If you think for a moment about the need for “motion” in order to understand observable phenomena, you may see that it isn’t necessary. In order for matter to exist, it must resist a change in its position. Begin by understanding that all matter (if it meets the human criteria for “existing”) must resist a change in its position – otherwise, it wouldn’t exist, or at least, could not be detected. Also, try believing that matter can only change its position by collision with other matter. In this world view, there cannot be a reference frame, so by position, what is meant is a physical location and with no reference frame it must also mean a location in time. So, for any two pieces of matter, they may exhibit a great amount of “motion” between them – however, of course, they are both at “rest” as far as they are concerned. When taken collectively, most objects move dramatically relative to each other, but since there is no reference frame, each object is happily at rest and locked in its own position in the universe. Inertia (and mass, being synonymous) is simply a term used to describe the tendency of matter to resist a change in its position.

This is what position is. It is a place in the universe and laws of motion can only be used to relate its position changing relative to some other position.

1. Everything has a fixed position and resists a change in that position.

2. A unit of resistance to change in position = inertia or mass

3. A change in position can only occur when two things collide. Let’s call thing #1 ‘mass1’ and thing #2 ‘mass2’.

4. When the position of mass1 is described relative to the position of a second mass2, then the relative position between the two things is described as motion (change of distance between the two things might equal zero, and therefore the distance physical coordinates are different but ‘time’ between the two is zero).

5. Acceleration is a mathematical relationship that describes the effect of collision, which is the only means that any mass can alter its position, and only by equally and oppositely altering the position of the mass it collides with.

6. If: position of mass1 minus position of mass2 = s (distance), then velocity is s as a function of time: v = ds/dt.

8. Acceleration is v as a function of time: a = dv/dt.

9. So, the position of mass1 minus the position of mass2 = a * t2.

10. Mankind has developed a concept of ‘force’ to deal with the observation of collisions. Since a thing has mass (resistance to change in position) and only a collision with another mass can overcome this resistance and can cause a change, what is really meant by the term ‘force’ is mass1 in a collision.

11. The motion equations then lead to mass1/mass2 = a or v/t.

12. Substituting the term ‘force’ for mass1, we arrive at a = F/mass2, or F = m*a.

Try to imagine a force caused by other than a collision. Of course, this theory leads directly to the notion of an aether.

 

[PeterDonis]

Anyway, this MOND-like force I mentioned is just my personal speculative idea anyway, and is not central to the topic of this thread.

Ok, so to get back to the central topic: would the following be a fair summary of how GR and Machian theories of the type you've mentioned deal with inertia?

(1) The "thing to be explained" is that bodies in certain states of motion feel no force (they are "freely falling"), while bodies in other states of motion do feel a force. In the latter case, the force felt is exactly proportional to the object's mass-energy (i.e., the equivalence principle holds).

(2) GR explains the above as follows: all of the mass-energy in the universe plays a role in determining the metric in any given region of spacetime. The local mass-energy influences the metric directly, through the local solution of the EFE; distant mass-energy influences the metric indirectly, because its effects on the metric on the past light-cone of the region under consideration propagate to that region. The latter can also be viewed as the global average mass-energy distribution of the universe determining the global solution of the EFE that governs its evolution, including the global average metric; local lumpiness in mass-energy then determines local corrections to the global average metric (equivalently, the global average metric determines the appropriate boundary conditions--e.g., asymptotic flatness--for the local solution).

Once the metric in a given region of spacetime is determined, in GR that automatically determines which states of motion are freely falling and which are not. The freely falling worldlines are geodesics; the "accelerated" worldlines (on which observers feel a force) are non-geodesics. The equivalence principle has to hold because the metric is just geometry; it doesn't "care" what kind of stuff is moving on a given worldline, it only "cares" what kind of worldline it is. (Or, viewed the other way, the fact that the equivalence principle holds is what justifies treating "gravity" as geometry, according to GR.)

Newton's gravitational constant, G, comes into play in GR as the constant of proportionality between mass-energy density and spacetime curvature; i.e., it tells you how much curvature is produced by a given amount of mass-energy. It is assumed to be constant in GR.

(3) A Machian theory explains the above as follows: any given test body (small lump of mass-energy) interacts gravitationally with all the other mass-energy in the universe. It is assumed that this gravitational interaction can be modeled as some kind of field on Minkowski spacetime. The equivalence principle arises automatically from the fact that the interaction is gravitational, since that makes it automatically proportional to the test body's mass-energy. (Note, however, that there would still need to be some kind of coupling constant for this field involved--see below.)

In the simplest case, where the test body is at rest relative to the average mass-energy in the universe, and there are no other bodies nearby, the test body feels no force because the forces exerted by all the individual pieces of distant matter cancel. By the principle of relativity, the same should be true for a body moving with a constant velocity v relative to the average mass-energy in the universe. (In Sciama's paper, he derives this result to order v/c by showing that the gravitoelectric field is proportional to dv/dt, and the gravitomagnetic field is zero, if there are no irregularities in the average mass-energy of the universe. Of course this was for a scalar-vector potential theory of gravity, not a tensor theory.)

