It has been a wonderful year, 1905, for Einstein who just published four papers in Physics. But, one of them on the Special Theory of Relativity rocked the foundations of our understanding of nature. Before jumping onto the postulates of the theory, we will look at the Maxwell’s equations for electromagnetic waves and derive the velocity of light(as given in the Wikipedia).

The Maxwell’s equations are as follows in the ‘Heaviside’ form of Maxwell’s equations where the fields are in free space without any charges or currents.

$$ \begin{aligned} \nabla \cdot \mathbf{E} &= 0 \newline \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \newline \nabla \cdot \mathbf{B} &= 0 \newline \nabla \times \mathbf{B} &= \mu_0 \varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t} \end{aligned} $$

Taking the curl of the second and third equations in $(1)$ $$ \begin{aligned} \nabla \times \left(\nabla \times \mathbf{E} \right) &= -\frac{\partial}{\partial t} \nabla \times \mathbf{B} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \newline \nabla \times \left(\nabla \times \mathbf{B} \right) &= \mu_0 \varepsilon_0 \frac{\partial}{\partial t} \nabla \times \mathbf{E} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} \end{aligned} $$

We use the following vector identity and also note that the divergence of the vecotrs $\mathbf{E}$ and $\mathbf{B}$ are zero.

$\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}$

Thus, we end up with $$ \begin{aligned} \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} - \nabla^2 \mathbf{E} &= 0\newline \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{B}}{\partial t^2} - \nabla^2 \mathbf{B} &= 0 \end{aligned} $$

Here we identify that the fields behave as a wave. The wave equation which describes a disturbance travelling along a medium with a velocity $\mathcal{v}$ with respect to the medium is

$$\frac{\partial^2u}{\partial t^2}=\mathcal{v}^2 \frac{\partial^2u}{\partial x^2}$$

Thus, we can conclude that the velocity of light is

$$c=\frac{1}{\sqrt{\mu_0 \varepsilon_0}}$$

But, if you observe the derivation clearly we see that the wave equation is with respect to the reference frame attached to the medium. What is such a medium for the electromagnetic field in equations $(1)$ ?

Taking an analogy from the wave equation, we should agree for one of the following two assumptions:

  • Maxwell’s equations should be correct in one reference frame only, which we do not yet know of
  • Maxwell’s equations are valid in all inertial reference frames.

Since, the results of the Maxwell’s equations have been thoroughly tested, there is no doubt that one of the above two assertions must be true(removing the possibility to assume that the Maxwell’s equations are incorrect). The first assumption points to the theory of ‘ether’, which is the one true medium through which the electromagnetic waves. This has been proved to be wrong by the famous Michelson-Morley experiment(which I will not describe here as the Wiki page will do a better job).

We are now forced to take the second assumption that the Maxwell’s equations are valid in all inertial reference frames, which implies that the speed of light is constant irrespective of the velocity of the observer. This is an attempt to understand ‘why and how’ of the second postulate of the Special Theory of Relativity. The rest of the theory is just algebra concerned with Lorentz Transformation and its implications.