Does General Relativity explain inertia?

[jcap]

As far as I understand it general relativity does not explain the origin of the inertial mass $m_i$ in Newton's law of motion $\vec{F}=m_i\ d\vec{v}/dt$ but rather it simply applies the concept to curved spacetime.

For example if we have a particle with inertial mass $m_i$ and charge $q$ moving in flat spacetime in an electromagnetic field $\vec{E},\vec{B}$ with relativistic 3-velocity $\vec{v}$ then its equation of motion with respect to its proper time $\tau$ is
$$q(\vec{E}+\vec{v}\times\vec{B})=m_i\frac{d\vec{v}}{d\tau}.\tag{1}$$
In curved spacetime the equation of motion $(1)$ becomes
$$q\ {F^\mu}_\nu\ v^\nu=m_i\Big(\frac{dv^\mu}{d\tau}+\Gamma^\mu_{\rho\sigma}\ v^\rho\ v^\sigma\Big)\tag{2}$$
where ${F^\mu}_\nu$ is the electromagnetic tensor, $v^\mu$ is the 4-velocity of the particle and $\Gamma^\mu_{\rho\sigma}$ is the metric connection.

Neither Eqn $(1)$ nor Eqn $(2)$ actually explain why it takes a force $\vec{F}=m_i\ \vec{a}$ in order to impart an acceleration $\vec{a}$ to an object with an inertial mass $m_i$.

Is this correct?

 

[vanhees71]

No physical theory explains why the physical laws are as they are, they just describe them more or less accurately. Everything rests finally on observations and building a mathematical description of the observations. Why there are inertial frames (global ones in Newtonian and special-relativistic physics, local ones in general-relativistic physics) nobody can tell, but it's a fact we know about due to observations and building mathematical models describing these observations. The very fact that this is so successful is quite an enigma (as Wigner put it).

 

[Dale]

As far as I understand it general relativity does not explain the origin of the inertial mass mi in Newton's law of motion F→=mi dv→/dt but rather it simply applies the concept to curved spacetime.

I would say it slightly differently. GR takes the concepts of inertial mass and gravitational mass and uses the concept of curved spacetime to explain why they are equal. Equivalently, you could say that the equality of inertial and gravitational mass stems from the weak equivalence principle.

Either way, you are correct that it does not explain the origin of either inertial or gravitational mass, only their equality.

You can supplement the weak equivalence principle with Newton’s 3rd law or the conservation of momentum to explain the origin of inertial mass, but I am not sure that using the conservation of momentum to explain inertia is not already sort of “cheating”. Of course, at least you could use Noether’s theorem to explain conservation of momentum. So maybe that would be the way.

 

[Herman Trivilino]

Neither Eqn (1) nor Eqn (2) actually explain why it takes a force F→=mi a→ in order to impart an acceleration a→ to an object with an inertial mass mi.

It's true by definition. Note that in special relativity there is in general no $m$ that will satisfy the relation $\vec{F}=m\vec{a}$.

 

[jcap]

Is it fair to say that general relativity explains the force of gravity as an inertial force but does not explain the origin of inertia itself?

 

[PeroK]

Is it fair to say that general relativity explains the force of gravity as an inertial force but does not explain the origin of inertia itself?

In GR there is no inertial mass. There is only gravitational mass. It's only when you consider the other forces of nature that there is an equivalence (or proportionality) between gravitational mass and inertial mass.

In any case, the mass of elementary particles is a slippery subject and ultimately leads to the Higgs field.

 

[jcap]

In any case, the mass of elementary particles is a slippery subject and ultimately leads to the Higgs field.

The Higgs field leads to the creation of rest mass but as far as I understand it does not explain the phenomenon of resistance to acceleration associated with rest mass.

 

[PeroK]

The Higgs field leads to the creation of rest mass but as far as I understand it does not explain the resistance to acceleration of inertial mass.