The presence of a nearby massive body changes the response of the test particle by changing the effective field acting on it; because of the nearby body, the test particle now feels no force if it is accelerating towards the massive body (because that is the state of motion in which all the forces on it--the force from the nearby body, plus the force from the rest of the universe--balance out and the net force is zero). The difference between the new state of motion that is now inertial for the test particle, with the massive body present, and the old state of motion that was inertial with the massive body absent, is the basis for what we measure as Newton's gravitational constant, G--basically, we equate the celeration of the test particle towards the massive body with the Newtonian formula, [itex]\frac{G M}{r^2}[/itex] (at least to lowest order), and solve for G using the known mass M of the massive body and distance r of the test particle from the body. But the value of G we obtain in this way will depend on how much mass-energy is in the rest of the universe and how it is distributed (though for some reasonable distributions the variation in G from place to place will be very small).

(Actually, what we measure as G would be a combination of this effect and the coupling constant that determines the strength of the "gravitational field" as a field on Minkowski spacetime, as mentioned above. The latter constant would presumably be universal, at least at low energy--at high enough energies quantum considerations would come into play.)

 

[PeterDonis]

In fact, the original topic of this thread did not involve Mach's principle.

Well, Mach's principle is certainly relevant, since if it's true then there is no need to view inertia as an innate property of objects; inertia is just the resultant of effects on the object from external sources.

As to the second question, about the origin of F=m*a: there are as far as I know not even clues where a deeper theory is to be looked for, if it exists at all.

If the equivalence principle holds (see my last post for some discussion of how a theory of inertia might entail this), then F = ma is a definition, not a deep question that needs an answer. Acceleration, defined as DU/D[itex]\tau}[/itex], the covariant derivative of 4-velocity with respect to proper time, becomes the more fundamental concept; "force" is simply defined as acceleration times rest mass.

Inertia is stunningly linear. If you double the force the acceleration doubles, there doesn't seem to be any saturation effect. Just how can a physical effect be absolutely linear?

If F = ma is a definition, as above, this is a non-issue.

 

[PeterDonis]

This is what position is. It is a place in the universe and laws of motion can only be used to relate its position changing relative to some other position.

How does all this square with the known facts relating to relativity? Specifically, we know to very high precision that the principle of relativity holds: there is no local way of distinguishing motion at a constant velocity from rest.

 

[Jonathan Scott]

Ok, so to get back to the central topic: would the following be a fair summary of how GR and Machian theories of the type you've mentioned deal with inertia? ...

I'm happy with almost all of your summary apart from one minor point; there's no additional coupling constant involved in the Machian theory (apart from perhaps a small dimensionless numerical constant), as it all depends entirely on ratios!

For the simplest class of such theories (in a scalar approximation) we have:

[tex]
G = \frac{n}{\sum{\left(\frac{m_i}{r_i c^2}\right)}}
[/tex]

where the sum is for all masses in the universe and n depends on the specific theory. This is known as the Whitrow-Randall relation. (The use of c in this form is somewhat over-simplified as the effective value varies with location, but there are ways to fix that by expressing it in energy units instead).

 

[Cleonis]

If the equivalence principle holds, then F = ma is a definition, not a deep question that needs an answer.

There seems to be some miscommunication here.

We can use calibrated force gages to measure the amount of force that is exerted upon objects. Let's say a thruster unit is moved towards ISS (International Space Station) and starts pushing ISS to another course. The amount of force that the thruster unit exerts upon ISS can be measured, and onboard accelerometer will measure the acceleration with respect to the local inertial frame. If the force exerted by the thruster unit on ISS is varied it is found that the amount of measured acceleration is proportional to the amount of impressed force.

That is what I have in mind (when I'm amazed by F=m*a).
It's not clear to me how that proportionality can be seen as a matter of definition, hence my suspicion that some miscommunication is at play.

Cleonis

 

[PeterDonis]

...there's no additional coupling constant involved in the Machian theory (apart from perhaps a small dimensionless numerical constant), as it all depends entirely on ratios!

I think that what I was referring to as the "coupling constant" is what you are referring to as "perhaps a small dimensionless numerical constant", depending on where you think the latter fits in. What I had in mind was that, as I said, the "gravitational force" has to somehow be modeled as a field on Minkowski spacetime, and there should be some coupling constant associated with that field. The value of that coupling constant would affect the numerical value found for G (in the formula you give, I would expect the field coupling constant to affect the value of n).

 

[PeterDonis]

The amount of force that the thruster unit exerts upon ISS can be measured, and onboard accelerometer will measure the acceleration with respect to the local inertial frame. If the force exerted by the thruster unit on ISS is varied it is found that the amount of measured acceleration is proportional to the amount of impressed force.

That is what I have in mind (when I'm amazed by F=m*a).
It's not clear to me how that proportionality can be seen as a matter of definition, hence my suspicion that some miscommunication is at play.

