Inertial vs Gravitational Mass "mystery"?

[Ghost117]

Hi everyone,

I read in a first year textbook (K&K) that the reason why "gravitational mass is proportional to inertial mass" is a big "mystery"...

Can someone please explain why this is a mystery?

Thanks

 

[A.T.]

Can someone please explain why this is a mystery?

Because we don't know it.

 

[dextercioby]

It's one of the postulates of classical Newtonian mechanics. Can't really do without.

 

[Ghost117]

Why is inertial mass "operationally distinct" from gravitational mass in the first place? Isn't mass just the amount of 'stuff' that makes up an object? All equations we input mass into require the same Kg value, don't they?

 

[A.T.]

Why is inertial mass "operationally distinct" from gravitational mass in the first place? Isn't mass just the amount of 'stuff' that makes up an object?

Replacing the word "mass" with the word "stuff" doesn't offer any new insight. You just have "gravitational stuff" vs. "inertial stuff".

 

[mfb]

Why is inertial mass "operationally distinct" from gravitational mass in the first place? Isn't mass just the amount of 'stuff' that makes up an object? All equations we input mass into require the same Kg value, don't they?

Compare it to electromagnetism, where the "gravitational charge" is replaced by the electric charge, and inertial mass is still inertial mass. There is no special reason why electric charge and mass should be proportional - and in fact they are not. Why is this different for gravity?

 

[Ghost117]

@A.T.

That's my point though, why make the distinction at all? But mfb's question is starting to clarify this...

@mfb
I think I'm starting to get it... So you're really asking: why is M present in both

and

while it's absent from the formula for electric charge?

 

[A.T.]

You're really asking: why is M present in both

and

while it's absent from the formula for electric charge?

Or, why is charge absent from the formula for gravitational attraction?

 

[A.T.]

That's my point though, why make the distinction at all?

Why don't make a distinction, because observation tells us they are the same. But it doesn't tell why they are the same.

 

[Ghost117]

"Or, why is charge absent from the formula for gravitational attraction?"​

Hmm... because Gravity isn't unified with Electromagnetism (yet)... ?

But even if it was, wouldn't the same question exist? Couldn't we just ask "why is M present in all 3 formulas" ?

... I guess maybe that's the mystery: why should the same value exist in completely separate formulas at all...

If that is the mystery, that's definitely interesting

 

[A.T.]

But even if it was, wouldn't the same question exist?

Yes, the "why?" is always a "mystery" in physics.

 

[Ghost117]

hmmm... yea, it's definitely one of those mysteries then...

Thanks everyone :)

 

[MrRobotoToo]

General relativity provides a resolution of sorts.

Q: Why do all objects fall identically in the same gravitational field, regardless of their mass?

A: Because they're following a geodesic through curved spacetime. A geodesic is a geodesic is a geodesic: it doesn't depend on the peculiarities of the objects that move along it.

 

[Ghost117]

General relativity provides a resolution of sorts.

Q: Why do all objects fall identically in the same gravitational field, regardless of their mass?

A: Because they're following a geodesic through curved spacetime. A geodesic is a geodesic is a geodesic: it doesn't depend on the peculiarities of the objects that move along it.

I guess I'll put this question on the shelf, till I get to GR...

 

[Matterwave]

General relativity provides a resolution of sorts.

Q: Why do all objects fall identically in the same gravitational field, regardless of their mass?

A: Because they're following a geodesic through curved spacetime. A geodesic is a geodesic is a geodesic: it doesn't depend on the peculiarities of the objects that move along it.

That gravity can be modeled by space-time curvature is a direct consequence of the equivalence principle (inertial mass = gravitational mass), so it's really more of a chicken-or-egg thing in GR.

 

[Buckleymanor]

Replacing the word "mass" with the word "stuff" doesn't offer any new insight. You just have "gravitational stuff" vs. "inertial stuff".

Ghost117 goes onto say in his post that all the equations we input mass into require the same KG values.May-be you missed that or don't consider that this allso offers any insight.
If you consider that inertial mass is measured by the KG values that we impose upon them which in turn is from the gravitational values we have found by weighing them using KG, then it's not really surprising that there is no difference between the two.
You might get some subtle arguments that say that inertial and gravitational mass could be different depending where in the universe they are measured.
It's doubtfull if the local unit of measurment is used whatever that might be:)

 

[harrylin]

"Or, why is charge absent from the formula for gravitational attraction?"​

[..] Couldn't we just ask "why is M present in all 3 formulas" ?
... I guess maybe that's the mystery: why should the same value exist in completely separate formulas at all...
If that is the mystery, that's definitely interesting

Ghost117 goes onto say in his post that all the equations we input mass into require the same KG values.[..] If you consider that inertial mass is measured by the KG values that we impose upon them which in turn is from the gravitational values we have found by weighing them using KG, then it's not really surprising that there is no difference between the two.[..]

