# Why is the speed of light in vacuum what it is?

Three-pronged question on the speed of light.

1. One: simply, do we know why the speed of light in a vacuum is what it is and why nothing is allowed to go faster?

2. Two: if we knew why the speed of light is what it is, would that give us insight on the nature of time, considering that they are tied together and a photon traveling at the speed of light, as it is seemingly compelled to do, experiences no sense of time?

3. Lastly: do we know why an object that has no mass, i.e. A photon the instant it is produced, is automatically compelled to travel at the speed of light?

• 1b, 2 and 3 all have the same answer: the Minkowski nature of flat space time. 1a is a question about the size of a dimensional constant and the meaning of those is less obvious that you might think. I know we have some links around here. – dmckee --- ex-moderator kitten Oct 2 '15 at 23:33
• Here are some on dimensional constants: physics.stackexchange.com/q/8373 physics.stackexchange.com/q/60941 – dmckee --- ex-moderator kitten Oct 2 '15 at 23:36
• The speed of light is equal to what Maxwell's equations say it has to be equal to. The reason why the speed is that and nothing else is that Maxwell's equations are what they are and nothing else. If you want a deeper understanding of "why", you must study Maxwell's equations. – WillO Oct 2 '15 at 23:59
• Two: the speed of light happens to have the value it has just by chance. It is a measure of how much space and time are mixed for two observers in relative motion. The larger the value of c, the smaller the effect. – user83548 Oct 3 '15 at 0:10
• Here is a similar question with answers: physics.stackexchange.com/q/3644 – Gary Godfrey Oct 3 '15 at 4:52

The speed of light can be derived from simple classical mechanics using the equations of Maxwell. This page offers such a derivation, utilising simple vector calculus. The speed of light in vacuum is seen to be $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$$ Where $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of vacuum.

The answer to the second part of your question as to why this speed seems to be the speed limit for all masses stems from Einstein's special theory of relativity. He provided an equation relating the moving (relativistic) mass of a body to it's rest mass. The equation predicted that the mass of a body traveling at the speed of light as observed by an observer in an inertial frame would be infinite. Obviously, this means that objects would require infinite work be done on it. This is not true for masless objects.

• μ0 and ϵ0 are not fundamental constants. – Count Iblis Oct 5 '15 at 18:23
• I agree. They aren't fundamental. They just happen to be the constants in the expression of $c$. – Praneet Srivastava Oct 6 '15 at 1:37
• well, who decided what are the true fundamental degrees of freedom the Universe is built on ? If you consider that c is fundamentally a property of the vacuum medium, then it's very tempting to see the analogy with mechanical waves travelling at $c = \frac{1}{\sqrt{\rho\beta}}$, i.e. the balance between something inertial and something potential. – Fabrice NEYRET Oct 30 '15 at 17:45

First, we do not know why the speed of light is what it is. Of course its numerical value depends on the particular unit system, but the basic fact remains that light has some particular speed, and we cannot explain that any further. This is similar to other physical constants such as $\hbar$ and $G$. Dimensionless constants such as $\alpha$, the fine structure constant, are more interesting because they are independent of the units, but we still do not know why they have the values they do.

(We could digress here about the fact that the speed of light is a conversion factor between space and time and can be set to 1 by a suitable choice of units, but in my opinion this is not a real explanation.)

I'm not sure what kind of insight would be gained by knowing this. There would probably be some, but physicists are so used to the idea that some constants just have to be measured that I don't know what it would mean to know where the speed of light comes from, and what kind of knowledge would come from that.

The answers to the rest of your questions depend on what you mean by "why". I could tell you that nothing can travel faster than light and that light is massless because special relativity demands it, but then you would ask why special relativity is true. I could tell you that Einstein saw that these are necessary consequences of the facts that the laws of physics look the same in every inertial system and so does the speed of light. And if you ask why these things are true, then all I can say is that all the experiments done to test them have showed that they are true, and that's as far as I can go. This is just the way the world works.

