What is mass? This is a fundamental question, which has recently, specially since the discovery of the Higgs boson five years ago, has often come up in questions or discussions on the IFT videos. So I’m going to explain what mass is. And… this has to do with the Higgs boson in the sense that the Higgs fields in the vacuum is the origin of the mass of particles with non-zero mass. But let’s cut the chase and let’s find out what is mass. To do that, we’re using the scientific method, which is as follows: First, choose some concepts which we consider fundamental, and preferably defined in a practical way. Next, study the relations among them, both theoretically and experimentally, to reach our conclusions, if any. And, if by doing more experiments, we find that our conclusions are wrong, we go back to point 1, and start all over again. Ok, let’s choose two concepts that we consider fundamental: space, which can be measured by distances, in practice measured with my own ruler; and time, which I can also measure with my watch. If we now take an object which is moving a given distance in a given time, with no external forces, we find that something, the quotient of distances and times, which is the velocity, is constant. Well, the velocity not only has a magnitude, which I call “v”, but also a direction, and is hence a vector, something that points in a certain direction. So let’s play a bit with vectors. I should remind you about how to add up two vectors. If I have a given velocity, which is a vector pointing in some direction, and which I call “v1”, and a second velocity, which I call “v2”, the question is what is the sum of “v1” and “v2”. Easy! To add up, we apply parallel transport to “v2” until its tail is a the head of v1 and the vector from the tail of v1 to the head of v2 is the sum “v1” + “v2” These are all the maths we need to go on. Ok, next thing is to do experiments in a billiard table. An ideal billiard table, with perfect and frictionless surface, with perfect balls, perfectly spherical, as ideal as possible. And then, two billiard champions are training by making pairs of balls collide; each one smacks on a ball, and they check what happens after they collide. These are vectors. So the balls bounce back with velocities in general different from the original ones, this with velocity v2′ (the ‘ meaning “after the collision) and this with velocity v1’. So what we find after many such experiments is the following: the sum of v1 and v2 before the collision equals the sum of v1 after the collision and v2 after the collision. And this holds for any collision. We also discover that v1 squared + v2 squared (these are numbers, so I can take their squares) equals v1 squared after the collision +v2′, after the collision, squared. Great! We’re on a good track, we have discovered, it seems, certains laws for collisions, certain laws of nature in the dynamics of particles. Let’s now change things and perform the same experiment, but with balls still of the same size, but made of different materials. In such case, we immediatly find that these formulas are wrong. To fix them, the most natural suggestion is to associate a quantity, which I’m calling “m”, to each of the two balls, “m1” for the first, and “m2” for the second. And we’ll find, after much experimenting, that the combination m1 v2 + m2 v2 is conserved. And also here if we also add these two things, m1 and m2. But this holds only for a given ratio of m1 and m2, a certain ration is the only one such that these laws are valid. Ok, we have then discovered two interesting things: one is the conservation of momentum, which for instance for particle 1 is m1 v1, and second, the conservation of enegy, the energy before and after is the same. Note that the convention, for reasons having to do with other conventions, such as what is a force and so on, the formula for the energy we have discovered is 1/2 m v^2. The enegry 1/2 m v^2 is a conserved quantity in particle collisions. Ok, it seems we all should be happy that these laws work out. But if we now do these experiments with particles whose speed is not much smaller than the speed of light, called “c”, if their speeds are comparable with that of light, then these conclusions are again wrong. And the quantity m1 v1, for the conclusion to hold, must be divided by the square root of 1 minus the particle velocity squared divided by the speed of light squared. This formula must also be modified, by replacing by c^2 here, removing the 1/2, and dividing by the same square root of 1-v^2/c^2. Once we do that the formulas for the conservation of energy and conservation of momentum are now correct. In the meantime we have also discovered another fact. I could rotate the billiard table without rotating the balls, or rotate the balls without rotating the billiard table, and independent of the orientation of this my reference system, the laws are still the same. Thus the basic laws of Mechanics are invariant under rotations. One last point Let me take units in which the speed of light is 1. Then using these formulas, we find that, for a given particle, E^2-p^2=m^2 The energy and momentum depend on the velocity with respect to the observed, but the mass does not. So energy and momentum are not relativistic invariants; but the mass we have introduced here, which is the intertial mass, is a relativistic invariant. When we introduce gravity and move on to Einstein’s Theory of General Relativity, it turns out that the mass sourcing gravity is the same as this. Inertial mass and gravitational mass are one and the same thing.