The force measured by the thruster unit is equal to the momentum transferred by it to the ISS per unit proper time, in the MCIF at the event at which the force is applied. That has to be true by conservation of momentum. The acceleration measured by the accelerometer is the change in the ISS's 4-velocity per unit proper time. Since 4-momentum is just 4-velocity times the rest mass of the object (in this case the ISS), the equality you're amazed at *has* to hold. The only caveat is that if the equivalence principle were false, you couldn't set up an MCIF in which, locally, the laws of SR hold (which is what allows you to say that 4-momentum is 4-velocity times rest mass).

 

[Cleonis]

The force measured by the thruster unit is equal to the momentum transferred by it to the ISS per unit proper time,

Naturally I agree that the propulsion from a thruster unit goes back to conservation of momentum: a rocket ejects small amounts of mass at extremely high velocity from its exhaust nozzle, resulting in acceleration of the rocket in the opposite direction.

We can attach that rocket to a larger object, such as ISS, to accelerate it. We can gage the force that is exerted upon ISS independently. For instance, we can use a force gage that works with a coiled spring. Or we can use properties of gas for force calibration. When the volume of a gas is halved its pressure doubles (to a first approximation, more accurate models are available). Bottom line: the way the force gage is calibrated is independent from the operation of the thruster unit.

Assume the thruster unit can vary the nozzle velocity of the exhaust (In other words, it can vary how hard the exhaust is accelerated.) We find that the force that is exerted is proportional to how hard the exhaust is accelerated. We find F=m*a

It's not clear to me how that proportionality could be seen as a matter of definition.

Cleonis

 

[Jonathan Scott]

I think that what I was referring to as the "coupling constant" is what you are referring to as "perhaps a small dimensionless numerical constant", depending on where you think the latter fits in. What I had in mind was that, as I said, the "gravitational force" has to somehow be modeled as a field on Minkowski spacetime, and there should be some coupling constant associated with that field. The value of that coupling constant would affect the numerical value found for G (in the formula you give, I would expect the field coupling constant to affect the value of n).

As a specific example, the value n = 1/2 combined with Marcel Brillouin's radial coordinate R = r - 2Gm/c² (where r is the Schwarzschild radial coordinate) gives a Machian expression for G which is still compatible with GR locally. That is, even if we assume that the global value of G varies in a Machian way, the Schwarzschild solution to the Einstein Field Equations still holds exactly for a single mass with G effectively being a constant. I previously described this result (which I found quite surprising) in more detail in this thread: https://www.physicsforums.com/showthread.php?t=206922"

 

[PeterDonis]

Assume the thruster unit can vary the nozzle velocity of the exhaust (In other words, it can vary how hard the exhaust is accelerated.) We find that the force that is exerted is proportional to how hard the exhaust is accelerated. We find F=m*a

It's not clear to me how that proportionality could be seen as a matter of definition.

In this case you would just do the same momentum balance between the thruster unit itself (more precisely, the nozzle, since that's what the exhaust pushes against) and the exhaust gas. Same result.

Let me try approaching this a different way. If you're surprised that F = ma always seems to hold, you ought to be able to describe a scenario, a way things could be, an alternate set of laws, in which F = ma would not always hold. Can you describe such an alternate set of laws, in which F = ma doesn't always hold, but conservation of energy and momentum still do? In other words, can you describe a way in which it could be possible that the two things I'm saying are connected (F = ma and conservation of momentum) become different?

 

[Cleonis]

Can you describe such an alternate set of laws, in which F = ma doesn't always hold, but conservation of energy and momentum still do?

That's an interesting angle. F=m*a is a linear law. Would any other law be compatible with the principle of relativity of inertial motion? The very essence of the galilean and Lorentz transformations is that they are linear transformations. Intuitively I'd say the principle of relativity of inertial motion rules out any non-linear law, leaving F=m*a as the only possibility.

Tying in F=m*a with the conservation principles is interesting. It's not explanation in terms of a deeper theory, but the interconnection is nice.

Cleonis

 

[Cleonis]

Well, Mach's principle is certainly relevant, since if it's true then there is no need to view inertia as an innate property of objects; inertia is just the resultant of effects on the object from external sources.

Then again, also without Mach's principle there is in itself no need to view inertia as an innate property of objects. As far as I can tell those are independent issues. (But maybe you and I have different things in mind when using the expression 'viewing inertia as an innate property of objects'.)

I'm not accustomed to considering Mach's principle. I know different versions of Mach's principle are in circulation. The idea never appealed to me, and since attempts to implement some version of Mach's principle have remained unsuccesful I saw little reason to give Mach's principle any thought.

In the outline of a Machian theory published by Dennis Sciama Maxwell's equations are used to get the exploration going.

I would like to check some things, to see if I understand them correctly.
- In Sciama's exploration each point mass in the Universe is taken as the source of a field. At each point in the Universe the local field that a test mass experiences is a superposition of all fields in the Universe. I will refer to that superposition of fields as 'the resultant field'.
- When the test mass has a uniform velocity with respect to the local resultant field all effects still cancel out. (Hence the resultant field features relativity of inertial motion.)
- When the test mass is accelerating with respect to the local resultant field there is an effect. To accelerate with respect to the local resultant field a force is required and Maxwell's equations (as implemented in Sciama's exploration) imply that the required force is proportional to the acceleration.

Those are the things I would like to check for correctness.

Cleonis