# If a photon has no mass why doesn't it have infinite speed? [duplicate]

Please help a naïve layperson understand -- if a photon has no mass, why is its velocity limited at all? Shouldn't a particle with no mass be able to travel at an infinite speed?

• Related: physics.stackexchange.com/q/3541 – David Herrero Martí Jun 27 '16 at 12:35
• Because massless things travel at the speed of light. – Rob Jeffries Jun 27 '16 at 12:41
• Possible duplicate of Why can't massless particle exceed speed of light? – ACuriousMind Jun 27 '16 at 13:18
• There is a curious assumption in the question: that a massless object should be able to move faster than a massive one. I wonder why a naïve layperson would think that. Common sense is not what it used to be... – Stéphane Rollandin Jun 27 '16 at 13:47
• @StéphaneRollandin well, for the same impulse the smaller mass goes faster, so taking the limit to zero mass.... – anna v Jun 27 '16 at 13:59

That light moves with a fixed speed in vacuum, in all reference systems is an experimental fact. Maxwell's equations fit so well all macroscopic electromagnetic data that the speed of light is fixed is not under question. It is inherent in the construction of the classical theory.

Light is made up by a zillion of photons. Photons are elementary particles in the standard model of particle physics. Elementary particles in motion with high velocities do not obey Galilean tranformations, which are implied in your question. These are true for macroscopic objects and velocities far away from the velocity of light. For the microcosm of elementary particles it is Lorenz transformations which define the velocities and the limits for transformations from one system to the other. In Lorenz transformations the maximum velocity achievable is the velocity of light, by all zero mass particles. Particles with mass can never reach it.

Thus, as the smallest particle of light is the photon, the photon must have zero mass since light's speed is limited by c.

• Thank you. I am thinking in too linear a fashion, I suppose. It seems as though the tail is wagging the dog, because I still don't see how a massless particle's speed can be limited by anything. But I appreciate your answer. – A Webb Jun 27 '16 at 13:19
• @AWebb. But then why do you feel that a massive particle's speed should be limited at all? – Stéphane Rollandin Jun 27 '16 at 14:01
• @StéphaneRollandin I am talking within Galilean transformations. Special relativitye which holds for elementary particles does not fall into this logic. The impuls itself means something different in SR – anna v Jun 27 '16 at 14:04
• @annav. I'm just trying to grok the OP's point of view. Your answer is fine IMO. – Stéphane Rollandin Jun 27 '16 at 14:06

There is no way to explain this without explaining relativity first. In Galilean universe (the "classical" physics, which is what most people intuitively assume and think about), light speed cannot be explained. Indeed, the Maxwell equations which is describe how light works, were the first clue that our understanding of space-time was flawed.

So the real question is: how to explain special relativity to a layperson. There are many approaches taken by documentaries and kids' books... I'd just say the following:

Velocity (as in, distance traveled, divided by time taken, measured both by the same person) is limited because the nature of our universe is such, that space and time are geometrically connected, similar to saying "why the largest possible angle in triangle is 180 degrees" or "why there's no north at the north pole". Forces cause acceleration and change momentum, but at large velocities, increasing momentum doesn't change the velocity all that much anymore.

There's sort of an analogy from real life... consider places A and B, and you want to get there in the shortest route possible. As long as people take very long and twisted ways around (the slow ways), then you can always find a better route and you don't even notice any limit (like our ancestors didn't need to care about the light speed)... but once you get to the really fast ways, you start realizing, there is a shortest way: the straight line. There's no shorter path than that, no matter how hard you try. It's pure geometry. There's no magical "speed limit" in the universe, we're just measuring a quantity that mathematically can't be larger than the speed of light. If you measure momentum, it can be as large as it likes.

Here's where the photons come in: in relativity, the relationship between momentum $p$, energy $E$ and velocity $v$ is $v=pc^2/E$. There's also the relation $E^2=(mc^2)^2+(pc)^2$. So, if you have mass, the energy will always be larger than $pc$, and the velocity will always be smaller than $c$. But for light, it's always $E=pc$, so $v=c$ (and can't be any different).

EDIT: how come the light speed isn't infinite? Because if it were, our universe would be much different in ways you'd most certainly notice. For wave equation to make sense, the wave speed has to be finite, so the fact that electromagnetic waves exist (Maxwell equations hold) proves it must be finite. If not, we'd have no induction, no radio, no interference.... it's quite natural for us to assume that changes in the environment take time to get noticed somewhere else, so much, that people struggled for a long time how gravity and such work (forces at a distance were problematic, and only resolved when we realized there are fields that carry this information through space). In that respect, finite speed of light made some things easier to understand. Instantaneous action at a distance would break a lot of physics.

• Can I point out, just for the sake of argument, that your answer here depends on the speed of light. When answering the question "Why is the speed of light a finite number?" it probably would be a better idea to answer with something other than "because light can't go faster than the speed of light". In the equation you wrote, all involve $c$. Setting $c$ to be infinite doesn't make this answer less true and, thus, should show that the question is still unanswered – Jim Jun 27 '16 at 13:34
• True, @Jim. $c$ in our universe happens to be finite and has to be measured to see how much is it in our universe... although, quite a lot of phenomena would severely break down if it were infinite. For one, there would be no wave optics. No Maxwell equations. There is a large gap between large but finite, and truly infinite. – orion Jun 27 '16 at 13:45
• I agree. I just wanted to point out the way your answer may or may not irk any of those who really want to know the "why" of it not being infinite (even though, IMHO, it isn't our job to know the answer to that question) – Jim Jun 27 '16 at 13:52

In the special relativity, as explained by Einstein, there are two possibilities. Either the speed of any particle is limited at all or not. Now if the speed is not limited, the Galilean theory pops out. But as we already know, that does not properly explain the transformation of velocities from one to another reference frame accurately when dealing with high velocities. So only other case is that the velocity is limited, if so what is it?? It comes out that this velocity is precisely the velocity of light.

So, any particle which is mass less travels at the speed c, which is why the speed of light is also c.

• Thank you for answering. I still don't get it, but it's nothing to do with the clarity of your answer. – A Webb Jun 27 '16 at 13:20
• Let me put it in this way. The speed of light c which we are talking about is not a property unique to light. It's just that the velocity c is the speed of causality as explained in the YouTube channel PBS space-time(By the way, it's a cool channel, do check it out). So anything with no mass travels with the speed of causality which is c. And so does the light travel with this speed. – lattitude Jun 27 '16 at 13:26

The speed limit c for massless particles is a characteristic of space.

In accordance with Maxwell's equation, speed of light which is the speed of massless particles is reflecting the characteristics of vacuum

• vacuum permittivity $\mu_0$ and
• vacuum permeability $\epsilon_0$.

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

Such a speed limit of Maxwell's equations is incompatible with Galileis relativity and led 1905 to Einstein's special relativity.