# Why is gravitational mass the same as inertial mass?

## Main Question or Discussion Point

Gravitational mass is the property of objects that determines how they interact via gravity. For example how the Moon rotates around Earth.

Inertial mass is an object's property that determines how much the object "resists" acceleration when force is applied to it.

And it seems both are one and the same property. That means if we had the Moon in the void of space and we had some reference point relative to which the Moon did not move, then we applied force to the Moon and measured how it accelerated then we could tell exactly how would it rotate around Earth, for example how fast it would have to rotate at given distance in order for its orbit to be circular.

What seems to me as a logical question is: Why are those two properties the same?

And I don't mean an answer like "because GR sais so". I may be wrong, but I think GR postulates them equal, it doesn't explain why.

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You are indeed correct that it is a postulate, not a theoretically predicted phenomenon. Experimental measurements have been undertaken for decades to verify the relationship to ever high precision. See www.livingreviews.org/lrr-2006-3

rbj
yeah, actually, it's not because GR says so, it's because a postulate of GR says so. so, to take this on, you have to go after an assumption that GR makes long before it gets to the Einstien field equation (or whatever it's called, the $8 \pi G$ thing).

Hurkyl
Staff Emeritus
Gold Member
What seems to me as a logical question is: Why are those two properties the same?

And I don't mean an answer like "because GR sais so".
If you don't accept GR as the theoretical foundation of our knowledge about gravity, then what, pray tell, do you accept as a foundation? (And why should that be a better choice?)

I may be wrong, but I think GR postulates them equal, it doesn't explain why.
The experimental evidence is overwhelming.

Folks,
some of this postulates about nature, can't be explained with maths. Einstein was asked, what he things about mass. He thoughts they are kinks in spacetime, concentrated spacetime lumps. Better explanation can be given by Higg's boson althought the latter is yet to be verified experimentally.

Still...

I'm not asking for evidence, I'm sure they are the same value.

I'm not questioning GR either - I'm only pointing out that such a fact lies in the core of it.

What I'm looking for is "why" are they the same value.

Dear Sten,
There is no 'proof' or mathematical explanation for this, afa I know. There are many expalanations given but all of them lead to more questions. The origin of 'mass' itself is not known unless Higgs boson is proven to exists. Let me explain the way I understand this. According to GR, Eintein started off withh assuming the two masses are same and derived that Gravity if actually caused by geometry of spacetime. Now, which is the cause and which is the effect ? If he started off with geometry of spacetime, we may logically conclude (or may be not) that two masses are same. actually accelaration itself tantamounts to having a proportaional gravitational field. In fact this led einstein to derive a geometric theory for EM, but he couldn't. The real explanation of mass would probably come from QM or the modern versions of the same - Strings, etc, which are yet to be proven experimentally.

Ich
What I'm looking for is "why" are they the same value.
There are three types of "mass": inertial mass, passive gravitational mass, and active gravitational mass. You're looking for the reason of the equality of inertial and passive gravitational mass.
The postulate of equivalence led Einstein to find out how gravity works: There is no gravitational force (proportional to gravitational mass) that accelerates bodies, which resist according to their inertial mass.
Falling bodies are not accelerated at all: Simply consider yourself (the stationary observer at rest on earth's surface) as being accelerated upwards by your interaction with the ground.
Then it becomes immediately clear that a) all bodies fall with the same acceleration (because it's actually you who is accelerating, not the bodies) and that b) if you weigh a body, you are actually measuring its resistance against acceleration.
So, in GR, it's not merely an equivalence between two properties of a body, there is rather only one property, aka inertial mass. Resistance against gravitation is physically the same as resistance against acceleration.

Hurkyl
Staff Emeritus
Gold Member
I'm not questioning GR either
I didn't say you questioned it. I said you rejected it as being a foundation. You do not think it's good enough to explain phenomena in terms of GR; you seek to have GR explained in terms of some other theory. So that begs the question -- what is that other theory? (And what makes it a better choice for a foundation?)

There is no 'proof' or mathematical explanation for this, afa I know. There are many expalanations given but all of them lead to more questions. The origin of 'mass' itself is not known unless Higgs boson is proven to exists.
Can you please point me to some of those explanations? Much appreciated!

turbo
Gold Member
Sten, you may want to look into the Athena project at CERN.
http://alpha.web.cern.ch/alpha/

Their goal is to produce, trap and test cold neutral antihydrogen, not only to test the postulate of CPT invariance, but also to test the Weak Equivalence Principle that says that a body's gravitational acceleration does not vary with the composition of the body. If CPT invariance holds, antihydrogen must have the same inertial mass as hydrogen. It is their intent to determine if the gravitational mass is also equivalent.

Can you please point me to some of those explanations? Much appreciated!
Mass, in a modern field theoretic context, is a property of the dispersion relation for excitations of the field. In particular, wave packets have the property that their group velocity is the derivative of the dispersion relation. Identifying mass as the proportionality between velocity and momentum, or between energy and momentum, it can be said that the mass of particles come from the dispersion relation. The Higgs mechanism can modify the dispersion relation of fields through a symmetry breaking mechanism --- this is why it is said that the Higgs mechanism gives mass to particles.

