The Cause of Inertia: Exploring Different Theories and Their Implications

[Athanasius]

I am curious to know the various opinions, what ideas you favor as to the cause of inertia? What could the reference frame(s) be that induces inertial effects, and how could those effects be caused? Could the reference frame be absolute space as Newton thought, or the more relativistic idea of the mass of the distant stars, as Mach thought? Any ideas as to why inertial mass is equal to gravitational mass? Do you believe that this indicates there is a relationship between the two? Lastly, would you say that Einstein incorporated Mach's ideas somewhat when he assumed that the effect of gravitation is equivalent to acceleration, because otherwise, how could acceleration produce an effect similar to gravity were it not for inertia?

Lastly, if Mach was correct that the mass of the universe is the cause of inertia, why is inertia not diminished as the universe expands, since the gravitational pull of the distant galaxies would grow progressively weaker? And if there is a Mach-friendly explanation for why inertia is not diminished as the universe expands, then why wouldn't the infinite mass of an infinite universe result in infinite inertia?

I am a physics layman, so please try to keep the replies to me non-technical, though I am willing to put forth the effort to struggle through more technical answers if need be. However, I will not at all begrudge you replying to each other in technical language if you want to.

 

[Matterwave]

The ideas of Mach, being more philosopher than physicist (due to his era), are problematic to implement in actual physics. This is because his ideas are, from a physicist's standpoint, very nebulous and without exact definition.

It is the current consensus in the physics community, that his idea as originally stated, is not correct and is not part of Einstein's general relativity. You can see, for example, the effort by Brans and Dicke to incorporate Machian ideas into general relativity by introducing a scalar field (which experiment has put severe limitations on).

However, some vestiges of Mach's ideas, perhaps in modified form, do show up occasionally in general relativity. For example, the Lense-Thirring (frame dragging) effect can be thought of as a "ghost" of Mach's idea in that the motion of an external object can in some way affect the local definition of inertial frames. Whether this effect is in accordance with the Mach's principle, is debatable depending on how loosely one interprets the words of Mach.

These ideas are somewhat technical however.

 

[PeterDonis]

if Mach was correct that the mass of the universe is the cause of inertia, why is inertia not diminished as the universe expands, since the gravitational pull of the distant galaxies would grow progressively weaker?

One way of looking at this is that, from the standpoint of, say, the solar system, the mass of the rest of the universe is basically distributed in a spherically symmetric fashion; so the space in which the solar system resides is basically an empty space surrounded by a spherically symmetric matter distribution, and GR has a theorem similar to the one in Newtonian mechanics that says the spherically symmetric mass distribution has no net gravitational effect on the empty space inside. So we can treat the solar system, to a good approximation, as an isolated system in an asymptotically flat spacetime, and this remains the case regardless of the expansion of the universe.

In other words, the gravitational pull of the distant galaxies on the solar system averages out to zero anyway, so the fact that each individual galaxy's pull decreases as the universe expands doesn't matter. More generally, the "effect on inertia" of all the distant galaxies cancels out as well, which is why inertia in our immediate local region of spacetime behaves the way one would expect it to in an asymptotically flat spacetime.

 

[Athanasius]

Thanks, Matterwave and PeterDonis for those clear explanations. Please take just a moment to join me in a thought experiment. Suppose our universe was finite, open and flat but much larger than the Hubble volume, and non-isotropic on a global scale, with an expanding edge far beyond the Hubble volume. Would you expect inertia to be more powerful in certain directions than others as we get nearer to the edge?

Lastly, can anyone point me to some of the best ideas as to why gravitational mass equals inertial mass?

 

[PeterDonis]

Suppose our universe was finite, open and flat but much larger than the Hubble volume, and non-isotropic on a global scale, with an expanding edge far beyond the Hubble volume.

Thinking of it as an "edge" might be a little misleading. What you're really saying is that the spatial topology is a 3-torus, with a finite spatial volume, instead of an infinite volume Euclidean 3-space. But assuming the space is homogeneous (which it should be if it's spatially flat), no point is different from any other, so you can't pick out any particular place and say that's the "edge". You just have a space where, if you travel far enough in any particular direction, you return back to your starting point.

Would such a space be non-isotropic? I don't think it would be, spatially. But there would be a "preferred" way to slice such a spacetime into space and time.