• Maxwell's equations predict the speed of light, do they not? – Gert Oct 3 '15 at 2:38
• Rotation of Minkowski space puts a limit on speed, and we believe (with evidence) that EM and gravitational field changes propagate at that maximum speed. – Bill N Oct 3 '15 at 3:55
• @gert, Maxwell's equations relate the speed of light to the permeability and permittivity of vacuum. So, the speed of light is what it is because those constants are what they are. Why are those constants what they are? It's just the same question wearing different clothes. Every time you ask why X is the way it is in physics, the best answer you can hope for is, "because Y is the way it is." – Solomon Slow Oct 5 '15 at 18:16
• Of course you can explain that further, all you have to do is consider why we chose some of these unit systems. E.g. if we use the meter and the second then that obviously ultimately derives from our own biology. – Count Iblis Oct 5 '15 at 18:21
• @CountIblis: This is what I refer to with the parenthesis on units. In my opinion, while it is true that the numerical value of $c$ depends on the historical choice of units, this just shuffles the question around but doesn't make it go away. By setting enough constants to 1 this could be turned into "why are the Planck units what they are?", and I think there's no answer to that. – Javier Oct 5 '15 at 18:53

We don't know why; it's a physical fact that's been established by experiment.

One could suppose after all, any finite speed was possible; and this is what Newtonian Mechanics allows.

• The speed of light in vacuum is a defined constant just like the number of liters in a gallon, or the length of a mile expressed in meters. it's not something to be determined by experiment (which doesn't mean that you can't do an experiment to measure it's value, just like nothing would stop me from bringing my meter stick to Britain and measure the distance between two mile indicators along some road). – Count Iblis Oct 5 '15 at 18:29

To make a long story short, when we write $c = 299792458$ meter/second, then the number 299792458, while in principle just an arbitrary number that defines the meter relative to the second, can be interpreted as a physical measure of the human body just like the number 25 appears in the expression 25 kg/meter^2 for the threshold BMI that separates overweight people from people with a normal weight. This is because the way we chose to define the meter in terms of the second such that it is compatible with older definitions that used the meter to express lengths, which was chosen to be a length of the same order has the human body. The modern definition of the second makes this time interval have pretty much the same value as the old definition that introduced the second as a small unit of time as perceived by us.

Now, the speed of light itself is not a physical constant from the point of view of fundamental physics, as Michael Duff explains in detail here. The general issue here is that given some set of laws of physics in the form of mathematical equations, you are always free to redefine the variables by multiplying them by constants, or perform more complicated mathematical transforms. The extra constants that then appear as a results of these transforms, obviously have nothing whatsoever to do with fundamental physics. But the transforms can be useful as a mathematical technique to describe certain scaling limits of the theory.

I show here how you can derive the classical limit of special relativity using a scaling argument. I work in natural units, I never depart from that, and yet a constant that I call c does appear, but it is a dimensionless scaling parameter. I then cannot put it in equations using dimensional arguments, because everything is dimensionless and remains so. Rather, I have to put it in the right places based on the scaling limit that I want to study.

The actual value of $x=299792458$ in the expression

$$c = x \frac{\text{meter}}{\text{second}}$$

is thus purely a matter of defining the meter and the second. The value of $c$ itself is 1 and that's not a mere convention, any more than measuring heights in the same units as distances parallel to the Earths surface is a "mere convention". Then since $c = 1$ means that:

$$x = \frac{\text{second}}{\text{meter}}$$

So, the SI value of $x$ is simply the factor by which the second is scaled relative to the meter (here we discard the notion that lengths and time intervals should have different dimensions). Now the choice of the two different units for distances in the temporal and the spatial directions is motivated by making relevant physical expressions for humans be of order unity. So, the meter is the length of a big step we can take and the second is one small step in the time direction that we can readily perceive. So, the value of $x$ is a measure of us humans just like the the average waist to height ratio is.

As far as perceptions are concerned, the second is perhaps more analogous to the millimeter, so the number $2.99792458\times 10^{11}$ is a measure of how much more resolving power we have in the spatial direction compared to the temporal direction.