Inertial vs Gravitational Mass "mystery"?

[jtbell]

if we could control the curvature of spacetime via the electrical charge of an object, then that should grant us control over gravity, no?

I think E&M and gravity would have to be described by two different kinds of spacetime. I suppose you could call them "parallel" spacetimes, but that might have too many science-fiction connotations.

 

[mfb]

Why?

Objects there follow straight lines in spacetime (geodesics). Those lines are just given by spacetime geometry, they are independent of the composition of objects flying through.

Do you have a references for corresponding measurements?

As soon as you find a reference how your proposed deviation is supposed to look like, otherwise it is unclear to me what the measurement would have to test.

For the effect of relativity on moving objects in general:
GPS - tons of papers
Gravity probe B was a satellite specifically designed to test relativistic effects on its test mass.
Earth is moving in the frame of the sun, and our solar system is moving in the frame of the galaxy. Apparently without any effect on gravity that could be seen locally.

 

[TurtleMeister]

The statement that I questioned you about in post #45 specifically states "as used in classical mechanics". Now you are saying:

They are too slow for a significant effect. That requires relativistic velocities.

Classical mechanics does not involve relativistic velocities, so what do you mean by "as used in classical mechanics"?

 

[DrStupid]

Objects there follow straight lines in spacetime (geodesics). Those lines are just given by spacetime geometry, they are independent of the composition of objects flying through.

The question was why a deviation in inertial and gravitational mass is impossible in GR. Could you please answer this question?

As soon as you find a reference how your proposed deviation is supposed to look like, otherwise it is unclear to me what the measurement would have to test.

https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf

GPS

- tons of papers

A single paper would be sufficient.

Gravity probe B was a satellite specifically designed to test relativistic effects on its test mass.

What was the results in regard to the equivalence of inertial and gravitational mass?

 

[DrStupid]

Classical mechanics does not involve relativistic velocities, so what do you mean by "as used in classical mechanics"?

What do you mean with "Classical mechanics does not involve relativistic velocities"? There is no speed limit in classical mechanics.

 

[TurtleMeister]

What do you mean with "Classical mechanics does not involve relativistic velocities"? There is no speed limit in classical mechanics.

No, but it's predictions are not accurate for those cases.

 

[TurtleMeister]

The link that you gave in response to mfb:
https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf
has nothing to do with what we are talking about. Active gravitational mass is an very different subject.

 

[mfb]

Objects there follow straight lines in spacetime (geodesics). Those lines are just given by spacetime geometry, they are independent of the composition of objects flying through.

The question was why a deviation in inertial and gravitational mass is impossible in GR. Could you please answer this question?

The quoted text is an exact answer to this question.

https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf

As far as I can see, this paper just follows GR. Where is the possible deviation?

A single paper would be sufficient.

Good, take the first google result.

What was the results in regard to the equivalence of inertial and gravitational mass?

No deviation from GR at all.

 

[DrStupid]

No, but it's predictions are not accurate for those cases.

That's one of the reasons why the classical definition of gravitational mass can't be used in GR. Thus again: What does it mean in GR?

The link that you gave in response to mfb:
https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf
has nothing to do with what we are talking about. Active gravitational mass is an very different subject.

You should read the whole paper and not only the title. The gravitational mass in the paper refers to Newtonian equations and in classical mechanics active and passive gravitational mass are identical.

The quoted text is an exact answer to this question.

The quoted text doesn't even mention inertial or gravitational mass. Please be so kind to answer the question.

As far as I can see, this paper just follows GR. Where is the possible deviation?

The paper results in a factor of 1+v²/c² between inertial and gravitational mass.

Good, take the first google result.

The abstract doesn't looks like it is about inertial and gravitational mass and I don't want to sign in for the full paper because proving your claims is not my responsibility.

No deviation from GR at all.

And what does GR say about inertial and gravitational mass?

 

[atyy]

As far as I can see, this paper just follows GR. Where is the possible deviation?

I think the problem is that Dr Stupid is taking "inertial mass=gravitational mass" to be exact. It is exact in Newtonian gravity.

However, in GR, inertial mass is not a fundamental quantity, and neither is gravitational mass. One could ask, if a mass follows a geodesic exactly, even if one includes backreaction of its mass on the background. If the backreaction is not included, then its mass is not gravitational.

Usually to avoid this problem, the weak EP is not formulated as "inertial mass=gravitational mass", but closer to what Dr Stupid is referring to as the Galilean equivalence principle. For example, http://relativity.livingreviews.org/Articles/lrr-2006-3/ section 2.1.

Also even if one uses universality of free fall, I think it is only true to some approximation in GR (due to backreaction problems). That's fine, since the EP is only local. However, if one states the EP as minimal coupling, then it is exact in GR.

