Inertial vs Gravitational Mass "mystery"?

In summary, the conversation discussed the mystery surrounding the proportionality of gravitational mass and inertial mass. Some argued that it is a big mystery while others pointed out that it is a postulate in classical mechanics. It was also mentioned that General Relativity provides a resolution by modeling gravity as space-time curvature, which is a consequence of the equivalence principle. However, some still find it mysterious that matter has both the properties of gravitation and inertia.
  • #36
Matterwave said:
That IS the weak equivalence principle. It means that two (small) objects will fall at the same rate when they are in the same gravitational field regardless of their mass.

Yes, of course this is the weak equivalence principle and in classical mechanics it does exactly what you say. But what does it mean in GR?
 
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  • #37
DrStupid said:
Yes, of course this is the weak equivalence principle and in classical mechanics it does exactly what you say. But what does it mean in GR?

? I'm not understanding your concern. In GR it means the same thing. GR is a physical theory, just like classical mechanics, and it has to be in line with the physical phenomena that we see and the physical experiments that we conduct. That we experimentally determined the WEP (inertial mass = gravitational mass) is theoretically embedded in the GR statement that all objects, regardless of inertial mass, fall along geodesics.
 
  • #38
DrStupid said:
Yes, of course this is the weak equivalence principle and in classical mechanics it does exactly what you say. But what does it mean in GR?

Actually, your point is a good point - I don't think we can really prove "equivalence principle = geometry".

However, there are 3 famous examples:
(1) Newtonian gravity
(2) Nordstrom gravity
(3) Einstein gravity.

These are all physically accurate theories of gravity in the Newtonian domain. Also, they can be written as fields on spacetime, or as spacetime curvature. Newtonian gravity is historically the first successful theory of universal gravity. After Maxwell and special relativity, it was realized that Newtonian gravity had to be modified to be consistent with special relativity. Nordstrom's theory was the first theory of gravity that was consistent with special relativity. Einstein, believing that the equivalence principle meant a geometric formulation was possible, together with Fokker reformulated Nordstrom's theory as a geometric theory. Later Einstein came up with general relativity which is a geometric theory. Cartan then tried to see if Einstein's idea could be applied to Newtonian gravity, since that too obeys an equiavlence principle. The resulting geometric formulation is Newton-Cartan theory.
 
  • #39
atyy said:
Actually, your point is a good point - I don't think we can really prove "equivalence principle = geometry".

However, there are 3 famous examples:
(1) Newtonian gravity
(2) Nordstrom gravity
(3) Einstein gravity.

These are all physically accurate theories of gravity in the Newtonian domain. Also, they can be written as fields on spacetime, or as spacetime curvature. Newtonian gravity is historically the first successful theory of universal gravity. After Maxwell and special relativity, it was realized that Newtonian gravity had to be modified to be consistent with special relativity. Nordstrom's theory was the first theory of gravity that was consistent with special relativity. Einstein, believing that the equivalence principle meant a geometric formulation was possible, together with Fokker reformulated Nordstrom's theory as a geometric theory. Later Einstein came up with general relativity which is a geometric theory. Cartan then tried to see if Einstein's idea could be applied to Newtonian gravity, since that too obeys an equiavlence principle. The resulting geometric formulation is Newton-Cartan theory.

What this Dr Stupid's point this whole time? I...did not get that from his posts.

Anyways, even though we perhaps can not say equivalence principle = geometry, we can say equivalence principle gives us motivation to look for a geometrical theory, while a lack of an equivalence principle (e.g. in E&M) discourages us strongly from looking for a geometrical (in the curvature of space-time sense, not in the tensor fields on top of background space-time sense) theory since different particles do not fall along the same trajectories/geodesics.
 
  • #40
Matterwave said:
I'm not understanding your concern.

What is gravitational mass?
 
  • #41
DrStupid said:
What is gravitational mass?

