Inertial vs Gravitational Mass "mystery"?

[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?

 

[DrStupid]

But only under conditions of relativistic velocities.

I wouldn't bet on it. However, a single counter example is sufficient for a falsification.

 

[Tom Potts]

The trouble with it is the word 'equivalence'. Einstein used the term because inertial and gravitational masses could be different properties. It looks very much like they are not equivalent but the same.
This is not a problem for GR as its built in as a core principle. Gravitational Mass === Inertial Mass. They are one and the same property but viewed from different human perceptions of reality.
There is no problem with it - it is how other scientific theories approach Mass that lead to some assuming there is a problem.

 

[haruspex]

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

You need to step back and think how the two are defined. It should become apparent that there is no clear connection between them.
E.g. imagine that only protons experienced gravitational attraction, but both protons and neutrons have inertial mass. Then masses would fall at different rates according to their proton/neutron ratios.

 

[brotherbobby]

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

Inertial mass is what appears in Newton's second law. Gravitational mass is what appears in Newton's equation of gravity. A priori, there is no reason to suppose that they are the same things. In fact, in analogy to electric charge, you can call gravitational mass as gravitational charge. And yet, experiements show that gravitational charge and inertial mass are the same things. There is no reason to suppose that two different things turn out to be the same (or proportional). That is the mystery.

 

[brotherbobby]

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

No, not true at all. Newton's gravity does not need gravitational mass be equal to inertial mass. It is not a postulate and it is not needed. That it turns out to be so is purely a coincidence in Newton's gravity. [Try putting mI not equal to mG and derive the time period of a pendulum for instance. The equation will be complicated, but that is not the point]

On the other hand, in Einstein's theory of gravity, such a coincidence is the starting point. If the two masses were not equal, Newton's theory of gravity would not care. Einstein's theory however would either have to be changed or abandoned - the equality being fundamental to the latter.

 

[sillyquark]

There is a thought experiment to help understand the equivalence of gravitational and inertial mass (and general relativity). Imagine you were motionless in an elevator (on Earth) and you activated a laser horizontal to you. Now imagine yourself in the same elevator, in space, infinitely far from everything else so that you and the elevator are an isolated system. Now imagine that the elevator started to accelerate with magnitude g. Now as the elevator is accelerating, shine the laser in the same way as before. In the first case the force you feel is the gravitational force, $F_{g}$, in the second case you feel the force of acceleration from the elevator, $F_{a}$.

Let $m_{i}$ be inertial mass and $m_{g}$ be gravitational mass. In the second case you are accelerating with acceleration g and you would expect to see the laser curve from the horizontal. Now in the first case what do you expect to see/feel? The force acting on you is $F_{g}= m_{g} g$, in the second case the force acting on you was $F_{a} = m_{i} g$. If these to forces are equivalent than $m_{g} = m_{i}$, this would also implies that the two situations are equivalent and that in the first case you expect to see the laser curve from the horizontal exactly the same as if you were accelerating. This tells us that gravity acts like a field of acceleration. At this point you may want to search for general relativity as that is what this thought experiment was setting up.

 

[LitleBang]

Isn't it possible the mechanism that causes gravity is also the mechanism that causes inertia?

 

[jtcapa]

Very interesting to read this discussion, and step back and reflect upon the terms being used in the various definitions such as "gravity". So much of this discussion is base upon the assumption we actually know and accept what Gravity really is. The original poster might have been interested in neutralizing this mysterious effect we call Gravity for all the obvious reasons, but we would have to have a complete and total understanding of this term. If there is a flaw or error in our understanding of Gravity and its effects, then everything that springs forth from it would also contain errors, based on flaws in our assumptions. It is easy to both love and hate gravity for this reason, it is very challenging to define with complete certainty.

 

[Ghost117]

@ haruspex, brotherbobby, sillyquark and jtcapa

You need to step back and think how the two are defined. It should become apparent that there is no clear connection between them.
E.g. imagine that only protons experienced gravitational attraction, but both protons and neutrons have inertial mass. Then masses would fall at different rates according to their proton/neutron ratios.

Ok let me try stepping back... I'll state my assumption first and then point out (what I think is) the flaw in it. If any of the following is incorrect, please let me know:

Assumption: The reason why a large planet has more gravity (higher value of g) then a small planet, is because it has more "inertial mass" (I know, different equations, but hear me out)... i.e. has more "stuff", which is responsible for "causing" gravity... So the more "stuff" (inertial mass) you have, the more gravity you will cause, and hence the inertial and gravitational masses are proportional...

