Why Is $m$ Special?

Our question this week was twofold:

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.)

and

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?

Let's tackle question 1 first. Simple answer - there is no necessary theoretical reason the inertial mass $m_i$ that appears in $F=m_ia$ is the same as the gravitational mass $m_g$! I can write down perfectly good physical theories where this is not so. The assumption that they are is the (weak) equivalence principle, and is one of the fundamental assumptions that goes into general relativity, Einstein's theory of gravitation. (There are different flavors of the equivalence principle, which is why that word weak is in there. Don't worry about the flavors unless you plan on going into gravitational physics as a career.) Einstein's general relativity is a beautiful theory, but just because it's beautiful doesn't mean it necessarily applies to nature. That it does, and that its assumptions are valid, are experimental observations. We experimentally determine that $m_i=m_g$ and derive theories based around that fact, we can't require a property of nature just because we have a pretty theory.

One can see how the equivalence principle is important to general relativity rather simply. Many of you have heard or read in popular science shows or books that general relativity is a theory of space and time, that masses cause a "curvature" of the spacetime, which we interpret as gravity, and that particles move along the shortest path between two points in this curved spacetime. Consider this last property. Put a particle with some mass, charge, etc. down at some point in space and let it go. Assuming that only gravity is acting on the particle (no other charges around for example) the particle is supposed to move along the shortest path between two points. But, the very notion of shortest path is a property of the geometry of spacetime, i.e. something completely external to the particle.

Therefore if general relativity is to be a valid theory, the motion of particle under the force of gravity must not depend on any property of the particle. Ever notice how in introductory Newtonian mechanics we just willy-nilly say that the acceleration of a particle on the earth's surface is $g$, without regards to what kind of particle it is? This is exactly saying that the motion of a particle under the influence of gravity is independent of the properties of the particle, i.e. the equivalence principle holds. The equivalence principle and the equality of $m_i=m_g$ is why we can treat gravity as a spacetime phenomenon that is universal for all particles, and you actually learned the equivalence principle in introductory physics when you did the 1-d kinematics of balls dropping!

Now for question 2. Let's first rephrase our statement of $m_i=m_g$ to be "if I turn off all other known forces, objects will accelerate in a gravitational field in the same way". This is true if $m_i=m_g$ using Newton's law for gravitation, and not true if $m_i\neq m_g$, hence we can use this statement equivalently.

Now, one can make up well behaved theories of our world that violate the equivalence principle, in that if I turn off all other known forces particles accelerate differently in a gravitational field, but don't have any other horrible diseases (a horrible disease in physics is something along the lines of particles being able to have infinite negative energy, generating energy out of nothing i.e. really bad stuff that makes your theory immediately badly behaved). However, one can't easily make such well-behaved theories using electric charge. Some of the students noticed this in the discussion, as they gave the great answer to question 2 that one can't have $F=qa$ because q is both positive and negative. If you had such a modified Newton's second law then you can create perpetual motion machines or generate infinite energy with a positive and negative charge side by side, as those charges would spontaneously accelerate off to infinity. Now this immediately means something - if you make up a theory that violates the equivalence principle and is well behaved, you really need to do it in some other way besides allowing electromagnetic properties of particles to change Newton's second law.

This means that you have to postulate that all particles have another type of quantity associated with them, call it the T value, besides mass and electric charge, and that $T>0$ so that one can't make perpetual motion machines, etc. This causes an additional problem. Intrinsic properties like T are usually associated with forces - a mass generates gravity and an electric charge generates an electric field. If you try to build physical theories where particles have some new T value, you usually wind up introducing an extra force into nature. And indeed, one of the ways we test the equivalence principle is to look for signals of a "fifth force" besides the usual gravitational, electromagnetic, strong and weak nuclear forces between different objects. If such a force was present, and all particles had various non-zero T value so felt the T-force, then certainly particles would not fall the same in a gravitational field. Hence, as it turns out, equivalence principle violation, while mathematically pretty easy, often comes with a heavy price - a brand new force.

Moral of the story: unless you have a really, really good reason to violate the equivalence principle, leave it alone and assume $m_i=m_g$ is a law of nature.