Difference between gravitational and inertial mass

[chipotleaway]

I can't seem to get my head around the difference between the two.
Inertial mass appears in F=ma and is a measure of an object's resistance to acceleration when being acted upom by a force/s. Gravitational mass appears in F=(GmM)/r^2 - what 'role' does mass play here?

 

[dev70]

there's a huge difference...inertial mass is found from Newton's 2nd law..nd it deals when a body moves horizontally..it does not change with speed (provided its speed is very less than that of light)..
whereas gravitational mass is the ratio of weight of the body acceleration due to gravity..as the acceleration due to gravity is not same everywhere on earth..so it changes..

hope that helps..

 

[Bandersnatch]

Inertial mass resists forces, gravitational mass is a source of forces.

I'm not sure what dev70 meant by saying that gravitational mass changes, though. It doesn't.

 

[ehild]

Inertial mass is defined by Newton's Second Law: Force causes acceleration of an object, and the ratio of the force and the acceleration is constant for the object.
You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same? The answer is not straightforward. Gravity can act differently on bodies of different material. The equivalence of the two kinds of mass was subject of Eötvös' experiments, for example. http://en.wikipedia.org/wiki/Eötvös_experiment

ehild

 

[dev70]

I'm not sure what dev70 meant by saying that gravitational mass changes, though. It doesn't.

well...i actually didnt mean it...mass remains same..but weight varies..that sounds better..thanks for pointing this out..

 

[TurtleMeister]

This is a source of confusion sometimes because there are actually two types of gravitational mass, active and passive. It's a pet peeve of mine when people use the term loosely without making it clear which one they're talking about. Passive gravitational mass is a measure of strength of a bodies interaction with a gravitational field. Active gravitational mass is a measure of strength of the gravitational field produced by a body. Usually the former is implied when the term "gravitational mass" is used by itself, but not always. And then sometimes people just simply get it wrong.

To give an example of what the differences are:

The inertial mass of your body could be determined by measuring how much force is required to accelerate yourself at a certain rate. Or in other words, Newton's second law.

The passive gravitational mass of your body could be determined by, well, simply standing on your bathroom scales.

The active gravitational mass of your body could be determined by using a torsion balance. And it would be a very delicate and difficult task. That's because the active gravitational mass of your body would be very small.

The equivalence principle tells us that the three methods of determining your body's mass, as described above, will give us the same result (within experimental error of course).

You may ask, how can active gravitational mass be equivalent to the other two types when I stated that it would be a very delicate and difficult task to measure in a human body. The answer is that the equivalence principle, as it applies to active gravitational mass, is a proportional equivalence and not a quantitative one.

 

[xAxis]

@TurtleMeister
What do you mean by active gravitational mass of your body would be very small. Isn't that mass same as a passive mass, but the field generated by it would be small?

 

[TurtleMeister]

@TurtleMeister
What do you mean by active gravitational mass of your body would be very small. Isn't that mass same as a passive mass, but the field generated by it would be small?

That's a good question. As I said, gravitational mass is a source of confusion. Sometimes it is described as the mass that produces the field, and sometimes it is described as a measure of the strength of the field itself.

Here is one definition from Wikipedia:
"

Active gravitational mass is a property of the mass of an object that produces a gravitational field in the space surrounding the object, and these gravitational fields govern large-scale structures in the Universe.

And here is another definition from the same Wikipedia page:

Active gravitational mass (* see below) is a measure of the strength of an object’s gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface).

I myself just consider it to be the same thing as the standard gravitational parameter, which is defined by Kepler's third law.

 

[chipotleaway]

Inertial mass is defined by Newton's Second Law: Force causes acceleration of an object, and the ratio of the force and the acceleration is constant for the object.
You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same? The answer is not straightforward. Gravity can act differently on bodies of different material. The equivalence of the two kinds of mass was subject of Eötvös' experiments, for example. http://en.wikipedia.org/wiki/Eötvös_experiment

ehild

Wow that's pretty unintuitive based on what I have learned so far. I guess I'm getting confusued because I feel like the mass of something determines how much it accelerates in response to a force and that same mass should also dictate the gravitational attraction between it and another mass.

 

[TurtleMeister]

Wow that's pretty unintuitive based on what I have learned so far. I guess I'm getting confusued because I feel like the mass of something determines how much it accelerates in response to a force and that same mass should also dictate the gravitational attraction between it and another mass.

There are a couple of things in ehild's post that you may have interpreted wrongly.

You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same? The answer is not straightforward.

The equivalence of inertial mass and passive gravitational mass (what ehild is talking about) has been experimentally verified to better than one part in 10¹³. Proposed satellite experiments will extend that to one part in 10¹⁸.

Gravity can act differently on bodies of different material.

