# Nature of light in Special Relativity

What is the nature of light in the context of Special Relativity? Is it a photon, or an electromagnetic wave, or something else?

I have doubts, because a photon seems to me a quantum mechanical concept, and Special Relativity is supposed to be a classical theory.

• SR doesn't say anything about anything other than the symmetry properties of space and time. – CuriousOne May 3 '16 at 21:44
• Special relativity is a purely "geometric" theory, it says nothing about what kind of matter might fill spacetime, all that matters is that it has mass. It is like Newton's three laws but without any expressions for particular forces. Those are provided by the Standard Model and general relativity, specifically quantum electrodynamics for photons. And quantum electrodynamics is a relativistic quantum theory, i.e. it subsumes both special relativity and quantum mechanics. – Conifold May 4 '16 at 1:32
• The comment made by @CuriousOne is very misleading. Special relativity is a theory about how observed events appear from different frames of reference, and light plays the absolutely critical role of the invariant path length under reference frame changes. To say that special relativity is a theory only about symmetries is rather reductionist, especially given that the theory is motivated and experimentally substantiated by experiments with light. – DanielSank Aug 7 '16 at 4:28
• @DanielSank: Yes, those are the symmetries that we are talking about. That's what the modern theory is, just like Galilean relativity was about the symmetries of Galilean space and time. That a world of effects follows from that, alone, is the beauty of it all. – CuriousOne Aug 7 '16 at 7:45

## 3 Answers

Classical electromagnetism is perfectly compatible with special relativity. In classical E&M, light is an electromagnetic wave and there is generally no useful formulation in terms of particles.

The most widely used technique to combine quantum mechanics with special relativity is relativistic quantum field theory. The relativistic QFT that corresponds to classical E&M is Quantum Electrodynamics (QED). The quantum nature of this theory gives rise to the concept of photons - quantized wave-packet excitations of the underlying "photon field." But this particle-like behavior is a purely quantum effect; in the limit where $\hbar \rightarrow 0$ and quantum effects can be neglected, the particle picture is not useful.

Caveat: in the limit of small wavelengths, classical E&M is well approximated by the simpler theory of "ray optics", where you can think of light as being a stream of "particles" moving in a straight line. But this is a general property of waves with small wavelengths and is in no way specific to light. When people talk about the "particle" nature of light, they're almost always referring to photons and quantum effects.

• Photons are quanta, not particles. There is no working particle theory of light. The old corpuscular theory can't even deal with classical diffraction. – CuriousOne May 3 '16 at 21:45
• @CuriousOne You're too partisan about this. The classical particle picture is simple and works just fine on length scales much greater than the wavelength. And isn't physics all about choosing an appropriate model? – knzhou May 3 '16 at 21:56
• @knzhou: You are not reading what I wrote. I didn't say that the classical model doesn't work, but that there is no working particle model for light. The two are not the same. The classical model for light is a wave model and, trough the Eikonal approximation, a ray model. Those have nothing to do with particles. As for quantum mechanics... it doesn't have particles, either, never did. – CuriousOne May 3 '16 at 22:23

To be clear, Maxwell's equations are known as "Lorentz-invariant" equations, which means that they take the same form in every Lorentz-transformed frame of reference. Special relativity actually came about from studying Maxwell's (classical) equations without charges or currents. Then we get:

$$\nabla \cdot \mathbf{E}=0$$ $$\nabla \cdot \mathbf{B}=0$$ $$\nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$ $$\nabla\times\mathbf{B} = \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}$$

Take the curl of Faraday's Law: $$\nabla\times(\nabla\times\mathbf{E})=\nabla(\nabla\cdot \mathbf{E})-\nabla^2\mathbf{E}=\nabla\cdot(-\frac{\partial \mathbf{B}}{\partial t})$$

And substitute Gauss's law for $\nabla\cdot\mathbf{E}$ and Ampere's law for $\frac{\partial}{\partial t}(\nabla\cdot \mathbf{B})$ and you'll find the wave equation for $\mathbf{E}$: $$(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2})\mathbf{E}=0$$ where $c=1/\sqrt{\mu_0\epsilon_0}\approx 2.998\cdot 10^{8}m/s$.

This wave equation is only for the case of what's called "Lorenz gauge" (not "Lorentz"), which corresponds to a frame of reference which is at rest compared to the medium. Back when people thought there was some kind of fluid or "ether" that electromagnetic waves traveled through, it makes sense that if you're moving relative to the fluid, then the velocity of waves in the fluid will change. The Michelson-Morley experiment helped to show that there was no "ether" that light travels through.

Einstein's insight was that electromagnetic waves travel at the same speed no matter what frame of reference you are using. This $\textbf{principle of relativity}$ is only satisfied if velocities don't add in the Galilean sense and rather follow a different set of rules about transforming frames of reference, called Lorentz invariance. It was in a sense luck that we discovered a theory of electromagnetism Lorentz invariant, but in another sense it was inevitable since the theory is inherently relativistic.

Notice that nowhere here do I discuss the particle nature of light. Special relativity really has nothing to do with classical versus quantum theory. It's all about the difference between Galilean invariance and Lorentz invariance.

Aside: When asked later on why he believed in special relativity, Einstein quoted Fizeau's experiment.

The nature of light itself is 'contextual.' An overly broad definition is true more often, but less true in each particular context; an overly specific definition is the most accurate within its own context, but likely less so in all others. Einstein does not define 'light' in the 'context' of SR...he assumes its theoretical validity as a concept representing a/the universal constant in accordance with other 'accepted/observed/proven' formulations of its nature/properties and then applies it to a theoretical construct of the universe, but never defines it in the conceptual linguistic sense. Obviously many critical aspects of a 'definition' can be found in/from the theory, but these are not the basis of the theory itself, nor are they required, in an abstract conceptual sense, for its validity.

• Speed of light needs to be an universal constant, because the Maxwell-equations have the surprising result that it needs to be the same in any reference frame. – user259412 Aug 7 '16 at 10:39

## protected by Qmechanic♦Aug 7 '16 at 5:33

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