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.
 A: 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.
A: 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.
A: 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. 