Problem: I've assumed that inertial mass (i.e. "stuff" that makes up the planet) is causing gravity, when they are two different concepts from two different equations. The correlation of more mass = more gravity does not prove any link, let alone causality, and yet the correlation exists! And this is the essence of the mystery... yes?

Inertial mass is what appears in Newton's second law. Gravitational mass is what appears in Newton's equation of gravity. A priori, there is no reason to suppose that they are the same things. In fact, in analogy to electric charge, you can call gravitational mass as gravitational charge. And yet, experiements show that gravitational charge and inertial mass are the same things. There is no reason to suppose that two different things turn out to be the same (or proportional). That is the mystery.

As far as their experimental equivalence goes, I was trying to follow the discussions in the preceding pages, but most of the material is beyond me for now... Although I did get the impression that this "equivalence" is not a universally accepted principle.

There is a thought experiment to help understand the equivalence of gravitational and inertial mass (and general relativity). Imagine you were motionless in an elevator (on Earth) and you activated a laser horizontal to you. Now imagine yourself in the same elevator, in space, infinitely far from everything else so that you and the elevator are an isolated system. Now imagine that the elevator started to accelerate with magnitude g. Now as the elevator is accelerating, shine the laser in the same way as before. In the first case the force you feel is the gravitational force, $F_{g}$, in the second case you feel the force of acceleration from the elevator, $F_{a}$.

Let $m_{i}$ be inertial mass and $m_{g}$ be gravitational mass. In the second case you are accelerating with acceleration g and you would expect to see the laser curve from the horizontal. Now in the first case what do you expect to see/feel? The force acting on you is $F_{g}= m_{g} g$, in the second case the force acting on you was $F_{a} = m_{i} g$. If these to forces are equivalent than $m_{g} = m_{i}$, this would also implies that the two situations are equivalent and that in the first case you expect to see the laser curve from the horizontal exactly the same as if you were accelerating. This tells us that gravity acts like a field of acceleration. At this point you may want to search for general relativity as that is what this thought experiment was setting up.

This is a really cool reminder that there is no way to tell the difference between feeling gravity and feeling acceleration, and looking at $F_{g}= m_{g} g$ and $F_{a} = m_{i} g$ for the two cases, i had something of a "light bulb" moment... BUT I'm confused about the laser curving part. Why would the laser curve in either case? In case #1 we're all just standing still, and in the second case, we're all accelerating uniformly. As far as I know, light only curves due to massive gravitational affects i.e. deep curvatures in 'spacetime', and even then it's not the light which is "curving" but space itself. p.s. General Relativity is definitely on my "to do" list... (in hopefully a few years)

Very interesting to read this discussion, and step back and reflect upon the terms being used in the various definitions such as "gravity". So much of this discussion is base upon the assumption we actually know and accept what Gravity really is. The original poster might have been interested in neutralizing this mysterious effect we call Gravity for all the obvious reasons, but we would have to have a complete and total understanding of this term. If there is a flaw or error in our understanding of Gravity and its effects, then everything that springs forth from it would also contain errors, based on flaws in our assumptions. It is easy to both love and hate gravity for this reason, it is very challenging to define with complete certainty.

I read this on NASA's website recently, it's relevant so I'll post it here... but this definitely deserves a separate thread:

"This is a snap shot of how gravity and electromagnetism are known to be linked. In the formalism of general relativity this coupling is described in terms of how mass warps the spacetime against which electromagnetism is measured. In simple terms this has the consequence that gravity appears to bend light, red-shift light (the stretching squiggles), and slow time. These observations and the general relativistic formalism that describes them are experimentally supported. Although gravity's effects on electromagnetism and spacetime have been observed, the reverse possibility, of using electromagnetism to affect gravity, inertia, or spacetime is unknown."
http://www.nasa.gov/centers/glenn/technology/warp/possible.html

p.s. Speaking of NASA, Interstellar is opening this weekend!

 

[haruspex]

The correlation of more mass = more gravity does not prove any link, let alone causality, and yet the correlation exists! And this is the essence of the mystery... yes?

Exactly. As I said, it is easy to imagine that there could be particles that have inertial mass but do not interact gravitationally.

There is a thought experiment to help understand the equivalence of gravitational and inertial mass (and general relativity).

It sounds to me like you are saying this somehow explains the equivalence. The significance of that thought experiment to Einstein, as I understand it, is that coupling it with the observation of equivalence led/helped him to formulate his general theory.