There is not really the concept of Newton's second law in quantum theory in terms of the interactions of elementary particles. Newton's laws emerge as the classical limit of quantum interactions.

 

[vanhees71]

I would say it slightly differently. GR takes the concepts of inertial mass and gravitational mass and uses the concept of curved spacetime to explain why they are equal. Equivalently, you could say that the equality of inertial and gravitational mass stems from the weak equivalence principle.

Either way, you are correct that it does not explain the origin of either inertial or gravitational mass, only their equality.

You can supplement the weak equivalence principle with Newton’s 3rd law or the conservation of momentum to explain the origin of inertial mass, but I am not sure that using the conservation of momentum to explain inertia is not already sort of “cheating”. Of course, at least you could use Noether’s theorem to explain conservation of momentum. So maybe that would be the way.

Well, the 3rd law is a bit problematic in relativity. I'd rather use it as an argument for why to use local field-theoretical descriptions for interactions between particles rather than action-at-a-distance concepts of Newtonian physics: In Newtonian physics there's no problem with the 3rd law for far-distant bodies. If the relative position of the bodies changes with time (because they are moving) there's no problem within Newtonian physics that the 3rd laws instantaneously holds for the interaction forces between these bodies, e.g., when thinking about motions of two celestial bodies changing their distance, the Newtonian gravitational interaction force instantaneously adapts such that the 3rd Law always holds.

That's problematic in relativity theory, because there should be retardation, as known in electrodynamics. Here we describe interactions as mediated by a field, which is a dynamical entity itself (as the masspoints are such dynamical entities). Any change in one charges position leads to changes of its electromagnetic field which propagates in space according to the retarded Green's function, i.e., with the finite speed of light. The force on the other charge due to that field is described by the local concept that it is given by the Lorentz force with the electromagnetic field taken at the position of this charge. The 3rd Law, or rather momentum conservation, always holds, but only if you take the momentum of the charges and the field together, and it's a local concept too, i.e., in a continuum description you get a local conservation law in terms of an equation of continuity.

In GR due to the (weak) equivalence principle inertial motion is the same as free fall, i.e., as it comes out from the formalism the motion of test masses in the gravitational field of another (heavy) body leads to the geodesic equation in the Lorentz manifold that describes spacetime at the presence of the heavy body and defines this gravitational field.

Another consequence of the (strong) equivalence principle is that not "gravitational mass" is the source of gravity, but all kinds of energy, momentum, and stress of the matter and radiation. The masses of particles enter just as parameters in the Lagrangian of the fields describing them (e.g., in terms of a hydrodynamical model of a gas) and are only correspondig contributions to the energy-momentum tensor of matter which provides the sources for the gravitational field on right-hand side of Einstein's field equation,
$$G_{\mu \nu}=\kappa T_{\mu \nu}.$$

 

[vanhees71]

The Higgs field leads to the creation of rest mass but as far as I understand it does not explain the phenomenon of resistance to acceleration associated with rest mass.

The Higgs field in this sense is the vacuum expectation value of a scalar field in the Standard Model of elementary particle physics. It leads to mass of the elementary particles, quarks and leptons, as well as through the Higgs mechanism the weak gauge bosons $W$ and $Z$ "mediating" the weak interaction. As @PeroK said above, the classical equations of motion for particles emerges as an approximation of the quantum theortical description.

 

[haushofer]

Just to add: Einstein hoped that his theory of General Relativity would be fully Machian, but it's not: Minkowski spacetime is a solution to the Einstein equations. So if you consider one testparticle in an otherwise empty universe, this particle has inertial properties while Mach would say it can't have those.

Mach would dictate that the inertial properties, i.e. the inertial mass of a particle m, should depend on all the mass M in the universe: $m=m(M)$. Furthermore, you'd expect on behalf of Mach's principle that

$lim_{M\rightarrow 0} \, \, m(M) = 0$

But that's basically how far Mach went; the exact relationship $m(M)$ is not given by the principle, one reason why eventually physicists started to loose their interest into it.

