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.

-
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
This question is a duplicate, and it's unfortunate that it's been accumulating a collection of answers that duplicate some of the incorrect answers to the previous question. –  Ben Crowell Oct 4 at 23:07
Any answer other than "we don't know" is false, IMHO :) –  DanielSank Oct 4 at 23:40
@DanielSank: We do know. It has the value it does because of our system of units. –  Ben Crowell Oct 5 at 0:22

6 Answers

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.

-
"This is simply a choice of units": no it's not. This drives me crazy. Equations in physics do not have units. Setting $c=1$ is a choice of simply dividing everything in the equation by $c$! –  DanielSank Oct 4 at 23:40
Yes, it is. Unfortunately, I find your claim that "equations in physics do not have units" too cryptic to reply to further. I don't understand what you think you're replying to or why you think this statement is relevant, much less what it means. –  Mark Eichenlaub Oct 5 at 1:07
@DanielSank Think a bit. What is a unit definition? Take length. The unit is "meter", we divide everything connected with length by "meter" to get a number. It is a choice of unit. When we put c=1 , we divide everything relevant to velocity by c in meters/second and change the number to units of c. –  anna v Oct 5 at 3:49
@DanielSank as for equations? Operators operating on the wave function give numbers in appropriately defined units for energy, mometnum ets. So equations must have units in order to relate to physical measurements. –  anna v Oct 5 at 4:05
@MarkEichenlaub: When I write $i\hbar d_t |\Psi\rangle = \hat{H}|\Psi\rangle$, what units is that equation in? None. That equation is valid in any system of units. Now I can divide both sides by $\hbar$, giving $id_t|\Psi\rangle = \hat{\Omega}|\Psi\rangle$ where now $\hat{\Omega}=\hat{H}/\hbar$. This is not "setting $\hbar$ to 1", it is dividing everything by $\hbar$ so that the Hamiltonian is now in dimensions of frequency. There aren't any units in sight. –  DanielSank Oct 5 at 4:30

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.

-
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
This question is a duplicate, and this answer is also a duplicate of a similarly fallacious answer to the earlier question: physics.stackexchange.com/a/3659/4552 –  Ben Crowell Oct 4 at 23:03

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.

-
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
This is mistaken; $c$ does not determine electric permittivity and magnetic permeability. Electromagnetism is described by two constants; if you don't wish to either of $\epsilon_0,\mu_0$ and you take $c = 1/\sqrt{\epsilon_0\mu_0}$ to be one of them, then a natural choice for the other would be the impedance of free space $Z_0 = \sqrt{\mu_0/\epsilon_0}$. So 19th century physics moved from three parameters to two, not three to one. –  Stan Liou Mar 15 at 11:34
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. Not true. Dimensionful constants have the values they do because of the units we've chosen. For similar reasons, experiments cannot determine whether a dimensionful constant changes over time. Only dimensionless constants have values that are independent of the system of units, and it only makes sense to search empirically for changes in dimensionless constants. See Duff, 2002, "Comment on time-variation of fundamental constants," arxiv.org/abs/hep-th/0208093 –  Ben Crowell Oct 4 at 23:06
What's not true? I said physicists were considering the issue, not that it was settled or true. The Duff paper is in fact a criticism of several other papers on the topic for the reasons you mention. If nobody was working on those lines Duff would have had nothing to write about. –  Charles E. Grant Oct 4 at 23:42
@CharlesE.Grant: Duff is correct. The people whose work Duff is criticizing are not very competent. If their results had been correct, then they could have been interpreted as a change in the dimensionless fine structure constant, but not as a change in $c$. (Their results didn't in fact turn out to be correct, which is a different issue.) –  Ben Crowell Oct 5 at 1:37

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 $c = 3 \times 10^8 \; \mathrm{m/s}$ 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.

-

The speed of light comes from the nature of space-time fabric. Any massless field carrier will move along at a speed equal to the space-time conversion. Oliver Heaviside showed that if a field moves at a finite speed, then a magnetic-like cofield exists, such that the maxwell equations arise, and the field travels at some velocity.

The principle of relativity then supposes that it would be possible to measure one's proper motion, unless all such speeds derive their speed from the spavetime metric.

-

Well i was thinking about the same question and i stumbled upon this explaination which seemed to be the most logical to 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.

Taken from Speed of light

-
Hi chossen-addict, this comes awfully close to plagiarism. Please be careful about quoting from other sites, especially when you quote from multiple users without attribution. –  Brandon Enright Apr 25 at 23:42
I think that I could quote someone while referencing them , I don't think its plagiarism but still I ll be more careful next time –  chossen-addict May 28 at 11:23
Before I edited it just now, this absolutely was plagiarism, because it wasn't quoted. I've edited it to be acceptable. Keep in mind that in order not to qualify as plagiarism, any material copied from another source must be quoted and must clearly identify the source. –  David Z Oct 4 at 22:47
Thanks for clarifying , will be careful from now on. –  chossen-addict Oct 17 at 13:07

protected by Qmechanic♦Oct 4 at 20:54

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

Would you like to answer one of these unanswered questions instead?