Indeed - it's also a mystery to me, why matter has both the properties of gravitation and inertia.
However, with that given, I cannot find a mystery in two stones weighing twice as much as one stone, and also resisting acceleration twice as much. Consequently Newton assumed (argued) that they are proportional. Thus it remains a mystery to me why, at least in classical mechanics, for some people the proportionality is a mystery - or even a "big mystery"...

 

[DrStupid]

That gravity can be modeled by space-time curvature is a direct consequence of the equivalence principle (inertial mass = gravitational mass)

Can you explain this within GR?

 

[Matterwave]

Can you explain this within GR?

What do you mean? Do you mean "can GR explain the equivalence principle?" In that case, no, the equivalence principle is a foundational principle of GR, and just like the principle of relativity is not explained by SR, but simply taken as experimental fact, the equivalence principle is similarly taken as experimental fact in GR.

However, I can explain why equivalence principle leads to space-time curvature model of gravity. This is because all things fall along geodesics. There can be no "gravity shielding" like there can be for the E&M force (neutral particles are not affected by it) and so the equivalence principle gives you the ability to formulate a theory in which all things fall in the same way. In this way, the theory of gravity can be modeled as intrinsic to the background gravitational field and independent of the object on which it is acting. If you tried, for example, to formulate a space-time curvature formulation of E&M, you would have to explain why different things fall differently...and so the space-time curvature would be different depending on the charge of the object that is falling through it. This would not be a very useful theory.

 

[Ghost117]

If you tried, for example, to formulate a space-time curvature formulation of E&M, you would have to explain why different things fall differently...and so the space-time curvature would be different depending on the charge of the object that is falling through it. This would not be a very useful theory.

But wouldn't that allow control over 'gravity'? I mean, if we could control the curvature of spacetime via the electrical charge of an object, then that should grant us control over gravity, no? ... Or maybe I'm misunderstanding what you said?

 

[DrStupid]

What do you mean?

I mean a closed line of argument from "inertial mass = gravitational mass" to "gravity can be modeled by space-time curvature" without using classical mechanics. You could start with the definitions of inertial and gravitational mass in GR.

However, I can explain why equivalence principle leads to space-time curvature model of gravity. This is because all things fall along geodesics.

Where does this comes from? Does it result from the weak equivalence principle and if yes how?

There can be no "gravity shielding" like there can be for the E&M force

ditto

and so the equivalence principle gives you the ability to formulate a theory in which all things fall in the same way.

Is the equivalence principle sufficient for this purpose or do I need something else?

 

[mfb]

But wouldn't that allow control over 'gravity'? I mean, if we could control the curvature of spacetime via the electrical charge of an object, then that should grant us control over gravity, no? ... Or maybe I'm misunderstanding what you said?

Do you have "control" over gravity just because you can move masses around? This is the same question.

 

[Ghost117]

Do you have "control" over gravity just because you can move masses around? This is the same question.

No, but if we could control the curvature of spacetime with electromagnetic forces, then we wouldn't need to move masses around, and this would give us control over gravity, no? Wouldn't this be a direct consequence of what Matterwave described as " a space-time curvature formulation of E&M" ?

 

[Matterwave]

But wouldn't that allow control over 'gravity'? I mean, if we could control the curvature of spacetime via the electrical charge of an object, then that should grant us control over gravity, no? ... Or maybe I'm misunderstanding what you said?

You misunderstood what I said. My point was not that the E&M force allows us to "control space-time curvature", it's that the E&M force is not a good candidate to be described by a "space-time curvature" theory. Unlike the gravity.

 

[Ghost117]

Thanks for clarifying Matterwave

 

[Matterwave]

I mean a closed line of argument from "inertial mass = gravitational mass" to "gravity can be modeled by space-time curvature" without using classical mechanics. You could start with the definitions of inertial and gravitational mass in GR.
Where does this comes from? Does it result from the weak equivalence principle and if yes how?
ditto
Is the equivalence principle sufficient for this purpose or do I need something else?

That gravity can be modeled by a "space-time curvature" theory requires the stronger form of the equivalence principle, either the strong equivalence principle, or at least the Einstein equivalence principle. You need not only that masses fall the same way, but that other fields behave the same way under the influences of local gravity as they do under acceleration. We must not be able to tell with local measurements whether our (boxed off from the outside) laboratory is floating in the vacuum of space or if its free-falling in a gravitational field. This requires things like light (in terms of single streaming photons), which has no inertial mass and no gravitational mass, to fall the same way when it is in a gravitational field.

