# The origin of the value of speed of light in vacuum

Meaning, why is it the exact number that it is? Why not 2x10^8 m/s instead of 3? Does it have something to do with the mass, size or behavior of a photon? To be clear, I'm not asking "how we determined the speed of light". I know there isn't a clear answer, I'm really looking for the prevailing theories.

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Related, but not quite a duplicate I think: physics.stackexchange.com/q/1383 – David Z Jan 23 '11 at 7:11
Light moves at the speed $c$, but "speed of light" is a bit of a misnomer, because other massless things move at this speed, and in fact the speed of light is physically significant even without referring to electromagnetism or anything that moves at $c$. You might try David Mermin's paper "Relativity Without Light" adsabs.harvard.edu/abs/1984AmJPh..52..119M (not free) – Mark Eichenlaub Jan 23 '11 at 7:39
I would point you to the @LubosMotl answer, which makes the critical point that the specific number is just an accident of units chosen. Meters are French, seconds as a concept are very old indeed (Babylonian?). So, if you mix those two you get a pretty arbitrary number. My own favorite c number is available on Google by entering the text in these brackets: [speed of light in furlongs per fortnight]. Clearly a much more fundamental number, that! – Terry Bollinger Dec 10 '12 at 7:07
– Ben Crowell Nov 1 '14 at 16:02

Tom, would you have asked the question "why is the speed of light 1 ls/s" if we happened to measure distance in lightseconds and time in seconds?

The true answer to your question is: the speed of light is 1 if you measure distance and duration in compatible units, and it is whatever your system of units defines it to be if you adopt units that are more cumbersome. Another way of explaining is that speed - loosely speaking - corresponds to an angle in spacetime. And angles are dimensionless.

I know, this is not seen as a satisfactory answer. But that is because you ask the wrong question. The right question is "why is everything around us so slow? Why are the speeds we typically encounter for material objects around 10^-8 level?"

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It doesn't have to be 1, you can pick any non-zero value. – Joe Fitzsimons Jan 23 '11 at 7:39
@Joe: there is no better number than 1 though :) – Marek Jan 23 '11 at 15:20
This just moves the goalposts to a new question, though. You'll always have other arbitrary constants - permittivity of free space, mass of an electron, planck's constant, and so on, and you can't get them to ALL be one. – spencer nelson Jan 29 '11 at 3:07
All dimensionful constants (c, hbar, G) can be made equal to one. That just boils down to working in Natural (Planck) units. There is no changing of goalposts. Just the fact that only dimensionless constants can be targeted as 'to be explained'. – Johannes Jan 29 '11 at 16:21
Sure, the units attached to c can change and the number can be boiled down to 1 depending on the units involved. Regardless, whether it's 186k miles/per s or 1 ls/s, the question remains. Why does light (or massless particles) move at the speed that it does? Or is this just a fundamental constant of nature that doesn't have an explanation? – Stephen Watkins Sep 8 '14 at 0:48

The particular value of $c$ depends on how long a meter is and how long one second is. If meters were longer, for example, the speed of light would be a smaller number, even though light would still be as fast. Viewed this way, physical measurements are ratios. In this case, it's a ratio of the speed of light to a rather arbitrary speed - one meter per second.

One meter per second is roughly a walking speed. So your question might be interpreted as, "Why is the speed of light three hundred million times faster than a walking speed?"

This question is very anthropocentric. It is a question about how large we are (how many atoms are in our bodies), how much power our muscles can exert (the energy involved in chemical reactions), and how strong our bones and ligaments are (the strength of materials).

Since we would like to stick to physics, it will be more insightful to look at the speed of light as a ratio of something else. We should look for some other speed set by nature, rather than a human-based speed, and compare the speed of light to that.

A typical candidate is to take Planck's constant $\hbar$ and the unit of electric charge $e$. These can be combined to create a velocity $e^2/\hbar = 2.2*10^6 m/s$. (In some systems of units, you need to include other "constants" like the permittivity of free space to convert the units.)

