# Why does the speed of light $c$ have the value it does?

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.

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I personally don't know why it is what it is. But, generally, what's wrong is how you got the information. I'm sure many scientists have asked and attempted to answer this question, and maybe some found the answer, but what's wrong here is that I was told the number, or told that it's 1 in natural units, but never told the why. I wasn't told, I don't know where to look, but I'm sure it's already been found or being researched. So 'science' isn't at fault. –  Noein Mar 15 '13 at 16:51
duplicate of physics.stackexchange.com/q/3644/4552 –  Ben Crowell May 18 '13 at 18:53

You've seen the speed of light quoted as roughly $3*10^8\, \text{m/s}$, 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

$$v = \sqrt{\frac{E}{m}}$$

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

$$E = \alpha^2 m_e c^2$$

where $\alpha = \frac{e^2}{\hbar c}$ is called the fine structure constant. Putting this together, we find

$$v = c \alpha \sqrt{\frac{m_e}{m_N}}$$

where $m_N$ 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 $v \approx c * 10^{-4}$

In other words, the speed of light is $10^4$ or $10^5$ 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 $10^{-5}$, 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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SPEED OF LIGHT:

This is a very interesting question. Going through the foundations of electromagnetism and the theory that led to Maxwell’s equations, there is an interesting element that can grab your attention. You can see that the speed of light is not as abstract and mysterious as it appears to be, but only if you look from a different perspective.

I will write Maxwell’s equations, in the vacuum, before their unification stage into one equation, the wave equation that proves the existence of electromagnetic waves, and which is a strong evidence of the unification between the electric and the magnetic field into the electromagnetic field. The equations can be found in any standard textbook on electromgnetic theory:

$\nabla\times{\bf E}=-\mu_0\frac{\partial{\bf H}}{\partial t}$

$\nabla\times{\bf H}=\epsilon_0\frac{\partial{\bf E}}{\partial t}$

$\nabla.{\bf H}=0$

$\nabla.{\bf E}=0$

These two equations lead to the wave equation for the EM field components:

$\nabla^2E_i=\epsilon_0\mu_0\frac{\partial^2 E_i}{\partial t^2}$

similarly for the magnetic field components.

The interesting feature in these wave equations is the factor $\epsilon_0\mu_0$ because they determine the speed of light

$c=\frac{1}{\sqrt{\epsilon_0\mu_0}}$

The history of these two constants is long and somewhat convoluted. The electric permittivity, $\epsilon_0$ and magnetic permeability, $\mu_0$, of free space can in principle, using modern technology, be determined by electrical measurements. For example $\epsilon_0$ can be measured using a plate capacitor. Measuring its capacitance and its geometrical features, we can then use the equation below to determine $\epsilon_0$

$C=\epsilon_0\frac{A}{d}$,

where $A$ is the area of the plates and $d$ the distance between the plates. Similarly, using the equation for the balance of a weight, $mg$, of some known mass, by the magnetic field force using the current balance exerimental arrangmet, we have

$mg=\mu_0\frac{I^2L}{2\pi a}$

where $I$ the electric current in the wire of the "current balance" and $L,a$ are parameters that are part of the experimental design.

Thus the question why the speed of light has the value it has, can be reduced to the question why these physical constants have the values they do. To make the discussion more interesting, the product of these constants must turn out to be independent of the reference frame in which they are measured, in order to ensure that the speed of light is the same for all observers.

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your last paragraph can be answered by reverse engineering from the experimental evidence for the special theory of relativity, i.e. c is constant therefore the product of the constants determining it must be invariant under lorenz transformations. You are right by referring why from problem number one to problem number 2: why these constants. Anybody studying physics knows that there is no ultimate unswer to "why" questions, but a reduction with demonstrating "how" to more esoteric relationships, and finally to axioms, for which there is no "why" answer except the circular one, consistency. –  anna v Mar 16 '13 at 7:35
A natural follow-up question would be: how far can this analogy be pushed? For wave propagation in materials the speed depends in large part on the stiffness. To create a capacitor with high capacity a material with a high value for the electric permittivity is placed between the capacitor plates. The electric field has a physical effect on that dielectric. In that sense it's a surprise that for a vacuum between the plates a non-zero value for the electric permittivity is measured. It's a very small value, but not zero. So: vacuum still has properties that allow propagation of EM-waves. –  Cleonis Mar 17 '13 at 11:42

Back in the early 19th century two of the natural constants were the electric permittivity and magnetic permeability of the vacuum. Then in the middle of the century Maxwell figured out that these two constants were uniquely determined by the speed of light. Moving from three apparently independent constants to one constant was one of the high points of physics in the 19th century.

Any number of physicists are working to find out if c, G, h, and e, are really constant and why they have the values they do. It's just that so far they haven't had any clear success. Nobody is satisfied with this, but apparently it's a very hard and deep problem.

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Actually, several theorists (Dirac among others) have suggested that the fundamental constants change with time. See for example physicsworld.com/cws/article/news/2010/sep/02/… –  Charles E. Grant Mar 15 '13 at 22:53
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 $c = 3 \times 10^8 \; \mathrm{m/s}$ 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 $3 \times 10^8 \; \mathrm{m/s}$? 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.