 

[DrStupid]

I think the problem is that Dr Stupid is taking "inertial mass=gravitational mass" to be exact. It is exact in Newtonian gravity.

Exactly. I have no problems with the Galilean equivalence principle. It is the Newtonian equivalence principle that confuses me in the context of GR. I assume it simply makes no sense in GR but i am not completely sure of that.

 

[Ghost117]

I think E&M and gravity would have to be described by two different kinds of spacetime. I suppose you could call them "parallel" spacetimes, but that might have too many science-fiction connotations.

As long as there ends up being a link from one to the other, so we can find a way to control one through the other (I really want flying cars already!)

 

[mfb]

The quoted text doesn't even mention inertial or gravitational mass. Please be so kind to answer the question.

I did. I explained why all objects (with a negligible mass) move on trajectories independent of their composition. This is equivalent to the equality of inertial and gravitational mass.

The paper results in a factor of 1+v²/c² between inertial and gravitational mass.

It does not calculate the gravitational mass of the passing object at all. It calculates the deflection an object sees in a specific setup and looks "if this would be an object with Newtonian gravity [WHICH IT IS NOT!], what would be its mass".

The abstract doesn't looks like it is about inertial and gravitational mass and I don't want to sign in for the full paper because proving your claims is not my responsibility.

Helping you to understand papers is not my responsibility.

And what does GR say about inertial and gravitational mass?

That they are identical, as mentioned multiple times.

 

[TurtleMeister]

Exactly. I have no problems with the Galilean equivalence principle. It is the Newtonian equivalence principle that confuses me in the context of GR. I assume it simply makes no sense in GR but i am not completely sure of that.

What's the difference between the Galilean equivalence principle and the Newtonian equivalence principle?

 

[Matterwave]

I think the problem is that Dr Stupid is taking "inertial mass=gravitational mass" to be exact. It is exact in Newtonian gravity.

However, in GR, inertial mass is not a fundamental quantity, and neither is gravitational mass. One could ask, if a mass follows a geodesic exactly, even if one includes backreaction of its mass on the background. If the backreaction is not included, then its mass is not gravitational.

Usually to avoid this problem, the weak EP is not formulated as "inertial mass=gravitational mass", but closer to what Dr Stupid is referring to as the Galilean equivalence principle. For example, http://relativity.livingreviews.org/Articles/lrr-2006-3/ section 2.1.

Also even if one uses universality of free fall, I think it is only true to some approximation in GR (due to backreaction problems). That's fine, since the EP is only local. However, if one states the EP as minimal coupling, then it is exact in GR.

Universality of free fall is technically untrue in Newtonian gravity as well, this is not a problem unique to GR. A larger mass attracts the Earth more than a smaller mass, and will fall ever so slightly faster to Earth due to this effect. Consider a Jupiter sized mass "falling to Earth". Only in the limit where the second mass can be ignored w.r.t. the Earth's mass can the strict adherence to any Equivalence principle be followed. But this is not a problem of the equivalence principle, it is an additional complication that we should be mindful about. Similarly, that objects follow geodesics in GR is only approximately true for small objects in a background (much larger) gravitational field. The real equation of motion in GR derives from the condition $\nabla T=0$, the local conservation of stress-energy. The two body problem is not solved analytically in general relativity like it is solved in Newtonian mechanics which gives us more complications.

I don't see any of this as being a problem for the weak equivalence principle; however, because practically speaking, the objects we drop in our experiments ARE negligible in mass (20-24 orders of magnitude smaller) as compared to the Earth.

 

[atyy]

Universality of free fall is technically untrue in Newtonian gravity as well, this is not a problem unique to GR. A larger mass attracts the Earth more than a smaller mass, and will fall ever so slightly faster to Earth due to this effect. Consider a Jupiter sized mass "falling to Earth". Only in the limit where the second mass can be ignored w.r.t. the Earth's mass can the strict adherence to any Equivalence principle be followed. But this is not a problem of the equivalence principle, it is an additional complication that we should be mindful about. Similarly, that objects follow geodesics in GR is only approximately true for small objects in a background (much larger) gravitational field. The real equation of motion in GR derives from the condition $\nabla T=0$, the local conservation of stress-energy. The two body problem is not solved analytically in general relativity like it is solved in Newtonian mechanics which gives us more complications.

I don't see any of this as being a problem for the weak equivalence principle; however, because practically speaking, the objects we drop in our experiments ARE negligible in mass (20-24 orders of magnitude smaller) as compared to the Earth.

But is WEP the universality of free fall, or is it "inertial mass = gravitational mass". If it is the latter, how are inertial mass and gravitational mass defined in GR?

 

[Matterwave]

But is WEP the universality of free fall, or is it "inertial mass = gravitational mass". If it is the latter, how are inertial mass and gravitational mass defined in GR?

I think I have made my position on this matter quite clear in my earlier posts. I do not wish to continue that argument, you can refer to my previous posts to see what I think about this issue.