 

[vanhees71]

I think one basic roblem with "Mach's principle" is that it is too vague. In which sense is it meant that "inertia of a body is due to the presence of all other bodies".

Concerning GR I'd say "inertia" is a fundamental concept as in Newtonian mechanics and special relativity, because GR takes this concept over from SR, but makes it local: In Newtonian mechanics and SR the principle of inertia is a fundamental assumption. Mathematically stated it just postulates that global inertial frames exist. How to formulate it in physical terms was worked out for Newtonian mechanics by Lange and it can also be used in SR, but I don't dare to discuss it here anymore ;-)). In any case GR takes the equivalence principle as a fundamental concept, and this says that there at any spacetime point there are Lorentzian reference frames, where local laws look as in SR, leading directly to the description of spacetime as a Lorentzian manifold.

I don't know whether GR can be taken as "Machian" or not, because the Mach principle is too vaguely formulated. In a certain sense I'd say you could reinterpret the vague Machian principle as being realized by GR in the sense that the distribution of energy, momentum, and stress of matter (and radiation) determines the concrete Lorentzian spacetime manifold through Einstein's field equations. Indeed, if you have no matter and radiation you get the Minkowski space as a solution, but as you say, there the principle of inertia is just a postulate, i.e., the assumption of the existence of a (in this case even global) inertial reference frame (and thus an entire class of inertial reference frames all moving with constant velocity wrt. each other).

 

[Dale]

Well, the 3rd law is a bit problematic in relativity.

Yes. I think it would be better to use momentum conservation. Particularly since that has an explanation in terms of Noether’s theorem.

 

[Dale]

Is it fair to say that general relativity explains the force of gravity as an inertial force but does not explain the origin of inertia itself?

I am not sure. I mean you could say that the EFE explains spacetime which brings the weak equivalence principle. It also explains the active gravitational mass as the stress energy tensor which is what bends spacetime. Then you can say that Noether’s theorem explains the conservation of momentum. Then the weak equivalence principle together with the conservation of momentum gives the equality of active gravitational mass with passive gravitational mass and inertial mass.

It seems like there are a lot of assumptions there and probably it could be claimed that some of those assumptions themselves already assume inertia. But it is not clear to me that the above line of reasoning couldn’t be modified and polished to really explain inertial mass.

Just to be clear, I have never read a source that claimed this line of reasoning, so I am not going so far as to say that it actually is valid. All I am saying is that it seems roughly plausible, so I am not certain that inertial mass has no explanation. It seems doubtful either way and I could see reasonable people disagreeing.

 

[PeterDonis]

In GR there is no inertial mass. There is only gravitational mass.

Actually I would put this the other way around: in GR there is no (passive) gravitational mass, there is only inertial mass. In other words, an object has a property (inertial mass) that says how much proper acceleration it experiences in response to a given force. But it has no property that says how it responds to gravity, because gravity is not a force in GR.

GR still has an analogue of the concept of active gravitational mass, but it is not a scalar, it's a tensor, the stress-energy tensor.

 

[haushofer]

Actually I would put this the other way around: in GR there is no (passive) gravitational mass, there is only inertial mass. In other words, an object has a property (inertial mass) that says how much proper acceleration it experiences in response to a given force. But it has no property that says how it responds to gravity, because gravity is not a force in GR.

Indeed, I'd say this is exactly one "Machian" aspect of GR ;)

 

[Sagittarius A-Star]

I think one basic roblem with "Mach's principle" is that it is too vague. In which sense is it meant that "inertia of a body is due to the presence of all other bodies".

I think in this sense, but it is difficult to prove or disprove it for the whole universe:

They consider an UR-scenario in which a bucket with water is at rest in a rotating universe, and a BR-scenario where the bucket rotates in a non-rotating universe
...
However, the spin of the bucket is conserved, so it will proceed to rotate in the BR-scenario even when no force acts upon it, and in the UR-scenario it will remain at rest in the rotating universe.
...
In the Kerr spacetime outside a rotating mass, for example, inertial frames are dragged around, and this very slow rotating motion of the inertial frames close to the Earth has now been measured with Lageos II and Gravity Probe B. Hence, according to the general theory of relativity inertial frames may be rotating.