Would you expect inertia to be more powerful in certain directions than others as we get nearer to the edge?

As I said above, thinking of it as having an "edge" is misleading; the space should be homogeneous and isotropic. I don't know if the presence of a "preferred" slicing into space and time would affect the behavior of inertia.

 

[PAllen]

Lastly, can anyone point me to some of the best ideas as to why gravitational mass equals inertial mass?

Well, Einstein's answer is: because they are the same thing. Something cannot be unequal to itself. This identity between (passive) gravitational mass and inertial mass is built into the geometric structure of GR: a geodesic defines inertial motion everywhere; gravity is not a force at all. For active gravitational mass, you could say things are more subtle: it is because the stress energy tensor is source of curvature, and, for an isolated, small, body, the SET in the body's local rest frame is determined solely by the inertial mass of the body. So, for active gravitational mass = inertial mass, Einstein built it into the field equation.

In effect, Einstein's response was that the observation that they were the same should foster a theory where they cannot be different because there is only one quantity used in the theory.

 

[pervect]

Suppose our universe was finite, open and flat but much larger than the Hubble volume, and non-isotropic on a global scale, with an expanding edge far beyond the Hubble volume. Would you expect inertia to be more powerful in certain directions than others as we get nearer to the edge?

I'm not sure I'm interpreting this correctly. Usually a finite universe is taken to have a finite spatial volume, in which case there is no need for an edge. The edge concept seems more applicable to an infinite volume, with the edge being where the finite volume of matter expands into infinity.

Lastly, can anyone point me to some of the best ideas as to why gravitational mass equals inertial mass?

This is assumed to be true by the principle of equivalence. Basically the point is that locally "gravity" is just another inertial "force", for instance the "force" that traveller's on Einstein's elevator would experience.

 

[Athanasius]

I'm not sure I'm interpreting this correctly. Usually a finite universe is taken to have a finite spatial volume, in which case there is no need for an edge. The edge concept seems more applicable to an infinite volume, with the edge being where the finite volume of matter expands into infinity.

True, and that is the much more commonly accepted model. But for the thought experiment I am imagining a much less commonly envisioned model, where omega equals one with a universe of finite mass (because the expansion is just enough to prevent eventual gravitational collapse). Then you would have a flat, finite, open, uncurved universe with an expanding edge. And my question was, do you think that inertia would be greater in certain directions as you neared the edge?

 

[PeterDonis]

I am imagining a much less commonly envisioned model, where omega equals one with a universe of finite mass (because the expansion is just enough to prevent eventual gravitational collapse). Then you would have a flat, finite, open, uncurved universe with an expanding edge.

No, you wouldn't. You would have a flat 3-torus universe with a finite volume but no edge. It's just the standard omega = 1 solution, but with spatial points in the usual infinite 3-space of constant time identified, so that the spatial volume is finite, the spatial metric is flat, but the spatial topology is a 3-torus, with no edge.

A flat universe with an actual edge to the space, AFAIK, is not even a physically reasonable solution of the Einstein Field Equation, because the edge would have to be associated with a discontinuity in both the spacetime curvature and the stress-energy tensor. (More precisely, the edge itself, and any neighborhood of the spacetime including the edge, is not a solution.)

 

[Athanasius]

Thanks for that explanation, PAllen and pervect. I did not realize that Einstein assumed inertial mass and gravitational mass to be the same thing. I thought that he simply considered them to have essentially identical effects, in the normal sense of equivalence, in which two things may be different but share the same quality. So in light of that, if a spaceship accelerated towards Mars at 1 G, and then decelerated at 1 G, there would be absolutely no difference between the effects of this and Earth gravity for the astronauts?

 

[PAllen]

Thanks for that explanation, PAllen and pervect. I did not realize that Einstein assumed inertial mass and gravitational mass to be the same thing. I thought that he simply considered them to have essentially identical effects, in the normal sense of equivalence, in which two things may be different but share the same quality. So in light of that, if a spaceship accelerated towards Mars at 1 G, and then decelerated at 1 G, there would be absolutely no difference between the effects of this and Earth gravity for the astronauts?

As far as the astronauts go, yes (except for the period of change of direction of acceleration, which is locally detectable). This is effectively the "strong equivalence principle" and, of a broad class of test theories of gravity, GR is the only one for which it is true.