This is, roughly speaking, the speed of an electron in an atom. An electron's energy is characterized by $E \approx e^2/r$, with $r$ the size of the orbit. Its angular momentum comes in units of $\hbar$, so $L \approx \hbar \approx mvr$. The virial theorem lets us write the energy as $E \approx mv^2$. Using these facts, we can look for a way to estimate the velocity. $v = mv^2/mv \approx E/(L/r) \approx (e^2/r)/(L/r) = e^2/L = e^2/\hbar$.

This "typical electron speed" is about $\frac{1}{140} c$. As a ratio, $e^2/\hbar c \approx \frac{1}{140}$. This is called the fine structure constant. It's very useful to know, because it's a number that describes the innate strength of the electromagnetic force.

Your original question becomes "why is the fine structure constant $\frac{1}{140}$?", or "Why is the speed of light $140$ when measured in fundamental units from quantum mechanics and electromagnetism?" Aside from a hokey invocation of the anthropic principle, I don't think there's an answer to this question, at least not yet. A physical "theory of everything" might hope to derive the fine structure constant from some more basic idea, but this has not yet been achieved, and it is unknown whether it ever will be.

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Nice demonstration of the fact that any question about magnitude can (and should) be translated into a question about dimensionless numbers. --- Just out of curiosity: why did you quote 1/140 (as opposed to ~ 1/137)? – Johannes Jan 23 '11 at 7:10
@Johannes Thanks. I used 140 because earlier in the post I quoted $e^2/\hbar$ to two significant figures, but I didn't have any special motivation for two sig figs in particular. According to Wikipedia it's known to 12 significant figures now. – Mark Eichenlaub Jan 23 '11 at 7:23
@Johannes btw, the number I actually have memorized is 1/137, same as everybody else. – Mark Eichenlaub Jan 23 '11 at 7:31

the speed of light was created by Nature to be one, the number whose multiplication influences nothing. But the primitive people who lived in spacetime and moved by speeds much smaller than $c=1$ - along small angles in the spacetime - were not able to see that their speeds were particular fractions of the maximum speed. The mankind remained that primitive until 1905 when Albert Einstein changed the story (with some marketing help by Hermann Minkowski in 1908).

So even though space and time are fundamentally the same quantity measured in different directions, the people chose different units for length and duration. Some particular people chose $1/24/3600$ of the solar day because the powers of $60$ and $12$ etc. were quite popular - a lot of random messy history of mathematical conventions. They called the units one second.

Other people chose one meter as $1/40,000,000$ of the circumference of a meridian.

In those randomly chosen units of distance and time - which were refined, to be more accurate, to the number of periods of various types of radiation - the speed of light $c=1$ could have been written as $299,792,458$ m/s. At least, the measurements became accurate enough so that the definition of one meter was changed in the 1980s to keep the speed of light in these units at least constant. So the speed I wrote is now actually exact, by definition.

Adult physicists who work with relativistic theories use units where $c=1$. Similarly, adult quantum physicists use units with $\hbar=1$, $k_B=1$, and sometimes $G=1$ when they study general relativity (or quantum gravity).

To summarize: the numerical size of the universal constants has nothing to do with fundamental physics - it is all about human conventions (the units).

Best wishes Luboš

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+1 for pointing out the only sane choice of units (none!). – Joe Fitzsimons Jan 23 '11 at 7:18
Actually, some adult physicists (including me) using $\hbar=2$ quite often ;-) – Frédéric Grosshans Jan 23 '11 at 12:22

Well, currently the speed of light is defined to be an exact number, with the second determined in terms of the electron transition times of cesium, and $c$ meters defined to be $c \times \left(1\, s\right)$. So, the trite answer for this is that we defined it to be so.