Source:
https://arxiv.org/ftp/arxiv/papers/0708/0708.0457.pdf

 

[PeterDonis]

I think in this sense, but it is difficult to prove or disprove it for the whole universe

The problem I see is that the term "rotating universe" is used without actually saying what it means. In GR, you can't just wave your hands and say "rotating universe": you have to specify which solution of the Einstein Field Equation you mean. (For example, you could specify the Godel universe.) And since that solution will obviously not be flat Minkowski spacetime, it is obvious in GR that the situation "non-rotating bucket in a rotating universe" and "rotating bucket in a non-rotating universe" are not physically equivalent, because the spacetime geometry is different.

Note that the term "non-rotating bucket" also needs to be given a specific meaning in GR; you can't just wave your hands and use it without saying what you mean. For example, you could say it means that the orthonormal tetrad describing the bucket's 4-velocity and spatial orientation is Fermi-Walker transported along its worldline. But this meaning will often not be the same as what one's pre-relativistic intuitions think "non-rotating" means.

For example, consider Gravity Probe B. Gravity Probe B's spatial orientation is "non-rotating" in the sense I gave above: its orthonormal tetrad is Fermi-Walker transported along its worldline. But it is, of course, "rotating" with respect to a hypothetical observer "at rest at infinity"--this is what "frame dragging" means. So the satellite is "non-rotating" in one sense, but "rotating" in another. Note that frame dragging is not the only effect that causes a mismatch between these two different senses of "rotating"--even Thomas precession in flat spacetime will do it, and de Sitter precession in Schwarzschild spacetime will also do it.

There are several Insights articles that discuss this whole issue in more detail; first, this one by Bill_K:

https://www.physicsforums.com/insights/precession-in-special-and-general-relativity/

And these three by me, doing the analysis in a somewhat different way and giving more discussion:

https://www.physicsforums.com/insights/how-to-study-fermi-walker-transport-in-minkowski-spacetime/

https://www.physicsforums.com/insights/fermi-walker-transport-in-schwarzschild-spacetime/

https://www.physicsforums.com/insights/fermi-walker-transport-in-kerr-spacetime/

Btw, the Gron paper is being extremely sloppy with terminology when it says "according to the general theory of relativity inertial frames may be rotating". By definition a frame that is "inertial" is always non-rotating in the Fermi-Walker transport sense, because an inertial frame cannot have any "fictitious forces" in it, and any such frame must be non-rotating in the Fermi-Walker transport sense. (It must also be centered on a worldline which is a geodesic, so that Fermi-Walker transport actually becomes parallel transport.) The sense in which the inertial frames in question are "rotating" is the "with respect to the observer at rest at infinity" sense, but in curved spacetime there are no global inertial frames, the inertial frames in question are only local (localized near the worldline of the object, in this case the Gravity Probe B satellite), so any behavior with respect to a distant observer is not usefully viewed as a property of the frame itself anyway. This is yet another issue I have with the Gron paper.

 

[Sagittarius A-Star]

The problem I see is that the term "rotating universe" is used without actually saying what it means. In GR, you can't just wave your hands and say "rotating universe": you have to specify which solution of the Einstein Field Equation you mean. (For example, you could specify the Godel universe.)

To my understanding, Gron does not mean a rotating Gödel universe. I think he means only a coordinate-rotation of the universe, if you use as reference frame the rest frame of rotating bucket. And then the idea is:

3. Centrifugal- and Coriolis acceleration as a result of inertial dragging

Source:
https://arxiv.org/ftp/arxiv/papers/0708/0708.0457.pdf

Here is another formulation:

Mach’s principle is the principle that states that the total rotation/acceleration of the Universe is unobservable (due to perfect inertial dragging) while the extended principle of relativity is the principle that allows us to consider the Universe as rotating/accelerating relative to something inside it, and due to the inertial “force” (created by the perfect inertial dragging) a real force is needed to keep the considered body “at rest”.

