Notes about the Speed of Light

I wanted to share this quote from Sean Carroll's The Particle at the End of the Universe because it's a handy way of understanding why nothing can travel faster than the speed of light (as opposed to arbitrary speed "barriers" like the speed of sound, which are really difficult engineering milestones only). The first two paragraphs are mostly to give a bit of context to what he is talking about.

When the Bevatron created antiprotons, it wasn't because there were antiprotons hidden in the protons and atomic nuclei they were working with. Rather, the collisions brought new particles into existence. In the language of quantum field theory, the waves representing the original particles set up new vibrations in the antiproton field, which we detect as particles.

In order for that to happen, the crucial ingredient is that we have enough energy. The insight that makes particle physics possible is Einstein's famous equation, E=mc^2, which tells us that mass is actually a form of energy. In particular, the mass of an object is the minimum energy that object can have; when something is just sitting perfectly still, minding its own business, the amount of energy it possesses is equal to its mass times the speed of light squared. The speed of light is a big number, 186,000 miles per second, but its role here is just to convert units of measurement from mass to energy. Particle physicists like to use units where speed is measured in light-years per year; in that case c is equal to one, and mass and energy become truly interchangeable, E=m.

What about when an object is moving? Sometimes discussions of relativity like to talk as if the mass increases when a particle approaches the speed of light, but that's a little misleading. It's better to think of the mass of an object as fixed once and for all, while the energy increases as it goes faster and faster. The mass is the energy that the thing would have if it was not moving, which by definition doesn't change even if happens to be moving. Indeed, energy grows without limit as you get closer and closer to the speed of light. That's one way of understanding why the speed of light is an absolute limit to how fast things can go - it would take an infinite amount of energy for a massive body to move that fast. (Massless particles, in contrast, always move at exactly the speed of light.) When a particle accelerator pushes protons to higher and higher energies, they are coming closer and closer to the speed of light, never quite getting there.

Chapter 4, p56-57


His primary goal here is not to explain the speed of light, but to give some background for what you are doing in a particle accelerator. You can create energy from mass and more importantly (for his interests here) you can create mass from energy, or heavy particles from lighter particles that have a ton of energy (traveling very fast). Since you have that constant of c^2, you can get a lot of energy from a tiny amount of mass, but it takes a ton of energy to create mass. So you might take relatively smaller particles and accelerate them close to the speed of light (you can't reach it without an infinite amount of energy) in a particle accelerator, and smash them together to get heavier particles.

Anyway, even though it wasn't his direct intent, I thought he gave one of the clearer explanations for why the reason the speed of light is uniquely a cosmic speed limit. To be honest, I'm not sure it even makes sense to consider the speed of light as a speed itself; it's more like it's part of the definition of time itself. The massless particles (like photons) all travel at the speed of light, with massive particles (that is, they have mass) being slower than that relative to their total energy. If you thought about a collection of cars you would have one that is fastest but really its speed is somewhat arbitrary. To suggest one of them is the fastest and it is impossible to go faster sounds like human hubris in the realm of automotive engineering. But light isn't merely the fastest of a set of speeds - it is like it is speed/time itself. If you did manage to have an infinite amount of energy somehow, you'd basically be stopped in time. And moving faster than the speed of light is equivalent to moving backward in time (that's not just my interpretation here, a lot of physicists have explicitly stated this).