Why does the speed of light have the value it does?

From old orange on the Physics Stack Exchange:

Why does light have the speed it does? why is it not considerably faster or slower than it is? I can’t imagine science, being what it is, not pursuing a rational scientific explanation for the speed of light. Just saying “it is what it is” or being satisfied saying it is 1 (c=1), does not sound like science.

Here are a couple of answers.

This one wow-ed me:

There’s a fundamental speed built into the fabric of spacetime called c. This speed c shows up all over the place in relativity calculations, and would be significant even if there happened to not be anything that actually traveled at that speed. Basically, if space and time are aspects of the same thing, then it should be possible to measure space with time units, or vice-versa, and if you were to do that, then speeds would be dimensionless. c is the speed that is equal to 1, in such units.

Well, one of the results of relativity is that any particle that has zero mass must travel at exactly c. Relativity itself is silent on the question of whether any such particles exist, but to the best of our ability to measure, the photon seems to be such a particle. So light travels at c.

Whoa.

Related to the above is this one; I liked the first half:

According to relativity, there are minimum and maximum speeds. Since photons are (believed to be) massless, they move at the maximum possible speed, hence the name “speed of light.” But the term really just denotes “the maximum possible speed.”

Moreover, relativity also theorizes that time and space are simply orthogonal directions in a larger manifold (with some conditions on the metric), so when we say c=3×108m/s we’re simply providing a unit conversion: one second is the same thing as three hundred million meters, just as 1 pound is the same thing as 453.6 grams.

So why is the value 3×108m/s? Well, the meter was chosen to be the easiest-to-define unit approximately equal to one yard (i.e., the length of an average person’s stride), and the second was chosen to be the easiest-to-define unit approximately equal to 1/(24*60*60) = 1/86400 of one Earth day. As it turns out, one second is then about three hundred million times longer than one meter.

Here’s Mark Eichenlaub:

You’ve seen the speed of light quoted as roughly 3108m/s, so the speed of light is very fast compared to one meter and one second. This is roughly a human walking speed, so your question could be interpreted as asking why light is few hundred million times faster than a walking speed.

The speed people walk is rather anthropocentric, though. Let’s choose something more neutral, like the typical speed of sound in a crystal. This is a few thousand meters per second. So the question we’ll investigate here is “Why is the speed of light about 10^5 times faster than the speed of sound in a crystal?”

Sound travels through solids as a compression wave. The atoms of the crystal are squeezed together somewhere, adding energy, and this sets up a traveling wave of compressions moving along the crystal. The stiffer the crystal is (more energy to squeeze), the faster the wave. The more inertia, the slower the wave. The only dimensionally-correct way to combine these to get a speed is

v=Em−−−√

where E is the energy per atom and m is the mass per atom. The mass just comes from the mass of particles. The energy in an atom comes from quantum mechanics, though. You can find it by balancing the electrostatic energy between an electron and a proton with the kinetic energy the electron has due to being confined to a region near the nucleus. So the energy depends on the strength of electric interactions, the electron mass, and Planck’s constant. Putting them together, you find that the energy is

E=α2mec2

where α=e2c is called the fine structure constant. Putting this together, we find

v=cαmemN−−−−√

where mN is the mass of a nucleus. Nuclei are some ten thousand times the mass of an electron and the fine structure constant is around .01, so that expression gives vc104

In other words, the speed of light is 104 or 105 times faster than the speed of sound in a crystal because the fine structure constant is small and because electrons are light compared to nuclei.

By the way, your aversion to setting c=1 is misplaced. This is simply a choice of units, not physics. In this unit system we would say that sound speeds are of order 105, so everything is the same as if we kept meters and seconds around.

To summarize: the speed of light is fast, but to make that meaningful we must specify what it is fast compared to. If we choose to compare it to everyday things like sound speeds, we find that the speed of light is fast because everyday things are made of atoms, and the energy in atoms is small. Sound speed isn’t special in this regard – you could take the thermal speed of gasoline you burned, for example, and it would be limited for roughly the same reasons. The energy in atoms is small because the fine structure constant is small and the electron is light compared to nucleons. There are no known reasons (to me at least) that the fine structure constant and ratio of electron to nucleon mass are small numbers.

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