I would think that the more careful answer would be that chemistry happens at very low energies compared to typical relativistic energies. Since the energies are low, this means that the fundamental time scales of everyday life are much longer (in relativistic terms) than the fundamental length scales. $c$ tells you how to convert from one to the other.

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This is a very thoughtful answer. – Inquisitive Jan 24 '15 at 3:54

The speed of light ("$c$") is really a conversion factor that converts space distances into time durations. It is part of the geometry of space-time and in particular it is used to calculate the invariant infinitesimal proper time, which in Minkowski flat space-time, is given by this formula:

$d\tau^2 = dt^2 - ( dx^2 + dy^2 + dz^2 )/c^2$

For any particle or object with a non-zero rest mass, the proper time is an invariant that all observers will agree on and this value will agree with the time recorded by a clock carried by the massive particle or object. So this is the real meaning of the constant "$c$" - it is a conversion factor between space and time in the 4 dimensional space-time geometry.

Now according to Special Relativity, a massless particle must always have 0 proper time ($d\tau^2 = 0$) which means:

$dt^2 = ( dx^2 + dy^2 + dz^2 )/c^2$

and therefore

$c = \sqrt{ (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 }$

which means that massless particles must always travel at speed "$c$". So "$c$" is really the speed of massless particles. The most obvious and well known massless particle is the photon - the quantum of the electromagnetic field. That is why "$c$" is the speed of light.

Theoretically gravitons would also be massless so they would also travel at speed "$c$". For a long time neutrinos were thought to be massless so they would have also traveled at speed "$c$" but now it is known that at least 2 and probably all 3 types of neutrinos have a very small but non-zero mass and therefore the non-zero mass neutrinos would have to travel at less than "$c$".

Thus the speed of light really happens to be the speed of all massless particles and it really is a conversion factor between space and time.

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A lot of the answers here seem to miss the thrust of the question, which is (I think), why do photons travel at the speed of light c, as opposed to some other speed. That is, given a definition of a meter as "a stick that is this long", why does light take the particular amount of time to cross that distance that it does.

The answer is that there is a set speed that any massless particle travels at, such as a photon, and that speed, c, is a fundamental property of our universe. Any time a physicist says something is fundamental, it means (s)he doesn't know why, it just is.

To be fair, you can explain why c is significant by appealing to relativity, the way we measure how time flows, the definition of the units we use when measuring it, etc. But, at the most basic level, c is a given, something we plug into equations, not something we get out of them. It is a property of light (and any massless particle), but it is one we have to observe the universe in order to find.

As a side note, the experiments used to determine the speed of light aren't especially unclear. There are several. Which one gave the first accurate answer is, perhaps, in some dispute, but the making of the measurement isn't a point of contention.

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Ultimately, the answer is that we don't know. And, at the same time, we don't know if we can ever know why it is this way.

We've measured $c$ with respect to a unit system that we humans have devised1, but we do not know if there is any reason for it to take that particular value; it simply is what it is.

1 We defined the speed of light as a constant in 1983, so technically the meter and second are determined from $c$, not the other way around as I have suggested here.

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c is just a fundamental property of our universe, the maximal speed of information propagation; yet it was defined to be exactly 299792458m/s to define a meter.

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The speed of light is not arbitrary. You can calculate the speed of propagation of small perturbations using Maxwell's equations, which gives $c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}$. Thus when these to constants are fixed, so is the speed of light.