 

[sillyquark]

This is a really cool reminder that there is no way to tell the difference between feeling gravity and feeling acceleration, and looking at Fg=mgg F_{g}= m_{g} g and Fa=mig F_{a} = m_{i} g for the two cases, i had something of a "light bulb" moment... BUT I'm confused about the laser curving part. Why would the laser curve in either case? In case #1 we're all just standing still, and in the second case, we're all accelerating uniformly. As far as I know, light only curves due to massive gravitational affects i.e. deep curvatures in 'spacetime', and even then it's not the light which is "curving" but space itself. p.s. General Relativity is definitely on my "to do" list... (in hopefully a few years)

In experiment two it should be clear why the light bends, or at least, I will try to make it clear. Imagine having a train car with an ice floor, and image putting some mass on the floor. When traveling at constant velocity the mass moves at constant velocity with the train, but once the train accelerates (which can be positive or negative), the mass is moving at a different speed then the train, in fact, the trains velocity is changing but the mass' velocity is not. For the momentum of the mass to change, there must exist some frictional force to act on the mass.

It is easier to think about situations with constant acceleration (no jerk) but we would still expect the light to bend in both cases. The light acts like a projectile once its fired, in the elevator there are no external forces, so the light should move in a straight line from your reference point. Once you start accelerating in the elevator, the light will have the same instantaneous velocity as you when it is emitted, but no forces act on it after it has been emitted so you accelerate and the light doesn't. Light has such a large velocity that it is easy to neglect the curve when accelerating with magnitude g.

Try to imagine accelerating a g10^100, it may be easier to see that your velocity is changing at a rate considerable different than the light.
If we go back to the situation when we are motionless in the elevator, we feel a force $F_{g}=m_{g}g$ , this is the same force $F_{a}=m_{i}g$ that we felt in the elevator (assuming the equivalence of $m_{g}$ and $m_{i}$. If these situations are then equivalent then we can conclude that gravity acts like a field of acceleration, and what we observe in the first case is identical to the second.

It sounds to me like you are saying this somehow explains the equivalence. The significance of that thought experiment to Einstein, as I understand it, is that coupling it with the observation of equivalence led/helped him to formulate his general theory.

It may come across that I am trying to explain it with this observation but I do not think that is the case. I fully understand that the equivalence between inertial and gravitation mass is both fundamental and certainly non-trivial. Einstein's thought experiment seems to answer how inertial and gravitational mass are equivalent, but not why. I do not know why, if I did I would probably be accelerated into a PhD program. I think that studying the observations can help build our own intuition and a partial understanding of the phenomenon, even though we are not answering why the phenomenon occurs.

 

[jtcapa]

the reverse possibility, of using electromagnetism to affect gravity, inertia, or spacetime is unknown."

I think this is one of many reasons that finding a unified theory of everything is impossible. There are simply to many variables which are most likely wrong, or our assumptions of them are wrong. We have all drank from the same tainted well of knowledge of these forces and it has colored our perceptions of what they actually are.

I played this thought game with a smart physicist for a period of several years, before I could eliminate the influenced perception of something as simple as gravity, as being something completely man made. Something we created to try and neatly compartmentalize this attraction force we are always immersed in. Newton is credited with inventing it, and then created a form of mathematics to support his invention. What if he was wrong? What if it is not about mass, or acceleration?

 

[Ghost117]

Exactly. As I said, it is easy to imagine that there could be particles that have inertial mass but do not interact gravitationally.

Perfect, the issue is clearer now

When traveling at constant velocity the mass moves at constant velocity with the train, but once the train accelerates (which can be positive or negative), the mass is moving at a different speed then the train,

Oh right, I was thinking in terms of constant acceleration only for both scenarios, now the bending makes sense =)

Newton is credited with inventing it, and then created a form of mathematics to support his invention. What if he was wrong? What if it is not about mass, or acceleration?

That's possible of course... I think Hubble's discovery that the universe is expanding should have made the scientific world question the validity of General Relativity, and consequently the entire conception of what gravity actually is... but instead we now have "Dark Energy" and they kinda just swept it under the rug...

 

[LitleBang]

If one you came up with a new theory to answer "Why" you wouldn't be allowed to post it here anyway so why bother?

 

[jtcapa]

That's possible of course... I think Hubble's discovery that the universe is expanding should have made the scientific world question the validity of General Relativity, and consequently the entire conception of what gravity actually is... but instead we now have "Dark Energy" and they kinda just swept it under the rug...

This always struck me as wrong. In nature what is true for the very small is most often true for the large, and seeing a complete reversal of the accepted effects of gravity in the expanding universe theory just seemed improbable at best. So we do what we do best, we embellish the truth with all kinds of fictional ideas rather than question the underlying facts. My children, when they were young, tended to do this same thing.

 

[Dale]

Closed pending moderation.

EDIT: the thread will remain closed.