 

[atyy]

I think I have made my position on this matter quite clear in my earlier posts. I do not wish to continue that argument, you can refer to my previous posts to see what I think about this issue.

Which specific posts?

 

[Matterwave]

Which specific posts?

See posts #35,37,41,44

 

[atyy]

See posts #35,37,41,44

If you define inertial mass and gravitational mass via the Eotvos experiment, are you assuming the Newtonian limit of GR?

 

[Matterwave]

If you define inertial mass and gravitational mass via the Eotvos experiment, are you assuming the Newtonian limit of GR?

Are you saying you expect the Eotvos results to perhaps not hold up if the experiment were conducted near a black hole?

I don't think that we are assuming a Newtonian limit of GR. That experimental physics can make a clear, unambiguous, and concise measurement of the validity of the weak equivalence principle, or "inertial mass = gravitational mass", I think it also provides us a clear, unambiguous definition of the statement "inertial mass = gravitational mass" no matter which theoretical framework you want to work with.

GR takes this principle, and others (incl. principle of relativity, etc.), and makes several conclusions based on them. That the WEP is embedded in the statement "all objects free-fall along geodesics" is clear I think. When you test the WEP, you are certainly testing GR as well, for if it fails, then GR has failed as well.

 

[atyy]

Are you saying you expect the Eotvos results to perhaps not hold up if the experiment were conducted near a black hole?

I don't think that we are assuming a Newtonian limit of GR. That experimental physics can make a clear, unambiguous, and concise measurement of the validity of the weak equivalence principle, or "inertial mass = gravitational mass", I think it also provides us a clear, unambiguous definition of the statement "inertial mass = gravitational mass" no matter which theoretical framework you want to work with.

GR takes this principle, and others (incl. principle of relativity, etc.), and makes several conclusions based on them. That the WEP is embedded in the statement "all objects free-fall along geodesics" is clear I think. When you test the WEP, you are certainly testing GR as well, for if it fails, then GR has failed as well.

No I am not suggesting that anything about the Eotvos experiment results. I would like to understand why you say the Eotvos experiment tests "inertial mass = gravitational mass". In GR, without the Newtonian limit, inertial mass and gravitational mass are not defined. In Newtonian gravity, inertial mass and gravitational mass are well defined. So I can understand that we have the Eotvos experiment, and the Eotvos experiment tests the equality of inertial and gravitational mass within Newtonian gravity. We can also have the Eotvos experiment be a test of GR. But given that without a Newtonian limit, inertial and gravitational mass are not defined in GR, how can the Eotvos experiment test the equality of inertial and gravitational mass in GR?

 

[Matterwave]

No I am not suggesting that anything about the Eotvos experiment results. I would like to understand why you say the Eotvos experiment tests "inertial mass = gravitational mass". In GR, without the Newtonian limit, inertial mass and gravitational mass are not defined. In Newtonian gravity, inertial mass and gravitational mass are well defined. So I can understand that we have the Eotvos experiment, and the Eotvos experiment tests the equality of inertial and gravitational mass within Newtonian gravity. We can also have the Eotvos experiment be a test of GR. But given that without a Newtonian limit, inertial and gravitational mass are not defined in GR, how can the Eotvos experiment test the equality of inertial and gravitational mass in GR?

As I have said, and reiterated, several times now, in GR the statement "inertial mass = gravitational mass" is embedded in the statement "all things free-fall along geodesics". Because GR took "inertial mass = gravitational mass" as its foundational principle, the WEP is automatically satisfied in GR. As I said before, if you don't want to say "inertial mass = gravitational mass", then fine, you can just replace it with the "Weak Equivalence Principle", but I think you will simply be confusing more people than you are helping.

A physical test, a physical experiment, like the Eotvos experiment is CONDUCTED and does NOT require ANY theoretical model to back it up. Such that the phenomena are observed is independent of any THEORY. As such, one can take the concepts tested for in the Eotvos experiments as the experimental foundation for the weak equivalence principle, the statement that I, and many others, prefer to call "inertial mass = gravitational mass". If you have objection to the terminology, then you are free to refer to it as something else as long as people will still be able to understand what you are saying.

 

[atyy]

As I have said, and reiterated, several times now, in GR the statement "inertial mass = gravitational mass" is embedded in the statement "all things free-fall along geodesics". Because GR took "inertial mass = gravitational mass" as its foundational principle, the WEP is automatically satisfied in GR. As I said before, if you don't want to say "inertial mass = gravitational mass", then fine, you can just replace it with the "Weak Equivalence Principle", but I think you will simply be confusing more people than you are helping.

What do you mean "embedded" and "founding principle"? Which quantity in GR is inertial mass? Which quantity in GR is gravitational mass?