Whether the equivalence (strong) is sufficient to derive a "space-time curvature" theory of gravity, depends on the exact specifics of what we mean by a "space-time curvature theory". It is not sufficient to derive GR, for example, because GR also relies on the principle of (local) relativity, as well as the principle of the constancy of the (local) speed of light (the postulates of SR). In addition, the constants within the Einstein Field Equations are derived from matching GR to Newtonian gravity in the weak field limit. So, the equivalence principle itself is certainly not enough to derive GR. However, the equivalence principle gives us a strong indication that some "space-time curvature" theory is possible. It won't narrow down which of a whole class of theories is correct; however. There are different "curvature" theories that obey the equivalence principle.

 

[DrStupid]

That gravity can be modeled by a "space-time curvature" theory requires the stronger form of the equivalence principle, either the strong equivalence principle, or at least the Einstein equivalence principle.

That means the original statement was wrong?

 

[David Carroll]

Is it a "mystery" because it is paradoxical that two massive objects attracted toward each other should both experience resistance to their respective accelerations due to their masses and experience increased attraction due to their masses? It's not paradoxical for the other fields because given a field strength, more mass resists acceleration quite simply, I imagine. Does inertial mass even figure into equations for force between two massive objects gravitationally attracted to each other?

 

[Matterwave]

That means the original statement was wrong?

If you insist that I was wrong when I said "equivalence principle" and not "the Einstein equivalence principle", then sure, I was wrong. :)

 

[atyy]

If you insist that I was wrong when I said "equivalence principle" and not "the Einstein equivalence principle", then sure, I was wrong. :)

Are there theories that satisfy the weak EP and are not metric theories?

 

[Matterwave]

Are there theories that satisfy the weak EP and are not metric theories?

I think...Newton's theory of gravity and his laws of motion? I mean that's where the weak EP originated anyways.

But I think people are reading too much into what I'm saying. I was trying to only say that the (strong) EP gives us an inclination that a space-time curvature theory is possible, and useful. I don't think I said the EP means that ONLY a space-time curvature theory is possible. This is certainly false, since one can model GR as a spin-2 tensor field with non-linear field equations on top of a flat background space-time... If I gave anybody that impression, then I apologize for being unclear.

 

[atyy]

I think...Newton's theory of gravity and his laws of motion? I mean that's where the weak EP originated anyways.

But I think people are reading too much into what I'm saying. I was trying to only say that the (strong) EP gives us an inclination that a space-time curvature theory is possible, and useful. I don't think I said the EP means that ONLY a space-time curvature theory is possible. This is certainly false, since one can model GR as a spin-2 tensor field with non-linear field equations on top of a flat background space-time... If I gave anybody that impression, then I apologize for being unclear.

I thought Newton-Cartan theory? http://en.wikipedia.org/wiki/Newton–Cartan_theory

Maybe you are "wrong", but not for the right reason?

 

[Matterwave]

I thought Newton-Cartan theory? http://en.wikipedia.org/wiki/Newton–Cartan_theory

Maybe you are "wrong", but not for the right reason?

I'm not sure what that means. XD

 

[DrStupid]

If you insist that I was wrong when I said "equivalence principle" and not "the Einstein equivalence principle", then sure, I was wrong. :)

That alone wouldn't be a problem. It was the addition "(inertial mass = gravitational mass)" that confused me in this context. I don't even know what that means in GR.

 

[Matterwave]

That alone wouldn't be a problem. It was the addition "(inertial mass = gravitational mass)" that confused me in this context. I don't even know what that means in GR.

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.

This is the form of the equivalence principle that is so famously tested in the Eötvös experiment:

http://en.wikipedia.org/wiki/Eötvös_experiment
http://en.wikipedia.org/wiki/Equivalence_principle#Tests_of_the_weak_equivalence_principle

 

[DrStupid]

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?

 

[Matterwave]

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.

 

[atyy]

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.

 

[Matterwave]

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.

 

[DrStupid]

I'm not understanding your concern.

What is gravitational mass?

 

[Matterwave]

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.

 

[DrStupid]

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?

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.

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.

 

[atyy]

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.)

 

[Matterwave]

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. :)

 

[TurtleMeister]

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?

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.

 

[DrStupid]

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.

 

[mfb]

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.

 

[DrStupid]

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.

Anyway, satellites are doing this.

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

 

[mfb]

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.

 

[DrStupid]

In GR a deviation is impossible

Why?

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?