Difference between gravitational and inertial mass

[D H]

Einstein said in his equivalence principle that gravitation and inertia are equivalent.

No, it doesn't. Here's the equivalence principle. It comes in three forms. The first is the weak equivalence principle, sometimes called the Galilean equivalence principle.

• The trajectory of a freely falling test body is independent of its internal structure and composition.

The second is the Einstein equivalence principle:

• The weak equivalence principle is valid.
• The outcome of any local non-gravitational experiment performed in a freely falling laboratory is independent of:
  • The velocity of the freely-falling reference frame in which it is performed and,
  • Where and when in the universe the experiment is performed.

The third is the strong equivalence principle:

• The weak equivalence principle is valid for self-gravitating bodies as well as for test bodies.
• The outcome of any local experiment is independent of:
  • The velocity of the reference frame in which it is performed and,
  • Where and when in the universe the experiment is performed.

With regard to the weak equivalence principle, this is what says that inertial and passive gravitational mass are the same. (Note that Newtonian gravity also asserts the equivalence of passive and active gravitational mass.) With regard to the Einstein equivalence principle, one key consequence of this statement is that the physics of a locally free falling frame is that of special relativity. The strong equivalence principle re-asserts the equivalence of pass and active gravitational mass, in a relativistic setting.

 

[DrStupid]

The first is the weak equivalence principle, sometimes called the Galilean equivalence principle.

• The trajectory of a freely falling test body is independent of its internal structure and composition.

[...]
With regard to the weak equivalence principle, this is what says that inertial and passive gravitational mass are the same.

As a consequence of the equivalence of inertial and gravitational mass the acceleration of a free falling body should be independent from it's velocity. This is not required by the Galilean equivalence principle. Therefore the Galilean equivalence principle differs from the weak equivalence principle.

 

[MikeGomez]

The first distinction can be found in Newton's Philosophiae Naturalis Principia Mathematica as a comment to his definition of "quantity of matter" (Definition 1):

"And the same [quantity of matter] is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter."

Newton clearly distinguished between weight and inertia but assumed them to be proportional due to corresponding experimental results. (I think he actually did it to keep the Galileian equivalence principle.) In classical mechanic there is no theoretical explanation for this proportionality. Even GR gives an explanation for bodies at rest only.

Using Newton’s point of view as an argument for distinguishing gravitational mass from inertial mass is not a very compelling argument.

Of course Newton (and many others before the time of Einstein) made a distinction between gravitational mass and inertial mass. It was what everyone thought before the equivalence principle. That all changed when Einstein came along. Newton was wrong. Einstein was right.

Regarding Olson and Guarino, they discuss increased gravitational mass due to the motion of bodies. That is a relativistic effect which increases the gravitational energy (the stress energy tensor). I don’t think it says anything about the conceptual nature of the mass. Mass is mass.

Einstein said in his equivalence principle that gravitation and inertia are equivalent. That means that you cannot distinguish between a frame of reference resting in a homogeneous gravitational field from an accelerating frame of reference. There are a lot of other wordings (e.g. equivalence between classical inertial systems and locally free falling systems or same trajectory for all bodies starting from the same point with the same velocity in a static gravitational field) but it is not identical with the weak equivalence principle. The equivalence of inertial and gravitational mass results from Einsteins equivalence principle for bodies at rest but not for bodies moving at relativistic velocities.

I agree with this paragraph with the exception of the last sentence which I have highlighted in blue. I have already indicated my opinion that bodies moving at relativistic velocities are not an example of any difference between gravitational and inertial mass, but I have another disagreement. You say that the equivalence of inertial and gravitational mass results from Einstein’s equivalence principle, and I believe it is the other way around. Einstein’s equivalence principle is derived from the equivalence of gravitational versus inertial mass.

I have two reasons for this. The first is that Einstein discusses the equivalence of gravitational and inertial mass before he proceeds to the equivalence of gravitational and inertial reference frames. Secondly, Einstein specifically states the dependence of the one on the other, which I will highlight in blue. Here I quote Einstein.

“According to Newton’s law of motion, we have

(Force) = (inertial mass) x (acceleration),

Where the “inertial mass” is a characteristic constant of the accelerated body. If now gravitation is the cause of the acceleration, we then have

(Force) = (gravitational mass) x (intensity of the gravitational field),

where the “gravitational mass” is likewise a characteristic constant for the body. From these two relations follows:

(acceleration) = (gravitational mass)/(inertial mass) x (intensity of the gravitational field).

If now, as we find from experience, the acceleration is to be independent of the nature and the condition of the body and always the same for a given gravitational field, then the ratio of the gravitational mass must likewise be the same for all bodies. By a suitable choice of units we can thus make this ratio equal to unity. We then have the following law: The gravitational mass of a body is equal to its inertial mass.

It is true that this important law had hitherto been recorded in mechanics, but it had not been interpreted. A satisfactory interpretation can be obtained only if we recognize the following fact: The same quality of a body manifests itself according to circumstances as “inertia” or as “weight” (lit. “heaviness”).”