I am assuming that ehild meant for this to be a question and not a statement, since it is followed up with a link to the Eotvos experiments. If it was meant as a statement then it is wrong (based on current theory and experimental evidence).

 

[Popper]

I can't seem to get my head around the difference between the two.
Inertial mass appears in F=ma and is a measure of an object's resistance to acceleration when being acted upom by a force/s. Gravitational mass appears in F=(GmM)/r^2 - what 'role' does mass play here?

As explained above there are three types of mass in Newtonian mechanics:

inertial mass - That which gives movng objects momentum (Weyl's definition of inertial mass)

active gravitational mass - The source of a gravitational field

passive gravitational mass - That on which a gravitational field acts.

It's very important to keep these things in mind and to note the difference between them. When you get into general relativity its essential to know that inertial mass density is different than the density of active gravitational mass while being the same for passive gravitational mass.

 

[TurtleMeister]

When you get into general relativity its essential to know that inertial mass density is different than the density of active gravitational mass while being the same for passive gravitational mass.

I have not studied general relativity, except for an occasional excursion into the "Special & General Relativity" forum here at PF. Does this difference in mass density between M_a and (M_p, M_i) have a constant proportionality in GR?

 

[Popper]

I have not studied general relativity, except for an occasional excursion into the "Special & General Relativity" forum here at PF. Does this difference in mass density between M_a and (M_p, M_i) have a constant proportionality in GR?

No. And it's not always possible to provide a definite quantity in generality which one can call the mass density. Mass is fully described by a tensor.

Using units for which c = 1, and letting the source be an ideal gas then

active gravitational mass density = energy density + 3xPressure

while

inetial mass density = energy density + Pressure

I should point out that different people define these terms differently than above.

 

[Dale]

I have not studied general relativity, except for an occasional excursion into the "Special & General Relativity" forum here at PF. Does this difference in mass density between M_a and (M_p, M_i) have a constant proportionality in GR?

To further what Popper said, in GR the active gravitational mass essentially doesn't exist. In Newtonian gravity the active gravitational mass is the thing which acts as the source of gravity. In GR, the thing which acts as the source of gravity is the stress-energy tensor.

In a situation where Newtonian gravity is a good approximation to GR then you can make the simplification Popper mentioned.

 

[ehild]

The equivalence of inertial mass and passive gravitational mass (what ehild is talking about) has been experimentally verified to better than one part in 10¹³. Proposed satellite experiments will extend that to one part in 10¹⁸.

I talked about classical mass. Each theory has got its own definitions and postulates. Classical physics postulated position, time, and interaction between objects. It also defined velocity and acceleration and Newton set up his laws. The second one is definition of the mass of an object: the inertial mass. There is also the weight of a body: you put a body on the scales and balance it with a standard "weight", then you say that the body has the weight equal to the standard one. That weight is proportional with the gravitational mass, as you balance the force of gravity. As inertial mass and gravitational mass were measured with different procedure, (one dynamic, the other static) it was not sure if they were really proportional or with properly chosen proportionality factor, were they identical. That was proved experimentally with very high accuracy since then, but the concepts themselves were not equivalent.

I am assuming that ehild meant for this to be a question and not a statement, since it is followed up with a link to the Eotvos experiments. If it was meant as a statement then it is wrong (based on current theory and experimental evidence).

It was a question, the masses proved to be equal experimentally with high accuracy. So we take them the same when determining acceleration or when we determine the force a big load exerts on a support.
...
You can make different theories on different models. You can make a theory where mass is postulated itself and no distinction exists between gravitational and inertial mass.

When you define a physical quantity you have to make it clear, how is it measured. In SR, rest length (proper length) is measured by comparing it with a metre stick, while you measure length with a different procedure from a moving frame of reference. The different measurement procedures result in different numerical values for the length of the same object.
The same with mass - rest mass is different from the moving mass.

And you can speculate what is mass, if it exist at all? ...

ehild

 

[ehild]

As explained above there are three types of mass in Newtonian mechanics:

inertial mass - That which gives movng objects momentum (Weyl's definition of inertial mass)

active gravitational mass - The source of a gravitational field

passive gravitational mass - That on which a gravitational field acts.

It's very important to keep these things in mind and to note the difference between them. When you get into general relativity its essential to know that inertial mass density is different than the density of active gravitational mass while being the same for passive gravitational mass.