A physical test, a physical experiment, like the Eotvos experiment is CONDUCTED and does NOT require ANY theoretical model to back it up. Such that the phenomena are observed is independent of any THEORY. As such, one can take the concepts tested for in the Eotvos experiments as the experimental foundation for the weak equivalence principle, the statement that I, and many others, prefer to call "inertial mass = gravitational mass". If you have objection to the terminology, then you are free to refer to it as something else as long as people will still be able to understand what you are saying.

This conception backs up the idea that you don't mean anything specific by "inertial mass = gravitational mass", since you are just saying this method doesn't require any theoretical concepts. Unless inertial mass and gravitational mass are such self-evident concepts that they are not theoretical, within this conception inertial mass and gravitational mass seem to be meaningless terms since they are theoretical.

 

[atyy]

Exactly. I have no problems with the Galilean equivalence principle. It is the Newtonian equivalence principle that confuses me in the context of GR. I assume it simply makes no sense in GR but i am not completely sure of that.

Here is an interesting comment in a paper by Di Casola, Liberati and Sonego that seems to agree with you. If I understand them correctly, inertial mass and gravitational mass are defined in GR in the Newtonian limit of GR.

http://arxiv.org/abs/1401.0030
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.084053
"Newton’s Equivalence Principle (NEP). In the Newtonian limit, the inertial and gravitational masses of a particle are equal. This formulation makes it possible to test NEP also for theories other than Newton’s. What really matters, in fact, is the notion of Newtonian limit, for the identification of an inertial and a gravitational mass is in general unambiguous only in those conditions."

 

[TurtleMeister]

Universality of free fall is technically untrue in Newtonian gravity as well, this is not a problem unique to GR. A larger mass attracts the Earth more than a smaller mass, and will fall ever so slightly faster to Earth due to this effect. Consider a Jupiter sized mass "falling to Earth". Only in the limit where the second mass can be ignored w.r.t. the Earth's mass can the strict adherence to any Equivalence principle be followed.

That's a common misconception. It's been discussed here at PF before. The UFF will hold true regardless of the mass difference between the two bodies in the two body problem. The misconception is caused by a confusion between the relative acceleration and the acceleration of each body relative to their common center of mass (barycenter). When you change the mass of either body, you also change the relative location of the barycenter. For example, if you increase the mass of a falling body, the barycenter moves closer to the falling body by just the right amount to keep it's acceleration (relative to the barycenter) the same.

It is the acceleration of each body relative to the barycenter that applies to the UFF, not the acceleration of one body to the other.

 

[Matterwave]

That's a common misconception. It's been discussed here at PF before. The UFF will hold true regardless of the mass difference between the two bodies in the two body problem. The misconception is caused by a confusion between the relative acceleration and the acceleration of each body relative to their common center of mass (barycenter). When you change the mass of either body, you also change the relative location of the barycenter. For example, if you increase the mass of a falling body, the barycenter moves closer to the falling body by just the right amount to keep it's acceleration (relative to the barycenter) the same.

It is the acceleration of each body relative to the barycenter that applies to the UFF, not the acceleration of one body to the other.

Fair enough. Since I have not studied closely the full 2-body problem in GR (which I am told has no analytic solution), I will stick to the approximation where one body has negligible mass.

 

[Matterwave]

What do you mean "embedded" and "founding principle"? Which quantity in GR is inertial mass? Which quantity in GR is gravitational mass?

You do not agree that Einstein founded his theory of general relativity starting with the weak equivalence principle?

This conception backs up the idea that you don't mean anything specific by "inertial mass = gravitational mass", since you are just saying this method doesn't require any theoretical concepts. Unless inertial mass and gravitational mass are such self-evident concepts that they are not theoretical, within this conception inertial mass and gravitational mass seem to be meaningless terms since they are theoretical.

Did I not say like 5 times before that if you don't want to say "inertial mass = gravitational mass "you can replace it with "the weak equivalence principle"? Are you saying "the weak equivalence principle" holds no theoretical ground in GR?

 

[atyy]

You do not agree that Einstein founded his theory of general relativity starting with the weak equivalence principle?

Sure, if you would like to talk history, not physics. He also used the principle of general covariance.

Did I not say like 5 times before that if you don't want to say "inertial mass = gravitational mass "you can replace it with "the weak equivalence principle"? Are you saying "the weak equivalence principle" holds no theoretical ground in GR?

What is the weak equivalence principle?

 

[Matterwave]

Sure, if you would like to talk history, not physics. He also used the principle of general covariance.

By this you are implying that general relativity does not obey the weak equivalence principle?

What is the weak equivalence principle?

I like this statement: The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.

Taken from wikipedia: http://en.wikipedia.org/wiki/Equivalence_principle#The_weak_equivalence_principle

 

[atyy]

By this you are implying that general relativity does not obey the weak equivalence principle?