 

[yogi]

As to why inertia would diminish with expansion , you might turn the question around and ask the reciprocal question - why would it not increase -Einstein arrived at the conclusion that inertial mass and gravitational mass were one in the same - now consider that each small g field of every local mass extends to the limit of the Hubble sphere - the volume of the negative g field grows with expansion - and the negative energy represented by each local field is equal to the positive energy mc^2 of the local mass from which the local g field derives. Therefore inertia under this line of inquiry might be expected to increase. Mach's principle in any event is a positive feedback scenario - any increase in the mass-energy of the universe adds to total inertia - and because of the equivalence principle, addition inertia is like adding mass - and more mass means greater inertia. Different musings lead to different possibilities - I liked your question

 

[Bill_K]

As to why inertia would diminish with expansion , you might turn the question around and ask the reciprocal question - why would it not increase -Einstein arrived at the conclusion that inertial mass and gravitational mass were one in the same - now consider that each small g field of every local mass extends to the limit of the Hubble sphere - the volume of the negative g field grows with expansion - and the negative energy represented by each local field is equal to the positive energy mc^2 of the local mass from which the local g field derives. Therefore inertia under this line of inquiry might be expected to increase. Mach's principle in any event is a positive feedback scenario - any increase in the mass-energy of the universe adds to total inertia - and because of the equivalence principle, addition inertia is like adding mass - and more mass means greater inertia. Different musings lead to different possibilities - I liked your question

"Musing", i.e. speculation, is always fun, yogi, especially since it doesn't require any math and hardly any physics. But here on PF, according to the Global Rules and Guidelines, we discuss only mainstream physics. Which the ideas you express above certainly are not.

 

[bcrowell]

Mach's principle is falsifiable and has been disproved by observation. http://physics.stackexchange.com/q/5483/4552

Mach was not just a philosopher. He did important work on the foundational issues of Newtonian mechanics. See The Science of Mechanics, 1902, http://www.archive.org/details/sciencemechanic00machgoog

 

[Creator]

inertia

... if Mach was correct that the mass of the universe is the cause of inertia, why is inertia not diminished as the universe expands, since the gravitational pull of the distant galaxies would grow progressively weaker? .

Athanasius,
This is a very insightful question, but it assumes that mass that is farther from the test object imparts LESS gravitational influence upon the test mass, when the opposite may actually true (assuming mach's Principle).
Not commonly realized is that in dealing with Mach's principle we are dealing with gravitational potential, Phi, which decreases as 1/R, more precisely; Phi = GM/R.
BUT, assuming a somewhat homogenous mass density in the universe (which is relatively spherically symmetric), we can see that the mass density of the universe actually increases as the CUBE of the distance as you move outward from the test mass...IOWs the amount of Mass increases VOLUMETRICALLY.

Thus the amount of mass at more distant radius increases spherically by the cube of the radius and thus in terms of grav. potential more distant mass actually has GREATER effect on the local test mass. In other words, The Grav. POTENTIAL , WHICH IS A SCALAR QUANTITY, increases BY THE SQUARE of the distance.

So to answer your question about expanding universe, IF Mach's principal is effective, and if there is a relation between inertia and grav. potential, then more distant mass dominates and there should be a GREATER growth in inertia as the universe expands.

Creator.
--" Consciousness...the annoying time between naps".--

 

[PeterDonis]

Not commonly realized is that in dealing with Mach's principle we are dealing with gravitational potential

Gravitational potential is only well-defined in stationary spacetimes. The universe as a whole is not stationary, because it is expanding. So there is no well-defined concept of "gravitational potential" in the universe as a whole.

assuming a somewhat homogenous mass density in the universe (which is relatively spherically symmetric), we can see that the mass density of the universe actually increases as the CUBE of the distance

I suspect you've made a typo here. A homogeneous mass density is roughly the same everywhere, which means it can't increase as the cube of the distance. I assume that what you actually mean here is that the total mass within a given distance increases as the cube of the distance, since you go on to say:

Thus the amount of mass at more distant radius increases spherically by the cube of the radius and thus in terms of grav. potential more distant mass actually has GREATER effect on the local test mass. In other words, The Grav. POTENTIAL , WHICH IS A SCALAR QUANTITY, increases BY THE SQUARE of the distance.

