Define the field strength tensor $F^{\mu \nu}(x)$ for an electromagnetic field specified by a 4-vector potential $A^{\mu}(x)$. How do the components of $F^{\mu \nu}$ change under a Lorentz transformation? Write down two independent Lorentz-invariant quantities which are quadratic in the field strength tensor.

[Hint: The alternating tensor $\varepsilon^{\mu \nu \rho \sigma}$ takes the values $+1$ and $-1$ when $(\mu, \nu, \rho, \sigma)$ is an even or odd permutation of $(0,1,2,3)$ respectively and vanishes otherwise. You may assume this is an invariant tensor of the Lorentz group. In other words, its components are the same in all inertial frames.]

In an inertial frame with spacetime coordinates $x^{\mu}=(c t, \mathbf{x})$, the 4-vector potential has components $A^{\mu}=(\phi / c, \mathbf{A})$ and the electric and magnetic fields are given as

$$\begin{aligned} \mathbf{E} &=-\nabla \phi-\frac{\partial \mathbf{A}}{\partial t} \\ \mathbf{B} &=\nabla \times \mathbf{A} \end{aligned}$$

Evaluate the components of $F^{\mu \nu}$ in terms of the components of $\mathbf{E}$ and $\mathbf{B}$. Show that the quantities

$$S=|\mathbf{B}|^{2}-\frac{1}{c^{2}}|\mathbf{E}|^{2} \quad \text { and } \quad T=\mathbf{E} \cdot \mathbf{B}$$

are the same in all inertial frames.

A relativistic particle of mass $m$, charge $q$ and 4 -velocity $u^{\mu}(\tau)$ moves according to the Lorentz force law,

$$\frac{d u^{\mu}}{d \tau}=\frac{q}{m} F_{\nu}^{\mu} u^{\nu}$$

Here $\tau$ is the proper time. For the case of a constant, uniform field, write down a solution of $(*)$ giving $u^{\mu}(\tau)$ in terms of its initial value $u^{\mu}(0)$ as an infinite series in powers of the field strength.

Suppose further that the fields are such that both $S$ and $T$ defined above are zero. Work in an inertial frame with coordinates $x^{\mu}=(c t, x, y, z)$ where the particle is at rest at the origin at $t=0$ and the magnetic field points in the positive $z$-direction with magnitude $|\mathbf{B}|=B$. The electric field obeys $\mathbf{E} \cdot \hat{\mathbf{y}}=0$. Show that the particle moves on the curve $y^{2}=A x^{3}$ for some constant $A$ which you should determine.