It depends on what you mean by weak equivalence principle. For example, I don't know whether you mean "inertial mass = gravitational mass" or some form of universality of free fall.

I like this statement: The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.

Taken from wikipedia: http://en.wikipedia.org/wiki/Equivalence_principle#The_weak_equivalence_principle

OK, that's basically universality of free fall. As far as I can tell Dr Stupid has no problems with this, which is basically near geodesic motion of a small body in GR. He is only asking whether "inertial mass = gravitational" mass has any meaning in GR. As far as I can tell, it does not have meaning in GR unless one takes the Newtonian approximation, which corresponds to the basic idea that "inertial mass" and "gravitational mass" are Newtonian concepts. However, what was confusing is that you did not agree that the Newtonian limit was necessary for these ideas to make sense in GR.

So if you don't defend "inertial mass = gravitational mass" as a GR concept even without a Newtonian limit, I believe you are agreeing with Dr Stupid.

I think he would say that WEP is UFF (exact, or very good approximation). And that Newtonian gravity uses the technical concept "inertial mass = gravitational mass" to enforce WEP. On the other hand GR uses the technical concept of "minimal coupling in the action" to enforce WEP. Since GR has Newtonian gravity as an approximation, then in that regime GR also has the concept of "inertial mass=gravitational mass".

 

[madness]

A physical test, a physical experiment, like the Eotvos experiment is CONDUCTED and does NOT require ANY theoretical model to back it up. Such that the phenomena are observed is independent of any THEORY.

Not really:

http://en.wikipedia.org/wiki/Theory-ladenness

 

[Matterwave]

OK, that's basically universality of free fall. As far as I can tell Dr Stupid has no problems with this, which is basically near geodesic motion of a small body in GR. He is only asking whether "inertial mass = gravitational" mass has any meaning in GR. As far as I can tell, it does not have meaning in GR unless one takes the Newtonian approximation, which corresponds to the basic idea that "inertial mass and "gravitational mass" are Newtonian concepts. However, what was confusing is that you did not agree that the Newtonian limit was necessary for these ideas to make sense in GR.

So if you don't defend "inertial mass = gravitational mass" as a GR concept even without a Newtonian limit, I believe you are agreeing with Dr Stupid.

I think he would say that WEP is UFF (exact, or very good approximation). And that Newtonian gravity uses the technical concept "inertial mass = gravitational mass" to enforce WEP. On the other hand GR uses the technical concept of "minimal coupling in the action" to enforce WEP. Since GR has Newtonian gravity as an approximation, then in that regime GR also has the concept of "inertial mass=gravitational mass".

Fine. I think my post in #41 was quite clear, and did not make any statements contrary to the statements you made here. I do not think this discussion should have taken this long, and I no longer think this discussion has any fruit to bear. If it was ambiguities in my language at fault, I apologize to Dr. Stupid.

 

[Matterwave]

Not really:

http://en.wikipedia.org/wiki/Theory-ladenness

A discussion of this will bring us far afield and into the philosophy of science, I fear. If you are not comfortable with my statement in that post with regards to a SPECIFIC physical consequence, then please bring it up. If you are just nit-picking my words because a deep discussion of the philosophy of science means that any scientific thought will likely be based on past paradigms and human philosophical assumptions, then I have no interest in discussing that.

My language is not perfect. I am not perfect. I am human, and am prone to make small lapses in coherence of argument. Are we good here?

 

[madness]

A discussion of this will bring us far afield and into the philosophy of science, I fear. If you are not comfortable with my statement in that post with regards to a SPECIFIC physical consequence, then please bring it up. If you are just nit-picking my words because a deep discussion of the philosophy of science means that any scientific thought will likely be based on past paradigms and human philosophical assumptions, then I have no interest in discussing that.

My language is not perfect. I am not perfect. I am human, and am prone to make small lapses in coherence of argument. Are we good here?

Well I don't agree with the nit-picking etc., but I agree not to go into philosophy, so yeah...

 

[atyy]

Fine. I think my post in #41 was quite clear, and did not make any statements contrary to the statements you made here. I do not think this discussion should have taken this long, and I no longer think this discussion has any fruit to bear. If it was ambiguities in my language at fault, I apologize to Dr. Stupid.

Basically, I agree with you to avoid mind-bending intricate discussions of the EP, but I just thought Dr Stupid had a point here, since some standard texts and even quite recent papers like the Di Casola paper I linked to do discuss these things. For example, IIRC MTW has a "medium-strength" EP, which I cannot remember what it is anymore, if I ever understood what the distinction was. I'd personally just go with spin 2 and *derive* the EP (ok, I don't actually know what that means, but apparently it can be done). :)

 

[TurtleMeister]

I've been doing a little research this evening and to the best of my ability it seems that all of the principles we have been discussing (weak equivalence principle, Galilean equivalence principle, and the Newtonian equivalence principle) include the concept of "passive gravitational mass = inertial mass"; possibly the most tested concept in physics.