Unfortunately, even if we leave aside the fact I noted above, that gravitational potential is not well-defined for the universe as a whole, this reasoning has a flaw in it. Since the matter in the universe is spherically symmetric about any given local volume of empty space, the net effect of it is *zero* on any point within that empty space, because a spherically symmetric mass distribution surrounding an empty space has no gravitational effect on any point in that empty space (roughly speaking, this is because every piece of matter outside the empty space has a corresponding one on the opposite side at the same distance, so their potentials cancel out). The increase in the total amount of mass present with distance doesn't change that.

 

[bahamagreen]

The closest thing I see to Mach's principle in regard to inertia is that in an expanding space universe two masses (galaxies, clusters, etc.) with wide separation distance are observed to be accelerating apart, but neither are experiencing an inertial mass resistance to the expansion of the space within which they are embedded, nor their parts, nor their constituents, etc... as if inertial mass resistance to acceleration is always "local" with respect to local accelerations, but with respect to the universal acceleration they don't resist but just accelerate "forcelessly" ro "inertialessly", like the way geodesics are traveled without "lateral acceleration components".

So really more like a "reverse" Mach principle?

Or maybe this has been examined and found to be expected?
I imagine that inertia is like acceleration - it should be "absolute" without regard to a distinction between local and universal acceleration?

 

[yogi]

"Musing", i.e. speculation, is always fun, yogi, especially since it doesn't require any math and hardly any physics. But here on PF, according to the Global Rules and Guidelines, we discuss only mainstream physics. Which the ideas you express above certainly are not.

Interesting - when I made a comment addressing the OP's question, I get slammed - then every other Tom, Dick and Harry comes along and post his 2-bits and its ok. Admittedly, the early statement of the Mach's Principle was vague - undeveloped and with little mathematical support ...Historically, the first promulgator of mass as mass creator was Bishop Berkeley. Bishop Berkeley's condemnation of Newtonian Space as a sideless box with no physical properties didn't influence Newton. But was he right? Dennis Schima developed an inertial theory of gravity based upon electrical analogies and addresses the issue I raised concluding it would lead to greater interaction for more distant matter. As generally understood, the Principle leads to synergistic reinforcement if it works at all. The OP introduced the subject with a question approach. He then slips in his own theory as part of the question. Good show - by tendering his own speculative viewpoint as a question, we get a reasonable discussion. IMO neo-Machian theory is very much alive, the whole idea of tying mass to the creation of mass has many interesting holistic consequences.

Some years back a fellow named (Garth) published a paper on these forums founded upon a version of Mach's Principle entitled "Self Creating Cosmology" It had some good ideas - but one part of the argument was falsified with the data from WMAP I believe. anyone interest should look in the archives.

Yogi

 

[bcrowell]

This whole thread reads like an attempt to reinvent the wheel. Brans and Dicke came up with a theory that did a pretty good job of embodying Mach's principle. It's called Brans-Dicke gravity. Brans-Dicke gravity was later falsified by solar-system tests. There is no need to indulge in vague speculation as if nobody had ever worked on these issues since the 19th century.

 

[yogi]

This whole thread reads like an attempt to reinvent the wheel. Brans and Dicke came up with a theory that did a pretty good job of embodying Mach's principle. It's called Brans-Dicke gravity. Brans-Dicke gravity was later falsified by solar-system tests. There is no need to indulge in vague speculation as if nobody had ever worked on these issues since the 19th century.

Experiments show that ω must in fact be quite high. The best current limit comes from the Cassini probe, which requires ω≥40,000 . Therefore, B-D gravity should be considered as falsified. So the modern, sensible answer to the OP's question is: Mach's principle is false, in the sense that experiments determine the universe to be no more Machian than GR -- which is not very Machian.

That is true - but B-D theory took a very narrow approach which involved adjusting only G.
A more general approach would be to adjust MG/R = c^2 = Constant. Since R increases with age - then the MG product should be allowed to increase - this allows G to have the same variance as Dirac's LNH, and M (the mass factor of the universe) can increase as the square of the radius as predicted by some versions of Mach's Principle.

My point is, only the restricted approach taken by B-D is effectively falsified, there are other avenues in which Mach's principle may still be a viable theory.

