What is mass?

The (most likely) Higgs boson discovery brings up a distinction in my mind that seems to be getting overlooked in most of the popular press discussions of the CERN work.  What do we mean as physicists when we talk about "mass"?  In classical mechanics, there are in some sense two types of mass.  There is gravitational mass - in Newtonian gravity, the gravitational force between two particles of masses \( m \) and \( M \) has the magnitude \(G m M/r^{2} \), where \( G \) is the gravitational constant and \(r \) is the distance between the particles.  The force is attractive and acts along the line between the two particles.  The Higgs boson has no (direct) connection to this at all.

There is also inertial mass, and this can be described in a couple of ways.  The way we usually teach beginning students is that a total force of magnitude \(F \) exerted on an object (in an inertial frame of reference, but that's a detail) produces an acceleration \(a \equiv d^{2}r/dt^{2}\) that is linearly proportional to \(F \).  Exert twice as much force and get twice as much acceleration.  The constant of proportionality is the inertial mass \( m \), and we write all this in one form of Newton's Second Law of Motion, \( \mathbf{F} = m \mathbf{a} \).  The more (inertially) massive something is, the smaller the acceleration for a given amount of force.

A more subtle way to define this would be to say that there is this thing called momentum, \(\mathbf{p} \), which we believe to be a conserved quantity in the universe.  Empirically, momentum is connected with velocity.  At low speeds (compared with \(c \), the speed of light), momentum is directly proportional to velocity, and the constant that connects them is the mass:  \( \mathbf{p} = m \mathbf{v} \).  (The full relativistic expression is \( \mathbf{p} = m \mathbf{v}/ \sqrt{1-v^{2}/c^{2}} \) ).  The more massive something is, for a given speed, the more momentum it has (and the more it's going to pack a whallop when it hits you).

The coupling of elementary particles to the Higgs field is supposed to determine this mass, the relationship between momentum and velocity (or equivalently, between momentum and energy).  As far as we know, the inertial mass and the gravitational mass appear to be identical - this is the Equivalence Principle, and it's supported by a wealth of experiment (though there are always ideas out there for new tests and new limits).