Personally, I don't see what the "mystery" is. To me the equivalence of active gravitational mass and inertial mass is where the mystery lies. From Wolfgang Rindler's book "Relativity: Special, General, and Cosmological: "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".

 

[atyy]

I've been doing a little research this evening and to the best of my ability it seems that all of the principles we have been discussing (weak equivalence principle, Galilean equivalence principle, and the Newtonian equivalence principle) include the concept of "passive gravitational mass = inertial mass"; possibly the most tested concept in physics.

Personally, I don't see what the "mystery" is. To me the equivalence of active gravitational mass and inertial mass is where the mystery lies. From Wolfgang Rindler's book "Relativity: Special, General, and Cosmological: "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".

Within GR, this fact is put in by hand as universal and minimal coupling between spacetime and matter in the action. Then the mystery becomes: why this universal and minimal coupling? Surprisingly, there is a claimed derivation of this form of the equivalence principle by considering gravity to arise from quantum spin 2. See http://arxiv.org/abs/1007.0435, section 2.2 "The Weinberg low-energy theorem", especially section 2.2.2 "Equivalence principle: the spin-two case".

 

[DrStupid]

What's the difference between the Galilean equivalence principle and the Newtonian equivalence principle?

Galilean equivalence principle: The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.

Newtonian equivalence principle: inertial mass = gravitational mass

And no, they are not equivalent. Within classical mechanics the Newtonian equivalence principle results in the same trajectory for every point mass starting from same position with same velocity. Therefore it full includes the Galilean equivalence principle. But the Newtonian equivalence principle also results in the same acceleration for every point mass at the same position independent from its velocity. The Galilean equivalence principle does not say or require something like that and therefore does not full include the Newtonian equivalence principle. Furthermore this additional consequence of the Newtonian equivalence principle can not be tested with the Eotvos experiment.

 

[TurtleMeister]

Within GR, this fact is put in by hand as universal and minimal coupling between spacetime and matter in the action. Then the mystery becomes: why this universal and minimal coupling? Surprisingly, there is a claimed derivation of this form of the equivalence principle by considering gravity to arise from quantum spin 2. See http://arxiv.org/abs/1007.0435, section 2.2 "The Weinberg low-energy theorem", especially section 2.2.2 "Equivalence principle: the spin-two case".

Thanks for the link atyy, but high energy physics is over my head.

But the Newtonian equivalence principle also results in the same acceleration for every point mass at the same position independent from its velocity.

I've never seen that before. Do you have a reference?

We may just have to agree to disagree DrStupid. I'll have to admit though that I'm not sure I've ever heard of the term "Newtonian equivalence principle" prior to this thread. I've always thought that Newton's experiments were just confirming Galileo's. I know from Newton's Principia that he did pendulum experiments using bobs made of different materials, which in effect would be the same as doing free fall experiments. And the UFF is a consequence of passive gravitational mass = inertial mass. Is there something else that he did that would set his equivalence experiments apart from Galileo's?

 

[DrStupid]

I've never seen that before.

I'm sure you have seen that before:

For constant inertial mass mi Newton's second law of motion results in

F = m_i \cdot a

and according to his law of gravitation the gravitational force acting on a point mass at position r with the gravitational mass mg (exerted by a point mass Mg at position R) is

F = \frac{{G \cdot M_g \cdot m_g \cdot \left( {R - r} \right)}}{{\left| {R - r} \right|^3 }}

With mg = mi this results in the acceleration

a = \frac{{G \cdot M_g \cdot \left( {R - r} \right)}}{{\left| {R - r} \right|^3 }}

which is obviously not only independent from the composition and mass of the body but also from it's velocity. In the publication I linked above Olson and Guarino demonstrated that this is not always the case. The description of the hyperbolic trajectory of a relativistic particle within classical mechanics requires a violation of the Newtonian equivalence principle whereas the Galilean equivalence principle still holds (because the relativistic trajectory is identical for all bodies starting from the same position with the same velocity). That's one of the reasons why the classical definitions of inertial and gravitational mass can't be used in GR.

Another (more fundamental) reason is the incompatibility of Newton's law of gravitation (which the classical gravitational mass is based on). It can't be used in SR because it is not consistent with Lorentz transformation and in GR it makes no sense at all because GR is a theory of gravitation itself and does not need any additional laws for gravity.

That's why I repeatedly asked for the definition of inertial and gravitational mass in GR but I didn't get an answer so far. Without such definitions "inertial mass = gravitational mass" is either false or pointless in GR.

I've always thought that Newton's experiments were just confirming Galileo's.

Of course they were. His experiments wasn't suitable for the detection of relativistic effects and within its scope classical mechanics is full consistent with the Galilean equivalence principle (I guess that's what the Newtonian equivalence principle was intended for).

