## Main Question or Discussion Point

First, I would like to say that this is my first post in this forum and that my knowledge of GR is weak. So I am hoping that my question can be answered in layman terms.

If I hold a 1kg object in my hand, I can easily feel it's inertia if I move it from side to side. I could also measure it's inertia by attaching it to a spring and timing the period of oscillation, or frequency. However, if I wanted to measure the strength of it's gravitational force (active gravitational mass), then I would have a hard time because the forces would be extremely weak. My question is, how can two quantities or properties that are so different be declared equal? If you were to tell me that the two quantities will always have the same ratio, then that would make sense. For example, I could double the inertial mass of the object and the gravitational mass (force) would also double. But to the best of my knowledge, the equivalence principle is not a ratio, it is an equality.

In what way is inertial mass and active gravitational mass equal?

By the way, I think this is called the strong equivalence principle (SEP). But please correct me if I'm wrong.

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Just to clarify, I am questioning the equivalence of the "inertial mass" and "active gravitational mass" of the 1kg object only. No other other objects, not even the earth, are involved.

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Dale
Mentor
If I hold a 1kg object in my hand, I can easily feel it's inertia. ... In what way is inertial mass and active gravitational mass equal?
The force on your hand is the same if you are holding a 1kg mass against 1g of gravity or if you are accelerating at 1g away from gravity.

Thanks Dale. Please just ignore earths gravity and pretend I am in near zero g, or that the 1kg mass is sitting on a frictionless table top. I edited my op to clarify.

Dale
Mentor
You can't ignore gravity when discussing the equivalence principle. The equivalence principle is (loosely speaking) about the equivalence between gravity and fictitious forces.

But I'm not ignoring gravity. The 1kg object has gravitational mass. But if the equivalence principle involves fictitious forces then that is something I was not aware of. Can you explain?

Dale
Mentor
Sure, the classical example of the equivalence principle is a rocket. If you are inside the passenger compartment of a rocket and only able to make local measurements then there is no experiment which you can perform that will allow you to determine if you are sitting on the ground in a 9.8 m/s² gravitational field under no thrust, or if you are in deep space accelerating at 9.8 m/s² under thrust.

In other words, if you are accelerating at 9.8 m/s² upwards in an inertial reference frame in deep space, then in your non-inertial reference frame there is a 9.8 m/s² fictitious force pointing downwards that is locally completely experimentally indistinguishable from gravity. This is the equivalence principle.

Thanks Dale. It appears you are describing the SEP as it is stated in Wikipedia. This is very confusing to me because I have read from numerous sources (one of them "Gravitation and Inertia" by Ignazio Ciufolini and John Archibald Wheeler) that Mi = Mgp = Mga, where Mi = inertial mass, Mgp = passive gravitational mass, and Mga = active gravitational mass. And this equivalence is said to be postulated in general relativity. But I have been unable to find any other explanations of this, other than a statement of their equality. Indeed, Wikipedia does not even mention it at all. That is the reason I stated in my OP that I am not even sure it is a part of SEP.

Please read over my OP again and notice that my question does not involve a comparison of accelerating and falling bodies. If Mi = Mgp = Mga then in what way are they equal? From my OP it is obvious that the magnitude of force involved in accelerating the 1kg mass (inertial) is hugely different from the magnitude of force generated by the 1kg masses gravitational field. How and in what way can they be equal?

DaveC426913
Gold Member
Why do you think that the gravitational force from the 1kg mass is important?

It is not. It is the gravitational force of the very large object in which the 1kg mass resides that is the equivalency being addressed.

You could do the experiment with a 1 microgram mass and 1 kg mass and a 1,000Gtonne mass simultaneously and you'd get all three saying the same thing. Their individual gravitational forces are irrelevant. They are all either under acceleration (via rocket) or in a gravitational field (via being on Earth) - and it is these two forces that are equivalent in all three cases.

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the equivalence principle is about the equivalence of passive gravitational mass and inertial mass. that is why relativity suggests that an object in a gravity field is simply following a straight line or geodesic or whatever its called.

Dave, I think you misunderstood my question. But I will continue to study your reply.

granpa said:
the equivalence principle is about the equivalence of passive gravitational mass and inertial mass. that is why relativity suggests that an object in a gravity field is simply following a straight line or geodesic or whatever its called.
So does GR not address the equivalence of active gravitational mass?

Ok Dave, I think what you are describing is the equivalence of inertial mass and passive gravitational mass. I have no problem with that. It's the equivalence of active gravitational mass that has me confused.