Source (end of chapter 3.3):
https://pdfs.semanticscholar.org/054a/25e98041a79cee78daecc09a63f77ca7b0eb.pdf

Both, Gron and Einstein think, that Mach's principle could be valid. According to Mach's principle, "pseudo-gravitation" is seen as an effect of real gravitation (perfect inertial dragging by the visible universe). That might be the reason, why both define the scope of "SR" such, that they regard calculations in accelerated frames not as part of SR, but only of GR. That differs from the viewpoint of most of today's physicists.

Source 1:

10.Conclusion
...
In order that both twins shall have the right to claim that they are at rest, we have to introduce the extended model of the Minkowski spacetime.Then the field of gravity experienced by B is due to perfect dragging induced by the accelerating cosmic shell.

Source:
https://oda.hioa.no/nb/the-twin-par...-relativity/asset/dspace:5303/1011415post.pdf

Source 2:

To be sure, the accelerated coordinate systems cannot be called upon as real causes for the field, an opinion that a jocular critic saw fit to attribute to me on one occasion. But all the stars that are in the universe, can be conceived as taking part in bringing forth the gravitational field; because during the accelerated phases of the coordinate system K' they are accelerated relative to the latter and thereby can induce a gravitational field, similar to how electric charges in accelerated motion can induce an electric field. Approximate integration of the gravitational equations has in fact yielded the result that induction effects must occur when masses are in accelerated motion.

Source:
https://en.wikisource.org/wiki/Translation:Dialog_about_Objections_against_the_Theory_of_Relativity

 

[PeterDonis]

To my understanding, Gron does not mean a rotating Gödel universe. I think he means only a coordinate-rotation of the universe, if you use as reference frame the rest frame of rotating bucket.

And if that's what he means, then he should specify the coordinate chart he is using and how it relates to standard inertial coordinates on Minkowski spacetime. For example, if he's using Born coordinates, he should say so.

However, if that is in fact what Gron intends, then his mention of "perfect dragging induced by the accelerating cosmic shell" makes no sense, since there is no such "cosmic shell" in Minkowski spacetime.

 

[Sagittarius A-Star]

However, if that is in fact what Gron intends, then his mention of "perfect dragging induced by the accelerating cosmic shell" makes no sense, since there is no such "cosmic shell" in Minkowski spacetime.

Yes. He proposes in another paper an extension of Minkowski spacetime:

8 Inertial dragging inside a rotating shell of matter
...
9 An extended model of Minkowski spacetime
...
We propose a generalized model of Minkowski space, i.e. of globally flat spacetime or the flat region of asymptotically flatspacetimes, where the space is completed by a faraway cosmic massive shell with radius equal to its Schwarzschild radius, representing the cosmic mass. Inside such a shell there is approximately flat spacetime and perfect dragging ([28, 29], and sect. 8 above).

Source:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.221.9088&rep=rep1&type=pdf

But I agree to that:

I think one basic problem with "Mach's principle" is that it is too vague.

 

[vanhees71]

I also don't see too much merit in Mach's philosophical ideas except that it has maybe triggered Einstein to think very deeply about the issue with how to define inertial reference frames (though this topic is not so easy to sensibly being discussed in this forum, and thus I abstain from giving my opinion on it again). The mathematical conclusion within GR is that there are always local inertial reference frames for any observer, defined by Fermi-Walker transported tetrads of this observer (see @PeterDonis 's posting #18).