 

[TurtleMeister]

Thanks for the clarification. I understand your argument now. The problem is that I've been talking classical mechanics and experimental evidence and you guys have been talking relativity and theory (which may be off topic because this is the classical physics forum). My knowledge in SR and GR is limited, so I'm reluctant to continue in that discussion.

I don't have much time during the week so I'll try to read through the publication that you linked to this weekend. Just looking through it quickly I see that it may be outside my ability to understand in a reasonable amount of time. But the subject matter is interesting to me, so I'll give it a try.

 

[TurtleMeister]

@DrStupid

I've finished reading the article that you linked "Measuring the active gravitational mass of a moving object". Briefly, here is what I got out of it: Particle m located at distance b below the x-axis is influenced by the active gravitational mass of particle M moving past it along the x-axis at relativistic velocity. This influence, or final velocity of particle m is first calculated using Newtonian equations. It is then calculated using relativistic equations.

I'm not sure how this applies to the equivalence principle. There was no mention of the equivalence principle by the author. A test particle was used for m because of the complications explained in section "III.Relativistic Equations". There were no material properties stated for m and M other than m<<M.

I do not find it surprising that the predictions of the relativistic equations differ from the predictions of the Newtonian equations. But the author states in section "IV.Definitions of mass in general relativity", "This is a natural problem to pose, and the result obtained is somewhat surprising". So maybe there is something I am missing.

 

[DrStupid]

I'm not sure how this applies to the equivalence principle. There was no mention of the equivalence principle by the author.

The paper summarises the result as follows:
"Roughly speaking, the gamma factor in Eq. (10) comes from special relativity and the Lorentz transformation, while the (1+ß²) factor comes from general relativity and the equations for geodesic bending. In the ultrarelativistic limit, the (1+ß²) factor approaches the value 2 and is then the same famous factor by which the general relativistic prediction for light bending exceeds the Newtonian prediction."

The information about inertial and gravitational mass is hidden in the last sense. According to Olsen & Guarino the gravitational mass of relativistic particles increases with (1+ß²)·gamma·m (in classical mechanics active and passive gravitational mass need to be equal). In order to get the correct result for the deflection of ultrarelativistic particles with classical mechanics their inertial mass must increase with gamma·m (this is usually meant by "relativistic mass"). In this special case "gravitational mass = inertial mass" turns into "gravitational mass = (1+ß²) x inertial mass".

That means if classical mechanics is used to describe the relativistic results than Newton’s equivalence principle must be violated (but not the Gelilean equivalence principle).
If classical mechanics is not be used than the classical definitions of inertial and gravitational mass must not be used either. Therewith we again return to the question for definitions of inertial and gravitational mass in GR.

There were no material properties stated for m and M other than m<<M.

Why do you limit "gravitational mass = iniertial mass" to bodies with different material properties only? The paper shows that the Newtonian equivalence principle can be violated for bodies with the same composition but different velocities without violating the Galilean equivalence principle as well as in accordance with experimental observations and the predictions of GR. I think that is what the authors was surprised about.

 

[TurtleMeister]

I will give a full reply to your post later. But for right now I would like to ask a favor. In the quote below you mention gravitational mass four times without specifying active or passive. Would you please insert the correct term?

Thanks

According to Olsen & Guarino the gravitational mass of relativistic particles increases with (1+ß²)·gamma·m (in classical mechanics active and passive gravitational mass need to be equal). In order to get the correct result for the deflection of ultrarelativistic particles with classical mechanics their inertial mass must increase with gamma·m (this is usually meant by "relativistic mass"). In this special case "gravitational mass = inertial mass" turns into "gravitational mass = (1+ß²) x inertial mass".

 

[DrStupid]

In the quote below you mention gravitational mass four times without specifying active or passive.

In this quote I also mentioned that active and passive gravitational mass need to be equal in classical mechanics. When Newton published his law of gravitation he didn't distinguished between these properties. He speaks about inertial mass ("quantity of matter") and gravitational mass ("weight") only (just to declare that he assumed them to be always equal due to experimental observations). But let's assume there are such things like active and passive gravitational mass in classical mechanics and the gravitational force acting on a body 1 in the gravitational field of a body 2 two would depend on the passive gravitational mass m_p1 of body 1 and the active gravitational mass m_a2 of body 2:

F_1 = G \cdot m_{p1} \cdot m_{a2} \cdot \frac{{r_2 - r_1 }}{{\left| {r_2 - r_1 } \right|^3 }}

In an analogous manner the gravitational force acting on body 2 depends on the passive gravitational mass m_p2 of body 2 and the active gravitational mass m_a1 of body 1

F_2 = G \cdot m_{a1} \cdot m_{p2} \cdot \frac{{r_1 - r_2 }}{{\left| {r_1 - r_2 } \right|^3 }}

Newton's third law of motion now requires that both forces adds to zero:

F_1 + F_2 = G \cdot \left( {m_{a1} \cdot m_{p1} - m_{p1} \cdot m_{a1} } \right) \cdot \frac{{r_1 - r_2 }}{{\left| {r_1 - r_2 } \right|^3 }} = 0

In order to guarantee this condition under all circumstances, the ratio

k = \frac{{m_{a1} }}{{m_{p1} }} = \frac{{m_{a2} }}{{m_{p2} }}

between active and passive gravitational mass must be identical for all bodies and Occam's Razor requires to include it into the gravitational constant. Therefore I do not need to specify active or passive gravitational mass.