Dale
Mentor
Dave is correct. Let's look at Newtonian gravity since they are equivalent there too:

f = GMm/r²
ma = GMm/r²
a = GM/r²

The acceleration is independent of the (passive) mass. That is what is meant by the equivalence of gravitational and inertial mass. Consider Coulomb's law

f = kQq/r²
ma = kQq/r²

The acceleration is not independent of either (passive) charge or mass. There is no equivalence between charge and inertial mass.

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Yes, I know that Dave is correct. As I have already stated, I have no problem with the weak equivalence principle. WEP concerns the equivalence of inertial mass and passive gravitational mass and that they are the same regardless of composition. I agree with, and understand, yours and Daves post. But that is not what this is about.

What I am questioning, and what I don't understand, is the equivalence of inertial mass and active gravitational mass. Mi = Mgp = Mga

DaveC426913
Gold Member
Can you point to a reference that claims this?

Yes Dave, but I have to leave right now. When I return I will give some references. Thanks to everyone for your input.

Dale
Mentor
Your question has caused me to reflect a bit, and I realized that I have never really understood the distinction between active gravitational mass and passive gravitational mass. If we think of the analogy of charge. Do we ever speak of active charge and passive charge?

GMm/r² = GmM/r²
kQq/r² = kqQ/r²

As far as I can tell there is no difference between active and passive gravitational mass any more than there is a difference between active and passive charge.

D H
Staff Emeritus
In what way is inertial mass and active gravitational mass equal?

By the way, I think this is called the strong equivalence principle (SEP). But please correct me if I'm wrong.
The equivalence of inertial mass, passive gravitation mass, and active gravitation mass is a consequence of the weak equivalence principle, not the strong version. The strong equivalence principle says a lot more than that. Among other things, it also says gravitation is purely geometrical in nature and that there is no fifth force.

The weak equivalence principle (equivalence of inertial and gravitational mass) is implicit in Newtonian physics. What Einstein did with the equivalence principle was to make this implicit assumption explicit -- and then take it several steps beyond that.

For example, I could double the inertial mass of the object and the gravitational mass (force) would also double. But to the best of my knowledge, the equivalence principle is not a ratio, it is an equality.
What's the difference, really?

What the equivalence of inertial and gravitational mass is saying is that mass is mass. There is no distinction between the three categories. If you know the inertial mass of some point mass, you do not need to concern yourself with the the composition of the object when it comes to determining either the gravitational potential of the object (active gravitational mass) or how the object behaves in the presence of some other gravitational potential (passive gravitational mass).

Since you are talking about force, you are implicitly bringing this back to the Newtonian domain. The equivalence of active and passive gravitational mass is implicit in Newton's universal law of gravitation. Moreover, even if Newton's law of gravitation is not correct, active and passive gravitational mass must be equal in Newtonian mechanics in order to conserve momentum / maintain Newton's third law.

Whether you like it or not doesn't really matter. All experimental evidence, and there is a huge amount of experimental evidence, points to the equivalence of inertial and gravitational mass.

DaleSpam said:
Your question has caused me to reflect a bit, and I realized that I have never really understood the distinction between active gravitational mass and passive gravitational mass. If we think of the analogy of charge. Do we ever speak of active charge and passive charge?
I'm not sure that you can make an analogy to "charge" because charge has polarity and gravity does not.

DH said:
The equivalence of inertial mass, passive gravitation mass, and active gravitation mass is a consequence of the weak equivalence principle, not the strong version.
Thanks for the clarification.

DH said:
TurtleMeister said:
For example, I could double the inertial mass of the object and the gravitational mass (force) would also double. But to the best of my knowledge, the equivalence principle is not a ratio, it is an equality.
What's the difference, really?

What the equivalence of inertial and gravitational mass is saying is that mass is mass. There is no distinction between the three categories. If you know the inertial mass of some point mass, you do not need to concern yourself with the the composition of the object when it comes to determining either the gravitational potential of the object (active gravitational mass) or how the object behaves in the presence of some other gravitational potential (passive gravitational mass).
Good point. So you're saying that the equivalence principle does not care about proportionalities. The three categories of mass are intrinsically linked so that if you know one then you know the other. And that is what is meant by the equality?

DH said:
Whether you like it or not doesn't really matter. All experimental evidence, and there is a huge amount of experimental evidence, points to the equivalence of inertial and gravitational mass.
Hahaha, I didn't realize I had expressed a like or dislike of anything. I am just trying to better understand gravity.

