Brian Bi

## Why is the speed of light very high?

As other answerers have already noted, the speed of light is a dimensionful quantity and therefore its size depends on the units you use to measure it. In other words, when you say the speed of light is high, you are comparing it to your unit of measurement, which is (maybe) 1 m/s. The speed of light is 299,792,458 times greater than this, so it seems very high. If you choose your unit to be the speed of light itself, then most other speeds just seem very low, rather than the speed of light seeming very high.

However, I'd also like to take another tack here, which I hope will be more illuminating, by reframing the question. The speed of light is much, much greater than the speeds we encounter in our everyday lives; why is this?

The answer has to do with energy densities. Let's consider an object of mass $$M$$, initially at rest. How much energy would it take to accelerate it to, say, 1% of the speed of light? This is still slow enough so that we can treat it classically; the answer is then about $$0.00005Mc^2$$, or 1/20000 of the object's rest energy. If the object is not being accelerated by external forces, all that energy has to come from within the object itself. That is, the object has to be in an excited state (even though we might not normally consider it in such), and has to transition to a lower energy state. The energy difference must be enough to make up the kinetic energy needed for the acceleration.

1/20000 doesn't seem like very much, but what kinds of energy densities can we typically achieve? Let's consider a simple chemical reaction, the combination of hydrogen and oxygen to form water. Two moles of hydrogen molecules and one mole of oxygen molecules have a combined mass of about 0.036 kg, corresponding to the rest energy of about $$8.99 \times 10^{16}$$ J. The free energy of reaction under SATP is about $$1.19\times 10^5$$ J [1]. We see then that the energy released in a typical chemical reaction is on the order of magnitude of a trillionth of the rest energy of the reactants. So forget about building a gas-powered car that can reach 1% of the speed of light—it's just not possible. Electric cars won't fare any better—a battery is just a device that converts a chemical energy gradient into an electrostatic one, and therefore you can't achieve a higher energy density that way.

We human beings are powered by chemical reactions. So are all other living things. So no known organism can propel itself to speeds anywhere approaching the speed of light, by the same token.

What about the nuclear scale? The fission of a 235U (uranium-235) nucleus generates about 202 MeV of energy [2]. The rest energy of a 235U nucleus is about 219000 MeV. So in a nuclear reaction you can recover about 1/1000 of the rest energy of the reactants. So what's stopping us from building a nuclear-powered vehicle that reaches 1% of the speed of light? Essentially, the fact that nuclear reactors are very big and complicated, so you have to drag around all that additional mass.

Finally, we ought to ask ourselves why these "excited states" we use (e.g., hydrogen and oxygen gas as an excited state relative to water, or 235U as an excited state relative to fission products) are so lame. Why is the energy difference we can extract from them only a tiny fraction of the rest energy? Basically, we're lucky we can even get that much. Because of entropy, excited states always decay, over a long enough period of time. 235U naturally decays; nuclear fission is just forcing it to relax more rapidly (though the products are not the same). Even hydrogen and oxygen gas, if you leave them in a sealed container for long enough (I don't know how long), will eventually completely combine to form water. We're lucky that there exist excited states that live long enough so that we can store them and use them as fuel. Contrast this with, say, the $$\Delta^{++}$$ baryon, which is an excited state of the proton. When it decays, it releases about 24% of its rest energy. If we could just have a tank of hydrogen where all the protons are replaced by $$\Delta^{++}$$ baryons, imagine how much propulsion we could get out of that! Unfortunately, the $$\Delta^{++}$$ baryon decays within about $$10^{-24}$$ s, and there is no way to force it to last any longer.