The distinction between active and passive gravitational mass are attributed to the approximation of the gravitational interaction with the help of "gravitational field". That is a proper approximation when the motion of one of the interacting bodies (the big one) is not influenced by the motion of the "small" body. It is said then, that a gravitational field exist around the big body, determined by its (active) mass. The intensity of this field multiplied by the (passive) mass of the small body is equal to the force acting on the small body, and that force determines its acceleration if divided by the (inertial) mass. That is an approximation which fails when the masses of the bodies are comparable. In case of a pair of twin stars, you can apply the law of gravity to figure out the motion of both starts, but you can not use the gravitational field approximation.

ehild

 

[Crazymechanic]

to the OP post I could use a gyroscope as an answer and a working model where inertial mass or the physical rotation of a given "mass" wheel counteracts the force exerted on the wheel by gravitational mass , by the way everyone can try to spin a bicycle rim and then try to move it in different angles while it spins.

 

[TurtleMeister]

@Popper or DaleSpam

Do you mean to say that in a situation where Newtonian gravity is a good approximation to GR that the stress-energy tensor will not necessarily be proportional to M_i and M_p? If this is true then how does the equivalence principle apply to GR?

I talked about classical mass. Each theory has got its own definitions and postulates. Classical physics postulated position, time, and interaction between objects. It also defined velocity and acceleration and Newton set up his laws.

Classical physics has changed over time. New terms and concepts are incorporated. For example, prior to the twentieth century the concept of mass being divided into the three types that we are discussing, did not exist.¹ Or at least there is no written record of it. In your post #4, you stated:

You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same?

emphasis mine

This refers to the equivalence of inertial mass and passive gravitational mass based on material composition. The experimental verifications that I mentioned are for the torsion balance experiments done by the Eot-Wash Group and the proposed satellite experiment called STEP. These experiments are just more modern versions of the Eotvos experiments that you linked to in your post.

¹Even though the concepts did not exist prior to the twentieth century, experiments involving them did, as evidenced by Newton's pendulum experiments using bobs made of various materials.

 

[inertian]

there is no difference between inertial and gravitational mass; however, how it is measured is the difference

 

[DrStupid]

The passive gravitational mass of your body could be determined by, well, simply standing on your bathroom scales.

The active gravitational mass of your body could be determined by using a torsion balance. And it would be a very delicate and difficult task. That's because the active gravitational mass of your body would be very small.

Why not using the passive gravitational mass of Earth to measure your active gravitational mass? That could be done by, well, simply standing on your bathroom scales.

 

[MikeGomez]

there's a huge difference

There is no difference.

inertial mass is found from Newton's 2nd law..nd it deals when a body moves horizontally

It can be any direction.

whereas gravitational mass is the ratio of weight of the body acceleration due to gravity..as the acceleration due to gravity is not same everywhere on earth..so it changes..

The attractive force decreases as distance between the masses increases. The masses don't change.

 

[Buckleymanor]

Inertial mass is defined by Newton's Second Law: Force causes acceleration of an object, and the ratio of the force and the acceleration is constant for the object.
You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same? The answer is not straightforward. Gravity can act differently on bodies of different material. The equivalence of the two kinds of mass was subject of Eötvös' experiments, for example. http://en.wikipedia.org/wiki/Eötvös_experiment

ehild

This is a bit confuseing, have you read the link you posted.From it I came to the conclusion that bodies of different materials act the same when gravity acts upon them.
As the experimental devices get more refined with time, the differences between the results of the different materials used, get less and less.
It appears to be a case of experimental error and refinement

 

[TurtleMeister]

Why not using the passive gravitational mass of Earth to measure your active gravitational mass? That could be done by, well, simply standing on your bathroom scales.

No, it's not that simple. The term active gravitational mass is somewhat of a misnomer because it's not really mass at all, it's a property of mass. It's a measure of the strength of a body's gravitational field. It is the same thing as the standard gravitational parameter and has the units of m³s^-2. It cannot be measured with bathroom scales. It can however be measured by a torsion balance, or if you're talking about a planet or moon, then it can be measured by using an orbiting satellite and applying Kepler's third law.

I have answered this question in good faith, believing it to be in line with current mainstream ideas.

 

[DrStupid]

The term active gravitational mass is somewhat of a misnomer because it's not really mass at all, it's a property of mass. It's a measure of the strength of a body's gravitational field. It is the same thing as the standard gravitational parameter and has the units of m³s^-2.

To my knowledge the strength of the gravitational field is measured in units of m/s² (e.g. round about 9.8 m/s² for the gravitational field of Earth at sea level).

It cannot be measured with bathroom scales.

In classical mechanics (and that's what we are talking about) the strength of a gravitational field can be determined by measuring the force acting on a body with known passive gravitational mass and the force acting on Earth within your gravitational field can be measured with bathroom scales.

It can however be measured by a torsion balance

What makes a torsion balance more appropriate than bathroom scales?

 

[jbriggs444]

To my knowledge the strength of the gravitational field is measured in units of m/s² (e.g. round about 9.8 m/s² for the gravitational field of Earth at sea level).