I would like to point out that the "huge amount of experimental evidence" you speak of is for tests of the equivalence principle as it relates to inertial mass and passive gravitational mass. I have only found one experiment that tests the equivalence of active gravitational mass: L. B. Kreuzer (1968) published in The Physical Review Second Series, Vol. 169, No. 5. This one and only test of the equivalence of active gravitational mass achieved a sensitivity of 5x10-5, far below that of the many, many tests done for the equivalence of inertial and passive gravitational mass (10-13). If anyone knows of any experiments done (after 1968) to test the equivalence of active gravitational mass then I would be interested in knowing of it. So despite the huge amount of experimental evidence for the equivalence of inertial and passive gravitational mass there is actually very little experimental evidence for the equivalence of active gravitational mass.

Here are a couple of google books references.

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Dale
Mentor
I'm not sure that you can make an analogy to "charge" because charge has polarity and gravity does not.
So what? The thing you are interested in has to do with "active" and "passive", not positive and negative.

DrGreg
Gold Member
What would it mean if active and passive gravitational mass were not the same? Consider two particles, one with active mass ma and passive mass mp, the other with active mass Ma and passive mass Mp. Let's stick to Newtonian gravity and forget relativity. The forces that the particles exert on each other are

$$\frac { G M_a m_p } {r^2}$$​

and

$$\frac { G m_a M_p } {r^2}$$​

If you believe in Newton's 3rd law, these are equal, so

$$M_a m_p = m_a M_p$$​

i.e.

$$\frac{M_a}{M_p} = \frac{m_a}{m_p}$$​

As this must be true for any pair of masses, this establishes that active mass is proportional to passive mass.

So why is the constant of proportionality equal to one?

Suppose not. Let's say active mass is redefined to be twice passive mass. That wouldn't make any difference as long as we also halve the value of G. We might as well just take the constant to be one to keep things simple.

So I would say the equivalence of active and passive mass isn't a testable hypothesis, it's a definition. And any "test of active mass" is really just another name for "measuring the value of G".

In what way is inertial mass and active gravitational mass equal?

By the way, I think this is called the strong equivalence principle (SEP). But please correct me if I'm wrong.

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Just to clarify, I am questioning the equivalence of the "inertial mass" and "active gravitational mass" of the 1kg object only. No other other objects, not even the earth, are involved.
from Clifford Will's book, Theory and experiment in gravitational physics, Cambridge University Press (1993):

Weak Equivalence Principle (WEP): Trajectories of uncharged test particles do not depend on their internal composition. (A test particle has by definition negligible mass.)

Local Lorentz Invariance (LLI): Outcomes of local non-gravitational experiments do not depend on the
velocity of the instruments.

Local Position Invariance (LPI): Outcomes of local non-gravitational experiments do not depend on
where or when they are performed.

The sum of the WEP, the LLI and the LPI is called the Einstein Equivalence Principle (EEP), and it
is central to modern gravitational theory.

The Strong Equivalence Principle (SEP) is now the requirement that the WEP, the LLI and the LPI
must be valid for particles with non-negligible self-gravity and for local gravitational experiments as
well.

If the SEP is fulfilled, you are guaranteed that active gravitational mass is equal to inertial mass and
passive gravitational mass.

Experimentally, the EEP is well-tested and seems solid. The SEP is not that well-tested, but at least the first part of it seems to hold up since observations of the Moon's orbit show that the WEP is valid for objects with non-negligible self-gravity (no Nordtvedt effect).

Dale
Mentor
What would it mean if active and passive gravitational mass were not the same? Consider two particles...
That is what I was trying to get across above, but you did a much better presentation and analysis.

Let's say that active and passive gravitational mass were not the same. Then Newton's law would need to have some form like:
$$\frac{G m_a^x m_p^y}{r^2}$$
where x and y were some different exponents. Or some other form where the two were not interchangeable.

So I would say the equivalence of active and passive mass isn't a testable hypothesis, it's a definition. And any "test of active mass" is really just another name for "measuring the value of G".
But the value of G may in principle vary in space-time, and this is indeed testable. Personally, I prefer to declare G to have a constant value by definition and allowing for active mass to vary rather than vice versa. The reason for this choice is that this makes it somewhat easier to sort out the traditional (hidden) assumptions made when considering observational consequences of variable G.

Al68
So you're saying that the equivalence principle does not care about proportionalities. The three categories of mass are intrinsically linked so that if you know one then you know the other. And that is what is meant by the equality?
Equal means more than just proportional. If inertial mass equaled gravitational mass times a variable, then they would only be proportional. But since inertial mass equals gravitational mass times a constant, and the value of the constant depends on how we define the units, our choice of units result in that constant being equal to one, so they are equal.

Anytime two variables are related by a constant function, we can simply define units for the variables which result in the constant being equal to one. There is simply no reason to define different units for inertial mass and gravitational mass which result in a proportionality constant different from one.

atyy