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Thinking Like a Theorist Number 3: Why is m special?

Update: My answer to this question.

There are lots of properties we give to various particles, like the electron or neutron. A fundamental physical particle has a mass, a spin, perhaps an electric charge, perhaps a color charge etc. etc. You are most familiar with the electric charge from Coulomb's law, that the force between two charged objects is given by

\[F=\frac{k q_1 q_2}{r_{12}^2}.\]

Similarly, there is a force between two massive particles in Newtonian mechanics

\[F=-\frac{G m_1 m_2}{r_{12}^2}.\]

From this 'force law' perspective there's nothing fundamentally different between the electric charge on an object and it's mass. In fact, the mass can just be thought of the 'gravitational charge' of an object.

Mass is special though, but not because there is anything special about Newton's Law of Gravitation. Rather, the special role of mass in physics is generated by one of Newton's other laws, namely his Second Law


In principle the m in Newton's second law and the m in Newton's gravitational law could be different as the F=ma mass is the "inertial mass" which plays a different role than the gravitational mass. We therefore have two questions:

  1. Is there a reason the inertial mass is equal to the gravitational mass? (And for anyone who is thinking "because the equivalence principle says so", that's a circular argument so you'll need to go further.)

  2. What would happen if Newton's second law was actually \(F=qa\), where q is the electric charge and force was measured in \(1 ~Newton=1~Coulomb~m/s^2\)? In other words, does the fact that \(F=ma\) place restrictions on what m can be for particles?

Note by David Mattingly
4 years, 1 month ago

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For the second question, one of the problems with \(F=qa\) is that \(q\) can have both positive and negative values. This means that applying a force on a negative charge would cause it to accelerate in the direction opposite the force (whether or not our current convention for the sign of charge would be the same is unknown). Also, many particles, such as neutrons, have no charge. This means that any force, including minuscule gravitational forces, would give a charge-less particle infinite acceleration, which implies that any charge-less object would always travel at \(c\), (assuming relativity is even valid!). This means that atoms would no longer be able to form, and the world as we know it couldn't possibly exist. I guess this means that \(F=ma\) means that \(m\) must never be negative, and almost always be greater than zero. The only known particles that have zero mass are gauge bosons: photons and gluons, which are force carriers. Ricky Escobar · 4 years, 1 month ago

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@Ricky Escobar This is a very important observation. Anyone know what horrible things (besides what's above) happen if we toss negative mass into our world? David Mattingly Staff · 4 years ago

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@David Mattingly Okay, I did a little research on the internet about the question you asked and came up with this list of horrible (horrible because they are weird!) things that would happen if we had negative mass.

1) Perpetual motion. Yes, if we had negative matter (or exotic matter as it is commonly called), we could have a perpetual motion machine. This is because exotic matter and matter would repel each other because of negative gravity but exotic matter would accelerate at the opposite direction of the force. That means the positive mass & the negative mass would accelerate in the same direction. So we could create a machine that would accelerate forever with zero input energy. This is known as 'negative mass propulsion'.

2) Time machines as they are seen in sci-fi movies. I read somewhere that Hawking proved that if you want to build a time machine smaller than the universe, negative mass is a requirement. So, negative mass would imply time travel.

Looking back on what I wrote, I just realized that I mentioned only two points. So this isn't much of a list. Feel free to add whatever you want. Mursalin Habib · 4 years ago

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In my understanding, mass is a measure of how much "matter" there is in an object. This is true both for inertial mass and gravitational mass.

Newton's gravitational law reflects the observed fact that matter attracts matter, so the force is proportional to the amount of matter there are in the objects in consideration i.e. to their gravitational masses.

On the other hand, Newton's second law interprets the superposition principle, that forces may be added (by vector addition) from all parts that composes an object to form the total force in this object. The force needed to accelerate a larger body is therefore the (vector) sum of the forces needed to accelerate the various components of this body, being therefore proportional to the amount of matter in the body i.e. the inertial matter.

Both "forms" of mass need not be equal to each other, only proportional for they represent the same physical concept (amount of matter). The proportionality factor is set to be one by the value of the gravitational constant G. What we have done is measure first the inertial mass of two bodies and the gravitational force between them for then calculate the value of G, forcing the proportionality constant to be unity (this is the first question).

For the second question, I think it couldn't be the electric charge to be in Newton's second formula because it measures another physical property and not the amount of matter in a body, which is the property that restricts the change in its motion.

It's true that electric charge also restricts the motion of a charged particle, but not by itself: it depends also on its velocity and this characteristic also "produces" its equivalent mass, the electromagnetic mass. Davi Silva · 4 years, 1 month ago

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@Davi Silva "In my understanding, mass is a measure of how much "matter" there is in an object. This is true both for inertial mass and gravitational mass."

This is true. In this language my question is: why is it the amount of matter that matters (no pun intended) in Newton's second law? Why does inertial mass even exist? Is there a theoretical reason, or is it just experimental? David Mattingly Staff · 4 years ago

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@Davi Silva Nice answer. I was thinking something along the lines of how e=mc^2 is linked with ke to get total energy in that pythag triangle thing, though you seem to have pretty much got it. Ali Rahemtulla · 4 years ago

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@Davi Silva i have a question, mass is the measure of how much matter there is in an object,no doubt but then can we say the mass of anti matter is negative? Snehdeep Arora · 4 years ago

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@Snehdeep Arora Nope! Antimatter has positive mass (and energy) like regular matter. Mursalin Habib · 4 years ago

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@Mursalin Habib right! ty Snehdeep Arora · 4 years ago

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Gravitational mass is the mass that is the number of particles making up an object which is responsible for force of attraction. Gravity is a long range force and is always attractive. Each particle is being attracted. Gravitational mass that is all the matter particles are responsible for the gravitational force. Inertial mass is the mass which resists change in its state of motion. If we try to change the state of the object, all the particles will resist it and hence, in both cases all the particles are involved in either attraction or in resistance. Hence, they should be the same because its the same particles constituting the object.