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@Joe F: all three are defined (i.e. exact numbers). – Johannes Jan 23 '11 at 6:44
@Johannes: that's not true. Their value in SI units is defined, but that is because they are used to define the units. It's not the same thing at all. – Joe Fitzsimons Jan 23 '11 at 6:49
@Joe: and that's why SI units are rational units. Measurement systems that don't recognize you can rotate yardsticks in space would define different units for depth and width. We all recogonize that as overcomplicating things. The same applies to rotations in spacetime. Yes, there are measurement systems that don't recognize you can rotate a yardstick for distance into a yardstick for duration, but there is no place for such systems in physics. – Johannes Jan 23 '11 at 6:56
@Johannes: That is not the point at all. The way units are defined in terms of setting specific values to physical constants doesn't mean those constants can't change. The consequence is simply that a better measurement of the speed of light means that my apartment is a different size in those units. This clearly does not mean c or $\mu_0$ or $\epsilon_0$ are fixed, simply that their measurement in SI units is. This is a quirk of the choice of units, not something fundamental. – Joe Fitzsimons Jan 23 '11 at 7:09
You miss the point. What I am saying is that if you accept invariance under the Poincare group as a fundamental property of nature, you can not maintain that the values for c, mu0 and eps0 are independent and unrelated. Your answer above suggests you are of that opinion. – Johannes Jan 23 '11 at 15:02

The speed of light is just a conversion factor from one coordinate direction to another. The proper unit is “one,” or $1.0ly/y$ and so forth. Now it has this funny set of units in cgs and so forth. However, if you were to argue the speed of light could be different in these units then the Planck units, such as $\ell~=~\sqrt{G\hbar/c^3}$ would all rescale accordingly and as a result so would our rods and clocks. This would make the rescaling completely unobservable.

Why $c~=~ 299,792,458m/s$ has to do with other constants of nature, such as the mass of the proton and so forth. We measure the speed of light according to physical objects and it has this large value due to the physical dimensions of rods, which depends on the Bohr radius which in turn depends on mass of electrons and so forth. The speed of light is so large, in part because gravity is very weak, and this really has to do with the fact elementary particles have little mass in comparison to the Planck mass. If this huge disparity did not exist the natural unit for the speed of light is one Planck length per Planck time, which is just a unity.

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The Planck length has nothing to do with the properties of rods and clocks. If that were the case, then we would need a theory of quantum gravity (which we don't have) in order to understand the properties of rods and clocks. – Ben Crowell Jul 27 '11 at 17:25

The speed of light is a property of spacetime, which has a Minkowski metric, which means it has a Null Space, which marries space and time into Velocity, which is invariant. One might have a look at David Hestenes "Space Time Algebra" and various of his papers, or visit a site on Geometric Algebra (ie, Clifford Algebra ). The geometry of spacetime is impervious to units of measurement - whatever floats your boat. One prof used furlongs per fortnight.

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Actually the speed of light can be different in different media, for example, it is higher than $c$ in Casimir vacuum and smaller than $c$ in a solid medium.

It seems the actual question here is why the speed of light in flat vacuum is the highest speed at which information can be transferred.

This comes from the Special Theory of Relativity where it was shown that if there was FTL information transfer, a casualty paradox would appear. This is because $c$ is used in Lorentz transforms.

And now naturally emerges a further question, why exactly the Special Theory of Relativity uses $c$ in its Lorentz transforms rather than any other speed. This is due to the idea that all inertial frames should be indistinguishable: the speeds of all processes, either electromagnetic, mechanical or gravitational should change similarly when changing a reference frame. If electric processes changes one way and say gravitational another way, we could determine whether we are in a moving or stationary frame.

It follows that all fundamental interactions should propagate at the same speed for this criterion to be met.

So actually if there somewhere exists a medium where speed of light is higher or smaller than c, then in such medium it is possible to determine whether the given frame is absolutely moving or stationary relative to the medium. Such medium does not have the main propertiy of our vacuum, which distinguishes it from Ether: that there is no difference between moving(without acceleration) in it and being in rest.

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As Kyle Kanos said in his answer, we just know it is there. It is calculated theoretically, and coupled with experimental evidence, we just accept that the speed of light is $c=3x 10^8 \frac {m}{s}$.