Here I skip a bit where he sets up the thought experiment of a man in an accelerated chest, who interprets his situation as under the influence of gravity instead of being accelerated. Here again I quote Einstein.

“… we can nevertheless regard the chest as being at rest. We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalized postulate of relativity.
We must note carefully that the possibility of this mode of interpretation rest on the fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass.”

 

[DrStupid]

Mass is mass.

And mass is not gravitational mass (unless for bodies at rest).

If now, as we find from experience, the acceleration is to be independent of the nature and the condition of the body and always the same for a given gravitational field, then the ratio of the gravitational mass must likewise be the same for all bodies.

But the acceleration is actually not independent from the condition of the body. It depends on it's velocity. That's the point of Olson's and Guarino's paper. As a consequence the acceleration of light in the gravitational field of the sun is as twice as high as predicted by Newtonian mechanics. This difference between Newton and Einstein was one of the most important experimental verification of GR.

Reversing your argumentation above this dependence of the acceleration from the velocity shows that the ratio of inertial and gravitational mass is not the same for all bodies under all conditions. It is limited to bodies at rest (or at least at non-relativistic velocities).

 

[MikeGomez]

But the acceleration is actually not independent from the condition of the body. It depends on it's velocity. That's the point of Olson's and Guarino's paper. As a consequence the acceleration of light in the gravitational field of the sun is as twice as high as predicted by Newtonian mechanics. This difference between Newton and Einstein was one of the most important experimental verification of GR.

Reversing your argumentation above this dependence of the acceleration from the velocity shows that the ratio of inertial and gravitational mass is not the same for all bodies under all conditions. It is limited to bodies at rest (or at least at non-relativistic velocities).

When Einstein says the acceleration is independent of the nature and condition of the body here, he is actually referring to an earlier statement he made which I left out for brevity. He said “Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not in the least depend either on the material or on the physical state of the body. For instance, a piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo), when the start off from rest or with the same initial velocity.” The italic emphasis is Einstein’s.

The Olson-Guarino paper states an increase in gravitational attraction “as though it had an increase in mass”. Italic emphasis is mine. This is a relativistic effect due to bodies with velocities comparable to the speed of light. They do use the term “gravitational mass”, but they are referring to this relativistic effect. I do not see any indication that they are trying to advance the idea that gravitational mass is different from inertial mass.

Please bear in mind that due to the equivalence principle this Olson-Guarino effect, if it be correct, would apply equally to an accelerated reference frame as to a gravitational frame. If you do the math properly, or if you could actually run the experiment in an equivalent manner, you would find the same results of “increased gravitational attraction” in the relativisticly accelerated frame.

 

[MikeGomez]

No, it doesn't.

D H, the implication that the equivalence of gravitational mass with inertial mass leads to equivalence of gravitation with inertia seems reasonable, and this probably comes up fairly often. It’s an interesting idea. When you say “no, it doesn’t”, do you mean that the equivalence principle doesn’t go as far as to say that, or do you mean that in a stronger sense, such that you have an argument which specifically contradicts this idea.

 

[DrStupid]

He said “Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not in the least depend either on the material or on the physical state of the body. For instance, a piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo), when the start off from rest or with the same initial velocity.”

In this wording - especially with the limitation to the same initial velocity - it is correct. The weak equivalence principle without this limitation is not correct.

I do not see any indication that they are trying to advance the idea that gravitational mass is different from inertial mass.

If they assume gravitational and inertial mass to be equal, how do they come to their conclusion "In the ultrarelativistic limit, the \left( {1 + \beta ^2 } \right) factor approaches the value 2 and is than the same famous factor by which the general relativistic prediction for light bending excess the Newtonian prediction."? This factor is nothing else than the ratio between gravitational and inertial mass as used in Newtonian mechanics and in the case of light it has already been proofed experimentally.

If you do the math properly, or if you could actually run the experiment in an equivalent manner, you would find the same results of “increased gravitational attraction” in the relativisticly accelerated frame.

Different accelerations at the same position will not become equal by changing the frame of reference.

 

[MikeGomez]

If they assume gravitational and inertial mass to be equal, how do they come to their conclusion "In the ultrarelativistic limit, the \left( {1 + \beta ^2 } \right) factor approaches the value 2 and is than the same famous factor by which the general relativistic prediction for light bending excess the Newtonian prediction."? This factor is nothing else than the ratio between gravitational and inertial mass as used in Newtonian mechanics and in the case of light it has already been proofed experimentally.

Apologies. I suppose that Olson-Guarino do propose the idea that gravitational mass is different from inertial mass. If you believe this also, and others on this forum do as well, then I suppose that I may be in the minority on that.

 

[D H]

That is an extremely suspect journal article. It has been in publication for almost thirty years and has been cited *twice* in that time, once in a Physics Essays article (we do not allow that journal at this site), and the other time in a Beyond the Quantum workshop paper (we generally don't allow those, either).