Maybe this is different in relativity but we will never know without the definition of gravitational mass in GR.

 

[TurtleMeister]

I am familiar with the equivalence of active and passive gravitational mass being a consequence of Newton's third law. But thanks for writing it out for me. By the way, there is a typo in your third line of equations. The part in parentheses should be $m_{a1} \cdot m_{p2} - m_{p1} \cdot m_{a2}$.
So, should I take the gravitational mass in the quote to be both active and passive? I'm just trying to understand the paper that you cited.

Why do you limit "gravitational mass = iniertial mass" to bodies with different material properties only? The paper shows that the Newtonian equivalence principle can be violated for bodies with the same composition but different velocities without violating the Galilean equivalence principle as well as in accordance with experimental observations and the predictions of GR. I think that is what the authors was surprised about.

I am not limiting the equality to material differences. I was just pointing out that this paper does not include that. Most equivalence principle experiments that I am familiar with do include it, so this is something new to me. I'm beginning to think that this is just a study of the difference between the Galileo version and the Newtonian version of the equivalence principle. The Newtonian version is simply a more complete version. It does not mean that there is some kind of conflict. Since the Galileo version does not include the velocity factor it should not be surprising that it's statement is not violated in the scenario of the cited paper. Interesting maybe, but not surprising. And it is not surprising that the Newtonian version is violated because we cannot expect it to be accurate when dealing with relativistic velocities.

Maybe this is different in relativity but we will never know without the definition of gravitational mass in GR.

I can sympathize with you on this. Maybe there is no definition for gravitational mass outside the Newtonian limit of GR. But like I said before, I am not very knowledgeable in SR and GR, so I should not even comment on it.

 

[DrStupid]

So, should I take the gravitational mass in the quote to be both active and passive?

Yes. I do not know why Olsen & Guarino limited their paper to the active gravitational mass. Maybe they wanted to point out that they considered the gravitational field of M only and neglected the gravitational field of the test mass. But that's just a speculation.

I'm beginning to think that this is just a study of the difference between the Galileo version and the Newtonian version of the equivalence principle.

Yes, that's what my original question is all about. I often read (not only in this thread or this forum) that one result from the other or that both are equivalent but that does not apply to classical mechanics and I have never seen a corresponding derivation for GR.

The Newtonian version is simply a more complete version.

The Galilean equivalence principle is complete enough. The Newtonian equivalence principle was just the easiest way to include the Galilean equivalence principle into classical mechanics but it overshoots the mark. That was no problem as long as the additional implications wasn't testable experimentally. But under relativistic conditions the Newtonian equivalence principle appears to be wrong whereas the Galilean equivalence principle still holds.

Since the Galileo version does not include the velocity factor it should not be surprising that it's statement is not violated in the scenario of the cited paper. Interesting maybe, but not surprising. And it is not surprising that the Newtonian version is violated because we cannot expect it to be accurate when dealing with relativistic velocities.

That the Newtonian equivalence principle can be violated within the limits of the Galilean equivalence principle might not be surprising but it shows that they are not equivalent.

 

[TurtleMeister]

Yes. I do not know why Olsen & Guarino limited their paper to the active gravitational mass. Maybe they wanted to point out that they considered the gravitational field of M only and neglected the gravitational field of the test mass. But that's just a speculation.

Yes, I believe that is what they intended. In the Olsen & Guarino's scenario, M plays the role of active gravitational mass and the test mass m plays the role of passive gravitational mass. This is the case because m<<M.

That the Newtonian equivalence principle can be violated within the limits of the Galilean equivalence principle might not be surprising but it shows that they are not equivalent.

But only under conditions of relativistic velocities. And that's the reason I do not find it surprising.

 

[William Jackson]

My lay understanding must be even poorer than I thought. I've understood Einstein's explanation to mean that what we experience as gravity is a form of inertia. And that the reason gravitational mass and inertial mass are the same, is that gravity and inertia are both instances of the same characteristic. And that some forms of mass-energy display this characteristic while others do not. Those that have that characteristic, we think of as "stuff" and those that do not, we think of as "energy".

Is that completely wrong-headed?