You see, we always expect everything to be as we like it. We want to have control over all around us, even nature. Our inherent conception is that we occupy a central, priveleged position in everything we do, from relationships to businesses to our understanding of the universe. The reality is that we are anything but priveleged. Science has repeatedly shattered this misconception of ours, but we still hang on to the ego. I suggest you watch Carl Sagan's famous speech The Pale Blue Dot which puts accross this message so powerfully.

In such a case, I think it is extremely unfair to expect the speed of light have a value that appeals to us, other than $c$, the universal speed limit. In your case, the question is not about why the speed of light has that particular value, but why we think it is not supposed to be so.

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""We calculated it theoretically, "" Who is this "we"? – Georg Jan 23 '15 at 17:36

There is a unique numeric value of c in whatever units (eg: m/sec , or miles/hour, or furlongs/fortnight) you are using that correctly converts velocity to $\phi$ radians for doing boosts with the Lorentz Group where $\phi$ is called the Lorentz Boost parameter. $$v/c=tanh(\phi)$$ The number of radians for a particular boost is not random because boosts do not commute. For $\phi << 1$ the following is true: $$[xBoost(\phi) , yBoost(\phi)] = zRotation(\phi ^2)$$ If you do not use the correct value of c to convert your boost velocity v to $\phi$ radians, then you will not calculate correctly the $\phi ^2$ radians of rotation that is experimentally seen. As long as Boosts are elements of the Lorentz Group, there is a unique numeric value of c for each choice of units. The value of c is not different in another universe, it is not set by God, nor can it be randomly set by Congress.

In some sense c is an artifact of how we measure the thing called velocity using sticks and ticks. If long go we were able to perceive velocity directly in $\phi$ radians, for example by seeing a fractional length or time contraction which $=cosh(\phi)$, then we would never talk about v or c.

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Suppose that there is a SuperEarth with a city called SuperParis. In this city, the inhabitants keep an iron rod with two marks on it. This distance is called a supermeter, it's the basic unit of length on superearth.

If by pure chance these people use the same unit of time as humans and if by pure chance 1 supermeter would be 299792458m here on earth - then what is the speed of light for the superpeople?

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That's just a change of units. You could similarly argue that the speed of light in mph is different. – Manishearth Mar 15 '12 at 7:10

We measure relations between objects that belong to our local environment. Our rulers are based on atomic properties.

c is a property of space or, as said above, is a property of the universe, not of the light, because I share the viewpoint of Israel Perez that matter and space are different expressions of the same entity, following Spinoza (Ethics).

c is a relation, a ratio, between length and time intervals. If you check the definitions on time and length units it becomes clear that they are not independently defined, one is defined with help of the other. Then one must conclude that c is really a fundamental property that express how perturbations can evolve within space.

Similar behaviour is watched in the waves in a lake. It is a property of the medium (vacuum, field, aether,space,...many names)

This explains why light propagates always in the same way regardless of the motion of the source or the receiver.

About the value of the measure of c:

I think that we can only measure the two-way speed of light, not the one-way (as Poincaré). We only measure the ratio (L/T) and this is an absolute constant.

Assume that in this world (or in the distant past) the atoms have the double of the size of our local atoms, then the relation L/T has to be the same, and physical laws keep invariant, (the electrons will take a longer time, compared to our local time, to evolve around the proton and the emmited light will be redened :-)

We do not make any 'absolute' measure on lenghts or time durations. Fundamentally we can not do it in another way.

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You can prove from Maxwell's equations that the speed of light is: $c=\sqrt{\frac{1}{\epsilon_0\mu_0}}$. Where $ϵ_0$ (vacuum permittivity) and $μ_0$ (vacuum permeability) are both constant.

If you want to fully understand how this equation works, I recommend you to read this proof: Maxwell's equations predict that the speed of light is constant.

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In cgs units, $\epsilon_0=1$ and $\mu_0=1/c^2$, so the relation is kinda trivial. That is to say, all three terms have exact values and does not at all explain why $c$ has the value it does. – Kyle Kanos Jan 22 '15 at 16:14