 

[mitochan]

The problem I see is that the term "rotating universe" is used without actually saying what it means. In GR, you can't just wave your hands and say "rotating universe": you have to specify which solution of the Einstein Field Equation you mean. (For example, you could specify the Godel universe.) And since that solution will obviously not be flat Minkowski spacetime, it is obvious in GR that the situation "non-rotating bucket in a rotating universe" and "rotating bucket in a non-rotating universe" are not physically equivalent, because the spacetime geometry is different.

System of rotating bucket in a non-rotating universe has boundary of $r=\frac{c}{\omega}$ beyond which all the physical bodies moves with speed more than c.

System of non-rotating bucket in a rotating universe does not have such a boundary and any part of universe has speed less than c in IFR for SR is applicable. Thinking about $v=r\omega<c$ and almost infinite radius of universe, I am afraid universe cannot rotate in a IFR.

 

[PeterDonis]

He proposes in another paper an extension of Minkowski spacetime

That's not an extension of Minkowski spacetime, it's a straightforward spherical shell spacetime; spacetime has flat Minkowski geometry inside the shell and Schwarzschild geometry outside the shell. That's a well-known solution of the Einstein Field Equation. But it doesn't fix the problem Gron is trying to fix: you still have the same issue of what happens at infinity, since the Schwarzschild geometry is still asymptotically flat.

Actually, though, even here Gron has another problem: it's physically impossible for the radius of the shell to be equal to the Schwarzschild radius for its mass, as Gron claims. Such a solution is not stable; the shell will collapse. In fact the shell can't even be at rest for an instant at that radius, since to do so even for an instant would require the matter in the shell to be moving at the speed of light.

I really can't recommend Gron as a source given all the issues with what he says.

 

[PeterDonis]

System of non-rotating bucket in a rotating universe does not have such a boundary

What do you mean by "rotating universe"? As has already been noted, that term is ambiguous.

 

[mitochan]

What do you mean by "rotating universe"? As has already been noted, that term is ambiguous.

As explained in post#23 I took it in your quoted explanation as rotation in IFR. If you mean it another way I would appreciate your correction.

 

[PeterDonis]

I took it in your quoted explanation as rotation in IFR

This doesn't answer my question. What spacetime geometry are you assuming?

If you are assuming flat Minkowski spacetime, and you are using an inertial frame, then the phrase "rotating universe" makes no sense as a description of that.

If you are assuming flat Minkowski spacetime, and you are using a non-inertial "rotating" frame such as Born coordinates, then this describes a rotating frame, not a rotating universe. The universe itself is not rotating. You've just chosen to use a rotating frame.

So the only way to make sense of the phrase "non-rotating object in a rotating universe" is if the spacetime geometry is something other than flat Minkowski spacetime. So which spacetime geometry is it?

 

[mitochan]

If you are assuming flat Minkowski spacetime, and you are using an inertial frame, then the phrase "rotating universe" makes no sense as a description of that.

That was my assumption and I wrote

Thinking about and almost infinite radius of universe, I am afraid universe cannot rotate in a IFR.

Thanks to

So the only way to make sense of the phrase "non-rotating object in a rotating universe" is if the spacetime geometry is something other than flat Minkowski spacetime. So which spacetime geometry is it?

I have no idea more than mine above written. Do we have the spacetime geometry candidtes for rotating universe ?

 

[PeterDonis]

Do we have the spacetime geometry candidtes for rotating universe ?

The only one I know of is the Godel universe.

 

[MikeGomez]

I have a question about a way of avoiding the rotating universe rabbit hole when discussing local inertial effects as conditioned by mass (energy) globally.

Isn't is possible to instead imagine a large, extremely dense, hollow cylinder, fixed at a location relative to the distant stars? The walls of the cylinder would be very thick, and the hollow in the center would be small but large enough to suspend a bucket of water, which itself would be held fixed and not rotating with respect to the distant stars.

Now if we hold the cylinder in a fixed location and bring it up to an extremely rapid spin, wouldn't it induce a sort of frame-dragging effect which would cause the water in the bucket to rotate (and rise) against the inside edge of the bucket?