Einstein is one of the most famous scientists in history, and one of the most famous equations of this most famous scientist is E=mc^{2} which, although famous, most people don’t know what it means. It is, of course, the equivalence of mass and energy. But what does that mean?

What E=mc^{2} actually means is that mass is a form of energy. Indeed, mass is just energy that is confined by the Higgs field. All particles are massless, as is seen with the Dirac equation:

A massless particle, though, does not experience time – the life of a photon, for instance, would “feel” like an instant from the point of view of the photon. The same would be true if it were possible for you to move at the speed of light: your lightspeed journey would take a literal instant. However, we know that particles like electrons (and neutrinos as well) experience time because their chirality flips: they evolve over time, flipping between right-handed chirality and left-handed chirality. Strangely enough, left-handed chirality is different than right-handed chirality – the two are not symmetric, resulting in parity violation. Left-handed chirality allows electrons, for instance, to interact not just with the electromagnetic force, but also with the weak nuclear force – when in the left-handed chirality, electrons have what is called weak hypercharge. The electron can only flip between left-handed and right-handed chirality by accepting and discharging (respectively) this weak hypercharge, and it picks up and discharges its weak hypercharge from and to the Higgs field, which takes on all values of the weak hypercharge at once, making it an infinite source and sink of the weak hypercharge. This constant exchange of the weak hypercharge with the Higgs field at infinitesimally small timescales causes it to slow down from light speed – the electron is made of pure energy that is trapped in a massless container of the Higgs field.

This idea of energy trapped in a massless container is an important intuition. The reason energy takes on the property of mass is because, if you imagine a massless container with perfectly reflective internal surfaces, with constantly reflecting photons inside, that is sort of like what matter is. The reason matter has inertia, for instance (why it is difficult to start moving it when you apply force in one direction) is because the pressure of the photons bouncing around inside the massless container will increase in the opposite direction you are applying force: more photons will be hitting the backside of the container and fewer will be hitting the front side, causing a net force acting against the force you are applying and thereby resisting the acceleration. Mass is literally the force that is the resistance to acceleration: if F=ma then by algebra we get m = F/a

We can also say that since E = Force*c where Force is roughly = m*c we can then say that E = (m*c)*c then we can say E=mc^{2} we can then use algebra to get m =E/c^{2} where E is equal to the energy of the photons in our massless box and *c* is the speed that they are traveling inside that massless box. And so, we can think of an elementary particle, like an electron, as the energy trapped within a massless container made out of Higgs field.

Something interesting about moving at the speed of light, though, as I have said, is that it would make everything happen in an instant. The reason for this is because, from our inertial frame of reference, a particle moving at the speed of light has its clock frozen – no time is passing. We can think of it like this: if a photon, a particle moving at the speed of light, inside our massless box is bouncing back and forth from the top and bottom of our massless box at a 90 degree angle to the surfaces, so that it remains bouncing off the same two spots, we can think of this like the container’s internal clock. There is a steady, reliable interval of time between when the photon bounces off the bottom, up to the top, and back down to the bottom – the distance and the speed of the photon remain constant. However, if the container begins moving sideways, relative to us, then the distance the photon must travel increases – it is now moving diagonal – but the speed of light is constant, so it takes longer for the photon to go from bottom to top and back down to the bottom again (from our inertial frame of reference):

However, from the inertial frame of reference of the container (we’ll call it container A), it is still at rest while at constant velocity. But, a photon-in-a-container clock that is at rest in our inertial frame of reference (we’ll call it container B) will be moving from container A’s frame from reference. If container A is moving at light speed, when it looks at the photon in container B, it is traveling perfectly horizontal – it is stuck in place and will never touch the top or bottom as long as container A continues moving at light speed. Thus, from the inertial frame of reference of container A, time is completely stopped everywhere in the universe, and therefore container A will arrive at its destination at exactly the same time that it departed – the journey is instantaneous.

In this spacetime diagram (apologies for sloppiness, I whipped it up in Paint) we can see that from the inertial frame of reference for the first container, the red photon reflects off the insides of the container in regular intervals, whereas in the second container, the blue photon never catches up with the side of its container because it is always moving away from it at the speed of light. That means that no time passes for the second container, but the contained red photon causes time to pass in the first container. Of course, from the inertial frame of reference of the second container, the blue photon is bouncing off the sides. This is technically true for the first container as well, but from that inertial frame of reference, time is standing still in the second container, so although it is in principle that the blue photon is bouncing off the sides, time as at a standstill and so it is not progressing any further than where it is stuck in the middle of its bounce (halfway between the two sides as shown in my diagram, but it could be the case that the blue photon is frozen in time anywhere inside the second container).

What this means is that the progression of time for matter requires the ensemble of the energy (the photons) and the massless container (the Higgs field). The flow of time – the “clock” by which matter “experiences” time – is the interaction of the photons with the massless container (the energy with the Higgs field as transfers of the weak hypercharge).

Something interesting, which may just be a confusion on my part, is that the massless container in the ensemble must be moving, such that the massless photons moving at the speed of light continue interacting with it without escaping. Yet, the massless container, since anything without mass moves at the speed of light, must also be moving at the speed of light. Since this massless container is the Higgs field, that must mean that the Higgs field is itself moving at the speed of light in some sense, and yet it must be stationary from the frames of reference of every massive particle, since all mass uses the Higgs field as its massless container. In what sense the Higgs field is moving at the speed of light while also being stationary from the frame of reference of every particle that has mass, I don’t know.

Let’s say that within our container we have two photons, both moving in opposite directions, such that they cannot both be made to have arbitrarily small total energy by changing frames, or by moving toward or away from either of them. In a two-photon system, the energy of one photon is decreased if you try chasing it, but the energy of the other photon – the one you are now moving away from – has an equal increase in energy. Two photons that are not moving in the same direction comprise an inertial frame where the combined energy is smallest, but not zero, called the *center of mass frame*. To get the photons to have equal energy while moving in opposite directions, you must move in the same direction and velocity as the center mass frame of the two photons. The total momentum of the photons will then be zero, since their momenta are equal and opposite. In this frame the two photons, as a system, have a mass equal to their total energy divided by the speed of light squared (m = E/c^{2}). This mass is called the invariant mass of the pair of photons together. It is only the invariant mass of a two-photon system that can be used to make a single particle with the same rest mass. Our particle, which is a container filled with photons (energy interacting with the Higgs field), has numerous photons moving in all different directions relative to one another, but it will have a single invariant mass.

However, if we take the point of view of a single photon, then how can the other photon be said to be moving away from it if it appears to be standing still in time? It is perhaps the case that instantaneous interactions are not actually instantaneous, but merely infinitesimal. If everything, from a photon’s inertial frame of reference, has no time passing, then this would also be true of the Higgs interactions: from the point of view of the photon, the next interaction with the Higgs field would never happen, because the Higgs field would be frozen in time. Two ways that this might not be the case: the infinitesimal time scales at which transfers of the weak hypercharge occur are so fast that they could be considered instantaneous (i.e. instantaneous is not instantaneous but infinitesimal); or, as I said previously, there is some way in which the Higgs field is both moving and stationary. So, is it perhaps the case that time itself is inherent in the Higgs field in some way? What would happen, then, if the Higgs field decayed to zero energy?