F=qa is not possible because as Ricky E. said that for neural particles, the acceleration for any force would be indetrminate, so it sort of becomes insensible to say that. Also, a particle may be very large but the net charge on it may be zero. So, in such cases for even large value of forces, the body will not move but by the formulae, F=qa, the body's acceleration would be infinity or undefined. Certainly, something wrong with this. Tushar Gopalka · 4 years, 1 month ago

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@Tushar Gopalka Probably there is one more perspective to relate both the terms "inertial mass" and "charge mass in case of a particle". Its the electrons that we are forgetting while looking for a proper description. The surface of every body is just a sea of electrons. Hence, whenever there is an application of force on a body(the surface applying the force being also constituted of electrons), there is a natural electron-electron repulsion providing for the inertial effect. (Perhaps, then friction is also a special case of this.) Hence, it is but the charge that provides for the mass and the gravitational effect as well. It is just our misconception of holding mass of macroscopic particles to be different from that of fundamental particles. Tamoghna Banerjee · 4 years ago

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It is because we exist. http://en.wikipedia.org/wiki/Anthropic_principle Ivan Stošić · 4 years, 1 month ago

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@Ivan Stošić Oh come on, really. Everyone knows that's weak. Kinda funny if it was meant as a joke though. Ali Rahemtulla · 4 years ago

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One interesting consequence of the second question:

qa=(Gm1m2)/(r^2) or a=-(Gm1m2)/(qr^2).

If m1 is the mass of the earth, and m2 the mass of the other things on it, and r the radius of the eath the acceleration of particles towards the earth(which normally is 9.8 m/s^2 irrespective of it's mass) would now depend on the particle's mass to charge ratio(m2/q).


Since our body contains a small charge of about 10 microcoulombs and if i take an average mass of 50 kg, the acceleration comes out to be 49*10^6 m/s^2. Which means even a small jump will kill us when we reach the floor! :P

In such a world, we will have to always keep ourselves highly charged. Which is certainly not possible. Only electric pokemon could survive in such a world! Alternatively, creatures with really small mass and high charge could. In a parallel universe, maybe. Mani Jha · 4 years, 1 month ago

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I have read in a book that the question 1 is the basis for Eistein to establish his theory of relativity Đinh Ngọc Hải · 4 years, 1 month ago

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@Đinh Ngọc Hải It's one of them! David Mattingly Staff · 4 years ago

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The kinetic energy of a particle classically is \(\int F dx =m \int \frac{d\frac{dx}{dt}}{dt} dx =m \int \frac{dx}{dt} d\frac{dx}{dt} =m \frac{(\frac{dx}{dt})^2}{2}=m\frac{v^2}{2}\). Evidently, replace \(m\) with \(q\) in Newton's second equation and kinetic energy is now \(q\frac{v^2}{2}\)[1]. Also, the rest energy of a particle is \(mc^2\), which would also change to \(qc^2\) (I think this is fundamentally because of the equivalence principle). Thus its total (non-potential) energy is \(q\frac{v^2}{2}+qc^2\)[2].

The result of this is that pair production of particles and antiparticles can occur with no external input of energy, as long as \(\sum q\frac{v^2}{2}=k\) ('kinetic energy' is conserved) and \(\sum q \mathbf{v}=0\) ('momentum' is conserved). Evidently, this does not happen, as we are not living in a giant plasmaball of death.

[1]Relativistically, I guess the kinetic energy \(mc^2 \left(\frac{1}{\sqrt{1-(v^2/c^2)}}-1\right)\) would change to \(qc^2 \left(\frac{1}{\sqrt{1-(v^2/c^2)}}-1\right)\) (not sure how to prove this), but my argument is pretty much the same either way.

[2] Or \(\frac{mc^2}{\sqrt{1-(v^2/c^2)}}\) A L · 4 years, 1 month ago

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There is no known reason for the equivalence of inertial and gravitational mass. This is a striking experimental fact as far as I know. Kikipoi Loisir · 4 years ago

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@Kikipoi Loisir can you give us any information, even a readable link, about how they test this? David Mattingly Staff · 4 years ago

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Sir Issac Newton did said that force is directly proportional to differential coefficient of momentum w.r.t time but there must be and equivalence constant such as k to express F = k *(\frac{dp}{dt} ) . where p = momentum so why is the value of k = 1 used??? Ayush Shaw · 4 years, 1 month ago

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@Ayush Shaw That equation can be arrived at using quantum mechanics. More precisely, the expectation value of the force equals the time derivative of the expectation value of the momentum. Marcell Simkó · 4 years ago

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Although I'm not too knowledgeable about general relativity, I understand that the equivalence principle (inertial mass and gravitational mass are equal) comes from the fact that the motion of objects is based solely off the geometry of space-time. David Zheng · 4 years, 1 month ago

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In my opinion, your second question is a little out of sense, because is a bit improbable to "acelerate" a eletric charge and obtain any sort of force. I think that will just cause a eletricmagnetic phenomenon or something. (Correct me if I am wrong with any physical argument, I'm not good at physics at all) Leonardo Cidrão · 4 years, 1 month ago

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I guess mass describes the property of any particle or body and is the fundamental property of nature Dipanjan Mondal · 4 years, 1 month ago

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@Dipanjan Mondal it a clear fact that " mass is the measure of inertia in a body " Ayush Shaw · 4 years, 1 month ago

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