 

[bcrowell]

That is true - but B-D theory took a very narrow approach which involved adjusting only G.
A more general approach would be to adjust MG/R = c^2 = Constant. Since R increases with age - then the MG product should be allowed to increase - this allows G to have the same variance as Dirac's LNH, and M (the mass factor of the universe) can increase as the square of the radius as predicted by some versions of Mach's Principle.

This seems like a half-baked personal theory. PF has rules against discussing personal theories: https://www.physicsforums.com/showthread.php?t=414380

 

[Athanasius]

No, you wouldn't. You would have a flat 3-torus universe with a finite volume but no edge. It's just the standard omega = 1 solution, but with spatial points in the usual infinite 3-space of constant time identified, so that the spatial volume is finite, the spatial metric is flat, but the spatial topology is a 3-torus, with no edge.

A flat universe with an actual edge to the space, AFAIK, is not even a physically reasonable solution of the Einstein Field Equation, because the edge would have to be associated with a discontinuity in both the spacetime curvature and the stress-energy tensor. (More precisely, the edge itself, and any neighborhood of the spacetime including the edge, is not a solution.)

Thanks for all of the replies! Peterdonis, can you elaborate on this a little more for me? What would force an omega=1 finite universe into a closed topology? And why a flat 3-torus? Why not one of the other finite flat closed topologies?

 

[Jonathan Scott]

Experiments show that ω must in fact be quite high. The best current limit comes from the Cassini probe, which requires ω≥40,000 . Therefore, B-D gravity should be considered as falsified. So the modern, sensible answer to the OP's question is: Mach's principle is false, in the sense that experiments determine the universe to be no more Machian than GR -- which is not very Machian.

That is true - but B-D theory took a very narrow approach which involved adjusting only G.
A more general approach would be to adjust MG/R = c^2 = Constant. Since R increases with age - then the MG product should be allowed to increase - this allows G to have the same variance as Dirac's LNH, and M (the mass factor of the universe) can increase as the square of the radius as predicted by some versions of Mach's Principle.

My point is, only the restricted approach taken by B-D is effectively falsified, there are other avenues in which Mach's principle may still be a viable theory.

I agree. Many classes of theory based on Mach's Principle have been falsified by experimental results, for example in that if the GR equations correctly describe the Lunar Laser Ranging experiment and certain types of supernova then this implies that G is not varying with time by anything like the amount that LNH-based theories would predict. However, this does not automatically rule out all possible theories based on Mach's Principle. So far I've not seen any particularly convincing one, and it would obviously need extraordinary strengths to compete with GR, but it is GR itself which provides some of the strongest suggestions in support of the idea that there must be something in it.

In particular, Dennis Sciama in his paper "On The Origin of Inertia" suggested a toy model for inertia in terms of the inductive effect of relative acceleration of gravitational sources, and a paper by Kenneth Nordtvedt about linear frame dragging (which I found some time ago on Google but I can't remember the details right now) shows that this effect appears within GR as well. The effect due to a single source is proportional to its potential and the acceleration, so if the Whitrow-Randall-Sciama relation is roughly right (that is, the effective sum of Gm/rc^2 due to every source in the universe is of order 1) then the calculated effect of assuming an accelerating body is at rest and it is the universe which is being accelerated the other way is physically similar to and of the same order as the effect of inertia, which is a very Machian effect.

So it seems that GR is locally surprisingly Machian in various ways (including both linear and rotational frame dragging, with the "sum for inertia" concept) and it is quite a puzzle that the standard theory says this has to be some sort of temporary coincidence as it can't be exact, and it is also clear that direct "Machian" modifications of GR, such as B-D theory, don't work experimentally.

 

[yogi]

I agree. Many classes of theory based on Mach's Principle have been falsified by experimental results, for example in that if the GR equations correctly describe the Lunar Laser Ranging experiment and certain types of supernova then this implies that G is not varying with time by anything like the amount that LNH-based theories would predict. However, this does not automatically rule out all possible theories based on Mach's Principle. So far I've not seen any particularly convincing one, and it would obviously need extraordinary strengths to compete with GR, but it is GR itself which provides some of the strongest suggestions in support of the idea that there must be something in it.