Ah, ok, that is a valid question. In this context, it is simply the "concept" associated with the rate at which (small) objects fall. Because "gravitational mass = inertial mass" is an assumption of the theory of general relativity, the concept of gravitational mass itself has been subsumed into the framework of the theory. It is manifest, as I mentioned earlier, in the statement that all objects (with the same initial four-velocity) fall along the same geodesics (if they started from the same position too of course). As such, within the theory itself, the assumption "gravitational mass = inertial mass" can not be questioned, it is embedded deeply in the concept of geodesics of space-time, it is rather like a mathematical axiom. However, in physics, we can't choose our axioms arbitrarily, we must test for them. A test on the equivalence principle, like the Eotvos experiment, is then also a test on GR. From an experimental point of view the statement "gravitational mass = inertial mass" is clear, and as such, it is also clear what it means for the theory of general relativity because GR must, in the end, be tested experimentally.

If you are not comfortable with the statement "gravitational mass = inertial mass", I would say simply think of the Eotvos experiment. That general relativity is in accord with the Eotvos experiment IS the statement in GR that "gravitational mass = inertial mass". We can define that statement operationally. If you are still uncomfortable with that statement, then simply switch it to "weak equivalence principle", but that just makes things harder to understand if you ask me.
 
  • #42
Matterwave said:
In this context, it is simply the "concept" associated with the rate at which (small) objects fall.

That sounds rather vague. The equation gravitational mass = inertial mass is meaningless without quantitative definitions of both properties. Such definitions are given in classical mechanics. But as the classical definition of gravitational mass is based on Newton's law of gravitation it can't be used in GR. So how is gravitational mass defined in GR?

Matterwave said:
Because "gravitational mass = inertial mass" is an assumption of the theory of general relativity, the concept of gravitational mass itself has been subsumed into the framework of the theory. It is manifest, as I mentioned earlier, in the statement that all objects (with the same initial four-velocity) fall along the same geodesics (if they started from the same position too of course).

How do you come from "gravitational mass = inertial mass" to the Galilean equivalence principle? In classical mechanics the latter one is full included into the first one but not vice versa. Can you show that the Galilean equivalence principle is a valid replacement for "gravitational mass = inertial mass" in GR.

Matterwave said:
If you are not comfortable with the statement "gravitational mass = inertial mass", I would say simply think of the Eotvos experiment.

As Eotvos experiment compares bodies with equal (or at least similar) velocities, it is sufficient to proof the Galilean equivalence principle. But as it does not compare bodies with very different velocities it is not sufficient to proof "gravitational mass = inertial mass" as used in classical mechanics. Whether this problem also applies to GR or not depends on the role of gravitational mass in GR. Therefore it is important to know the quantitative definition of gravitational mass within GR.
 
  • #43
Matterwave said:
What this Dr Stupid's point this whole time? I...did not get that from his posts.

Anyways, even though we perhaps can not say equivalence principle = geometry, we can say equivalence principle gives us motivation to look for a geometrical theory, while a lack of an equivalence principle (e.g. in E&M) discourages us strongly from looking for a geometrical (in the curvature of space-time sense, not in the tensor fields on top of background space-time sense) theory since different particles do not fall along the same trajectories/geodesics.

I'm not sure what his point is, but I think they are generally valid points based on an ultra-refined classification of equivalence principles. It's interesting if one doesn't have a theory of gravity that works, but since physicists believe in GR so much nowadays, they (and me) tend to take it as the axiom, rather than the ultra-refined classification of equivalence principles that may guide one if one doesn't have a candidate theory. (In fact, I think many would even go with quantum spin 2 as axiomatic, rather than classical GR.)
 
  • #44
Well, guys, if you all think I'm wrong in saying inertial mass = gravitational mass, then fine, I'll accept that. I am not here for a debate on the intricacies of the (several different) equivalence principles and such. It is my opinion that a purely operational definition of the concept "gravitational mass = inertial mass", as elucidated by the Eotvos experiment, is quite clear and unambiguous, and since it is a part of experimental science and not theoretical science, it does not depend on which (valid) theory you are looking at, but if you do not accept it as such, then that is your choice. :)
 
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  • #45
DrStupid said:
As Eotvos experiment compares bodies with equal (or at least similar) velocities, it is sufficient to proof the Galilean equivalence principle. But as it does not compare bodies with very different velocities it is not sufficient to proof "gravitational mass = inertial mass" as used in classical mechanics.
Why do you think that a comparison between bodies with different velocities is required to test the equivalence of passive gravitational mass and inertial mass?