If the postulate that the gravitational and inertial mass of a test body are equal is broken, then test bodies of of different inertial masses will have different accelerations in a gravitational field. Therefore, Einstein wouldn't be able to abstract the gravity field as a property of the body creating it and the space-time around it, independent of the test body. Hence GR and the language of manifolds won't work - it simply won't contain enough info to solve for the motion of a test body of arbitrary inertial mass. A concept like geodesic won't exist - different inertial masses will follow different trajectories.

So the assumption of equal (or proportional, depending on choice of units) inertial and gravitational masses is actually what prompted Einstein to consider manifolds as mathematical description of gravity.

Quantum Field Theory does not explain the inertial masses of the elementary particles - mass is just a parameter in the Lagrangian that you adjust to match experimental data. The Higgs field doesn't explain the mass of elementary particles either - the necessary mass parameter is obtained by adjusting another parameter - the coupling of the particle to Higgs.

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Gravitational mass is the property of objects that determines how they interact via gravity. For example how the Moon rotates around Earth.

Inertial mass is an object's property that determines how much the object "resists" acceleration when force is applied to it.

And it seems both are one and the same property. That means if we had the Moon in the void of space and we had some reference point relative to which the Moon did not move, then we applied force to the Moon and measured how it accelerated then we could tell exactly how would it rotate around Earth, for example how fast it would have to rotate at given distance in order for its orbit to be circular.

What seems to me as a logical question is: Why are those two properties the same?

And I don't mean an answer like "because GR sais so". I may be wrong, but I think GR postulates them equal, it doesn't explain why.
Is there a such thing as "Gravitational mass" in GR?

In another thread https://www.physicsforums.com/showthread.php?t=225573 several posters have suggested that a particle moving horizontally will fall faster than a particle with no horizontal motion relative to the massive body. Now if that is true, does it not violate the equivalence principle? Does it not require the gravitational mass of the particle to somehow be different to its inertial mass when moving horizontally in order to fall faster?

Imagine this experiment in a rocket far fronm any significant gravity. A particle is fired horizontally simultaneously with releasing a stationary particle. A moment later the rocket accelerates at 1g in a direction tranverse to the moving particle. The rocket can have no influence with particles until they hit a wall or the floor. An observer inside the rocket sees that the particles hit the floor of the rocket at the same time. An observer outside the rocket that does not undergo acceleration also agrees they the floor hits both particles simultaneously. Both observers agree the particles "fall" at the same rate irrespective of the horizontal motion.

Now we duplicate this experiment with a massive gravitational body that minimises curvature and tidal concerns. In the other thread a long cylindrical body (possibly infinitely long) with a gravity strength similar to the Earth was suggested. This is as close as it is possible to duplicate the rocket experiment to test the equivalence principle. Suggestions that the horizontaly moving particle falls faster than the vertically falling particle in this "flat" gravitational field contradict the equivalence principle. Arguments that the clocks at the top and bottom used to time the falling rates will be out of sync to their hight is just a smokescreen because the difference in the clock rates in a Earth like gravitational field is less than one part in a billion. Sure the IS a difference, but the adjustment is insignificant compared to the suggested increase in falling rate due to horizontal motion at relativistic speeds. No one in the forum has given a formula for the increase in a horizontally flat gravitational field, but I found a paper by Matsas that suggests the gravitational force acting on a particle with a horizontal velocity of v is increased by $1/\sqrt{1-v^2/c^2}$. Does that actually translate into greater acceleration and reduced falling time? If it does, then the equivalence principle has been discarded.

Still...

I'm not asking for evidence, I'm sure they are the same value.

I'm not questioning GR either - I'm only pointing out that such a fact lies in the core of it.

What I'm looking for is "why" are they the same value.
A better question is how could they not be?

For instance, you could ask the same question this way: Why does heavy objects fall at the same speed as lighter objects? You could the consider two 1 kg balls and it makes perfect sense that they would fall side by side at the same speed. So if we glue the two balls together for 2 kg and it's really no different than falling side by side. Now suppose we crushed one ball and put it inside the other. The only reason to suppose that it would fall faster that way is to assume there is something like an ether pushing back less on it because it is denser, like air does.

If you need a resistance, such as an ether, proportionate to density for inertial and gravitational mass not to be equivalent a failure of this equivalence would be harder to explain. The only explanation needed is that there is no ether.

For instance, you could ask the same question this way: Why does heavy objects fall at the same speed as lighter objects? You could the consider two 1 kg balls and it makes perfect sense that they would fall side by side at the same speed. So if we glue the two balls together for 2 kg and it's really no different than falling side by side.
If you bring two 1kg balls together the total mass is less than 2kg.

GR is not a linear theory, the gravitational field couples to itself. Two masses A and B at some distance will come together but the fact that they come together influences the gravitational mass.

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If you bring two 1kg balls together the total mass is less than 2kg.

GR is not a linear theory, the gravitational field couples to itself. Two masses A and B at some distance will come together but the fact that they come together influences the gravitational mass.
The GR effects you are speaking of is so tiny as to not even be measurable from the binding energy of two 1 kg balls. We are talking the principle of equivalence here.

Even if we consider it the extra mass is not going to change the rate of acceleration in free fall, per the principle of equivalence.

Is this a dead end? I see the definition of active gravitational mass as having the units of cubic meters per seconds squared, i.e. squared velocity of light times lambda, a wavelength or a "communication" distance between two masses!)