In particular, Dennis Sciama in his paper "On The Origin of Inertia" suggested a toy model for inertia in terms of the inductive effect of relative acceleration of gravitational sources, and a paper by Kenneth Nordtvedt about linear frame dragging (which I found some time ago on Google but I can't remember the details right now) shows that this effect appears within GR as well. The effect due to a single source is proportional to its potential and the acceleration, so if the Whitrow-Randall-Sciama relation is roughly right (that is, the effective sum of Gm/rc^2 due to every source in the universe is of order 1) then the calculated effect of assuming an accelerating body is at rest and it is the universe which is being accelerated the other way is physically similar to and of the same order as the effect of inertia, which is a very Machian effect.

So it seems that GR is locally surprisingly Machian in various ways (including both linear and rotational frame dragging, with the "sum for inertia" concept) and it is quite a puzzle that the standard theory says this has to be some sort of temporary coincidence as it can't be exact, and it is also clear that direct "Machian" modifications of GR, such as B-D theory, don't work experimentally.

Jonathon, I am curious as to your assessment of the following - the usual reason for dismissing the variance of G is based upon the failure to detect long term changes in the orbits planetary moons. The tests however are only capable of measuring the invariance of the MG product. Machian theory (although clearly not developed by Mach to a point of falsifiability) can be interpreted as gradually acquired inertia - at least this is the way it is sometimes introduced (Harrison -Cosmology 2001 edition at page 160 - the bootstrap argument). My point is, gradual inertia compliments decreasing G - so the lunar orbits would show no change even though G could be changing as 1/R as would be required to comply with Diracs LNH. That the MG product is always constant is not seriously in doubt - but IMO that should not be used to discredit Mach's principle.

 

[Jonathan Scott]

Jonathon, I am curious as to your assessment of the following - the usual reason for dismissing the variance of G is based upon the failure to detect long term changes in the orbits planetary moons. The tests however are only capable of measuring the invariance of the MG product. Machian theory (although clearly not developed by Mach to a point of falsifiability) can be interpreted as gradually acquired inertia - at least this is the way it is sometimes introduced (Harrison -Cosmology 2001 edition at page 160 - the bootstrap argument). My point is, gradual inertia compliments decreasing G - so the lunar orbits would show no change even though G could be changing as 1/R as would be required to comply with Diracs LNH. That the MG product is always constant is not seriously in doubt - but IMO that should not be used to discredit Mach's principle.

I agree that ideas involving variable M but a constant GM product (which was the case you previously mentioned) are among those which are not directly falsified by the current experimental results. However, any theory which allows fundamental constants to become variable has many other challenges to cover both at the local level (for example conservation laws may require a changing overall time rate, with various other implications) and at the cosmological level (for example relating to redshift and supernova timing). Frequent attempts to allow constants such as particle masses and c to be variable end up instead as unhelpfully complex redefinitions of exactly the same physics, or alternatively end up failing self-consistency checks.

This is not the place to speculate about the viability of specific alternative theories, but I think it is important to be aware that alternatives exist, and to be able to compare and contrast their approaches with GR. Although GR is the best theory we have so far, it is very puzzling that it seems to deny Machian ideas in its overall theory while supporting them to high accuracy in local frame-dragging calculations confirmed by experiment.

 

[yogi]

I agree that ideas involving variable M but a constant GM product (which was the case you previously mentioned) are among those which are not directly falsified by the current experimental results. However, any theory which allows fundamental constants to become variable has many other challenges to cover both at the local level (for example conservation laws may require a changing overall time rate, with various other implications) and at the cosmological level (for example relating to redshift and supernova timing). Frequent attempts to allow constants such as particle masses and c to be variable end up instead as unhelpfully complex redefinitions of exactly the same physics, or alternatively end up failing self-consistency checks.

This is not the place to speculate about the viability of specific alternative theories, but I think it is important to be aware that alternatives exist, and to be able to compare and contrast their approaches with GR. Although GR is the best theory we have so far, it is very puzzling that it seems to deny Machian ideas in its overall theory while supporting them to high accuracy in local frame-dragging calculations confirmed by experiment.

Thanks for your response. I fully agree with everything you said except the word "fundamental" as it applies to G. But that can be the subject of another post, or perhaps a book, or even a lifetime of inquiry.