Matterwave said:
It is my opinion that a purely operational definition of the concept "gravitational mass = inertial mass", as elucidated by the Eotvos experiment, is quite clear and unambiguous, and since it is a part of experimental science and not theoretical science, it does not depend on which (valid) theory you are looking at, but if you do not accept it as such, then that is your choice. :)
I agree. This concept might be one of the most tested concepts in physics. It wasn't just Eotvos. Newton was the first when he experimented with pendulum periods using bobs made of different materials. Today we have The EotWash Group, and in the future we will have satellite experiments.
 
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  • #46
TurtleMeister said:
Why do you think that a comparison between bodies with different velocities is required to test the equivalence of passive gravitational mass and inertial mass?

Because you can't be sure that the equivalence of gravitational mass and inertial mass is valid for bodies with different velocities without testing it. That's how physics work.
 
  • #47
DrStupid said:
Because you can't be sure that the equivalence of gravitational mass and inertial mass is valid for bodies with different velocities without testing it. That's how physics work.
How would a deviation look like?

Anyway, satellites are doing this.
 
  • #48
mfb said:
How would a deviation look like?

In classical mechanics a velocity dependent ratio between inertial and gravitational mass results in a velocity dependent acceleration of a free falling body. I can't say what it will look like in GR because I do not know what "gravitational mass" meas there.

mfb said:
Anyway, satellites are doing this.

They are too slow for a significant effect. That requires relativistic velocities.
 
  • #49
DrStupid said:
In classical mechanics a velocity dependent ratio between inertial and gravitational mass results in a velocity dependent acceleration of a free falling body. I can't say what it will look like in GR because I do not know what "gravitational mass" meas there.
In GR a deviation is impossible, so that's pointless.

They are too slow for a significant effect. That requires relativistic velocities.
Our measurements are precise enough for a significant effect. Relativistic effects (both from motion and gravitational time dilation) are measured with high precision.
 
  • #50
mfb said:
In GR a deviation is impossible

Why?

mfb said:
Our measurements are precise enough for a significant effect. Relativistic effects (both from motion and gravitational time dilation) are measured with high precision.

Do you have a references for corresponding measurements?
 
  • #51
Ghost117 said:
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. :eek:
 
  • #52
DrStupid said:
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.
 
  • #53
The statement that I questioned you about in post #45 specifically states "as used in classical mechanics". Now you are saying:
DrStupid said:
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"?
 
  • #54
mfb said:
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?

mfb said:
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

mfb said:
GPS
mfb said:
- tons of papers

A single paper would be sufficient.

mfb said:
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?
 
  • #55
TurtleMeister said:
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.
 
  • #56
DrStupid said:
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.
 
  • #58
DrStupid said:
mfb said:
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.
 
  • #59
TurtleMeister said:
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?

TurtleMeister said:
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.

mfb said:
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.

mfb said:
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.

mfb said:
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.

mfb said:
No deviation from GR at all.

And what does GR say about inertial and gravitational mass?
 
  • #60
mfb said:
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.
 
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  • #61
atyy said:
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.
 
  • #62
jtbell said:
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. :eek:

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!)
 
  • #63
DrStupid said:
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".
DrStupid said:
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.
DrStupid said:
And what does GR say about inertial and gravitational mass?
That they are identical, as mentioned multiple times.
 
  • #64
DrStupid said:
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?
 
  • #65
atyy said:
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.
 
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  • #66
Matterwave said:
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?
 
  • #67
atyy said:
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.
 
  • #68
Matterwave said:
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?
 
  • #69
atyy said:
Which specific posts?

See posts #35,37,41,44
 
  • #70
Matterwave said:
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?
 

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