That's the field strength right here at the Earth's surface. If you want a single number that reflects the overall gravitational field of the earth, you do not want that number to depend on where you took the measurement.

Gravity follows an inverse square law. So you have to multiply the measured local acceleration by the square of the radius where you took the measurement to arrive at the standard value for the field strength. That result should match Newton's universal gravitational constant (big G) times the mass of the earth.

See "The GM Product" on http://en.wikipedia.org/wiki/Gravitational_constant.

 

[DrStupid]

So you have to multiply the measured local acceleration by the square of the radius where you took the measurement to arrive at the standard value for the field strength.

OK, that wuld mean I need the passive gravitational mass and the radius of Earth to measure my active gravitational mass using bathroom scales.

 

[D H]

In classical mechanics (and that's what we are talking about) the strength of a gravitational field can be determined by measuring the force acting on a body with known passive gravitational mass and the force acting on Earth within your gravitational field can be measured with bathroom scales.

First off, you have the sense of "passive" and "active" backwards. More on this later.

Your bathroom scale does not measure gravitational force. There is no local experiment that can measure gravitational force. A spring scale is a kind of a local experiment. What a spring scale measures is the upward normal force exerted by the scale on you. Since you are standing still on the scale, you can use that measurable upward normal force as a stand-in for gravitational force. Suppose you take that scale on a zero g airplane ride and use it to weigh yourself. You will notice that what the scale registers varies a lot even though the gravitational force only changes by a tiny amount as the plane climbs and dives.

What makes a torsion balance more appropriate than bathroom scales?

The distinction between active and passive gravitational mass.

A bathroom scale indirectly measures the Earth's gravitational force on you. Here it's the Earth that is the active gravitational body and you are the passive body. Suppose you connect a small test mass to an extremely sensitive spring and place that test mass near you. The test mass will feel a feeble gravitational attraction toward you. Now you are the active gravitational body and the test mass is the passive one.


Note very well: The distinction between inertial mass, passive gravitational mass, and active gravitational mass borders on fringe physics. There is no distinction whatsoever between the three concepts in Newtonian mechanics. That inertial mass and passive gravitational mass are one and the same is also a central tenet of general relativity.

What about active gravitation mass? That gets very tricky in general relativity because of nonlinearities in the theory. Gravity begets gravity: http://www.einstein-online.info/spotlights/gravity_of_gravity. Einstein was very careful in developing his equivalence principle to avoid these nonlinearities. That's why he wrote about test masses (very small masses) that steer clear from those nonlinear behaviors.

 

[DrStupid]

Since you are standing still on the scale, you can use that measurable upward normal force as a stand-in for gravitational force.

That's sufficient.

A bathroom scale indirectly measures the Earth's gravitational force on you. Here it's the Earth that is the active gravitational body and you are the passive body.

And vice versa.

Suppose you connect a small test mass to an extremely sensitive spring and place that test mass near you. The test mass will feel a feeble gravitational attraction toward you. Now you are the active gravitational body and the test mass is the passive one.

Here again both bodies are active and passive at once. I am also attracted toward the test mass. Everything else would be a violation of the third law.

The distinction between inertial mass, passive gravitational mass, and active gravitational mass borders on fringe physics. There is no distinction whatsoever between the three concepts in Newtonian mechanics.

There is a distinction between inertial and gravitational mass but Newton assumed them to be equal due to corresponding experimental results. But you are right in regard to active and passive gravitational mass. There is nothing like that in the original definition of Newtons law of gravitation. Even the mathematical formula shows that both masses play the same role.

That inertial mass and passive gravitational mass are one and the same is also a central tenet of general relativity.

There is no such thing like gravitational mass in GR. Transferring the results of GR back to classical mechanic shows that the weak equivalence principle is valid for bodies at rest only. At relativistic velocities happens something like this: http://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf

What about active gravitation mass? That gets very tricky in general relativity because of nonlinearities in the theory.

As I already wrote above there is no gravitational mass in GR. In GR the source of gravitation is the stress-energy-tensor. Gravitational mass is used in Newtons law of gravitation only. That's why I pointed out above that this discussion is about classical mechanics only.

 

[MikeGomez]

There is a distinction between inertial and gravitational mass but Newton assumed them to be equal due to corresponding experimental results.

It was Einstein who said that inertial and gravitational mass are equal in his equivalence principle. Where do you find a distinction between the two? The link you provided does not work.

 

[DrStupid]

It was Einstein who said that inertial and gravitational mass are equal in his equivalence principle.

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.

Where do you find a distinction between the two?

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.

The link you provided does not work.

Olson DW, Guarino RC. Measuring the active gravitational mass of a moving object. American Journal of Physics. July 1985. 53(7):661-3

 

[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.