# Why is the speed of light in vacuum a universal constant? [duplicate]

While getting familiar with relativity, the second postulate has me stuck. "The speed of light is constant for all observers". why can't light slow down for an observer travelling the same direction as the light?

• A postulate is an experimental fact that is accepted as an axiom of the theory. It cannot be justified within the theory itself. – Omar Nagib Mar 15 at 12:36
• there is no "right" way, even if the second one seems more intuitive. And nature is way, it is not what our intuitions tell us to be – Wolphram jonny Mar 15 at 12:37
• Sure, light could slow down for an observer traveling in the same direction as the light. But that's not what we observe. – PM 2Ring Mar 15 at 12:38
• BTW, we have lots of questions here on this topic, eg physics.stackexchange.com/q/192891/123208 – PM 2Ring Mar 15 at 13:53
• A key detail to add to any one of the answers here: Light doesn't. While the question of why it can't may be one for the philosophy teachers, practically speaking, every time we have measured the speed of light for various observers, every single study we have done has come to the conclusion that it is constant for all of them. – Cort Ammon Mar 15 at 17:46

The constancy of speed of light was first predicted by Maxwell. He had discovered for equations, which we now call Maxwell equations.

## Maxwell equations

$$\nabla\cdot{E}=\frac{\rho }{\epsilon_{o}}$$ $$\nabla\times{{E}}=-\frac{\partial B}{\partial t}$$ $$\nabla\cdot B=0$$ $$\nabla\times B=\mu_o j+\mu_o\epsilon_o \frac{\partial E}{\partial t}$$ These four equations represent the Maxwell equations. First let’s see what are the conditions/constraints of the vacuum equations. In a vacuum there is no charge, so ρ just becomes zero. Also the is no change in current in vacuum, which means $$\frac{\partial j}{\partial t}$$ will be zero

## Electromagnetic Waves and Maxwell Equations

Integrating the second Maxwell equation with respect to time we get

$$B=-\int{\nabla\times{ E}}{dt}$$

Now let's put this expression for B into the fourth equation. Note we are allowed to interchange the positions of the integral and curl in this case.

$$-\int\nabla\times{\nabla\times E}dt=\mu_o\epsilon_o\frac{\partial E}{\partial t}$$

Now this equation simplifies as

$$-\frac{\rho}{\epsilon_o}+\nabla^2 E=\mu_o\epsilon_o\frac{\partial^2 E}{\partial t^2}$$

Now as we have written the equations for vaccum, $$\rho =0$$ hence our equation just becomes

$$\nabla^2 E=\mu_o\epsilon_o\frac{\partial^2 E}{\partial t^2}$$

Now this equation is very similar to the standard wave equation which is

$$v^2\nabla^2 \phi = \frac{\partial^2 \phi}{\partial t^2}$$

Thus by comparison, we get

$$c=\frac{1}{\sqrt{\mu_o\epsilon_o}}$$

## Alright but what does this have to do with the constancy of speed of light in vaccum?

The fact is that $$\mu_o$$ and $$\epsilon_o$$ are independent of reference frame, and hence, the speed of light in vaccum is a fundamental constant irrespective of the reference frame in question. This was the start of Theory of Relativity as proposed by Einstein.

## Experimental Evidence

After the predictions, Michaelson and Morely performed the famous experiment which proved that the speed of light was to be independent of reference frame or else the Earth wouldn't be moving

• This is the correct answer. However it must be said that your expression for $c$ was known for a long time, and yet people didn't believe the speed of light was independent of the reference frame. To reconcile that result with the Galilean way of adding velocities, people assumed light was traveling in a very peculiar medium, aether. – lcv Mar 15 at 19:01

It is actually better to work from the general principle of relativity, that

• Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them

which is an expression of Hume's Principle of uniformity in Nature,

• the fundamental behaviour of matter is always and everywhere the same.

Without this principle, there would be no science because if laws could change observing them would be meaningless.

It follows that all observers can set up coordinate systems in exactly the same way. There are then two possibilities. Either there is, or there is not, a maximum speed in nature. If there were not, the laws of physics would be different from those we observe (e.g. Newtonian relativity would hold for classical electrodynamics and numerous empirically confirmed predictions of relativity would have been false). Hence there is a maximum speed which is necessarily constant for all observers.

Relativity depends on this logical argument, not strictly on the physical speed of light. It just happens that, to the accuracy of measurement, the speed of light is equal to the maximum speed.

• "If there were not, the laws of physics would be different from those we observe (Newtonian relativity would hold for classical electrodynamics)" - Could you expand on that a little? – Antimon Mar 15 at 18:08
• @Antimon, difficult to expand much without going into chapter and verse on numerous results, which does not seem appropriate here. Essentially, Maxwell's equations and Newtonian mechanics were not empirically compatible. Observation supported Maxwell's equations, and relativity showed how to modify Newtonian mechanics to match observation. – Charles Francis Mar 15 at 18:18
• @Antimon If there were no speed limit to information transfer then it'd be almost impossible to make confident predictions. See physics.stackexchange.com/a/230915/123208 for details. – PM 2Ring Mar 15 at 18:39
• Thanks to both of you! That link seems to hit the nail on the head. – Antimon Mar 16 at 13:51

It does seem strange at first, it sounds like you're asking "If I'm moving towards a source of light won't I see the light travelling faster than c?". If you picture yourself standing still, watching someone drive past you on a car, if they throw a tennis ball in the direction they're driving the speed of the tennis ball will be whatever velocity they throw it at from their point of view PLUS the speed of the car from your point of view. If you repeated this experiment with a torch instead surely you would see the light travelling at $$c$$ PLUS the speed of the car? The actual answer is you would not, and in fact the tennis ball doesn't even travel at exactly the speed as outlined above. In special relativity everyone measures the speed of light to be the same value regardless of where they are.

You can show that length contraction and time dilation are two derived concepts from the second postulate.

We have no answer to the question why the speed of light is a constant for all observers. Physics describes the laws of nature and not why nature does it this way. Thus, this question is beyond the scope of physics.

• No answer to the deep question of why, but back in the 19th century, James Clerk Maxwell showed that the constancy of the speed of "electromagnetic waves" was a consequence of laws of electricity and magnetism that were known at that time. en.wikipedia.org/wiki/… Took some time after that to establish that "electromagnetic waves" and light were the same thing, but by now, we know it to be true. – Solomon Slow Mar 15 at 19:03
• I would argue the other way round: Maxwell's equations are Lorentz invariant, because the speed of light is a constant. – Semoi Mar 15 at 19:30
• Other way round from what? I'm not saying that Maxwell told us why the speed of light is constant (deep reason). I'm only saying that he told we'd have to re-think a lot of what we thought we already knew about electricity and magnetism if it was not constant (shallow reason.) – Solomon Slow Mar 15 at 19:49
• I probably misinterpreted your the "... consequence of ..." part. Sorry! – Semoi Mar 15 at 20:08

The short answer is, because if we assume the speed of light is constant then our equations predict experimental outcomes with greater accuracy.

The electromagnetic answer is, because if you plug the measured permittivity and permeability of free space into Maxwell's equations, that is the speed you get.

The Relativistic answer is, the speed of light is assumed to be a law of physics, and therefore a constant for all observers.

It is the speed of causality. If there was no limit in how fast information can travel all would happen at once. The universe is a causal place because information needs time to travel a distance. Your question is probably much deeper than you intended it to be.

Speed of light is constant in all inertial frames in a vacuum, this is a postulate of Special theory of Relativity. There was no assumption that Speed of light is the fastest traveling speed in the Universe but if you study Special theory of Relativity closely, you will understand that particles having zero mass can only have the highest speed, like approximate massless particles, neutrinos travel really fast but never reach the speed of light, because light particles are massless. Now, following from Maxwell's theory of Electromagnetism, you can understand that EM waves/fields are traveling with very fast (not yet the speed of light or photons or massless particles), you will observe that EM fields/waves don't change their speed when the reference frame is changed. This was the most important observation by Maxwell and Faraday knew this will revolutionize Physics which happened when Einstein took this as a postulate and then as people were working on experiments to find massless particles and they found that they already knew that these are EM waves or photons or LIGHT!