Lorentz boost to obtain parallel E and B fields?

[Xavier1900]

Homework Statement

Suppose given an electric field $\vec{E}$ and a magnetic field $\vec{B}$ in some inertial frame. Determine the conditions under which there exists a Lorentz transformation to another inertial frame in which $\vec{E} || \vec{B}$

Homework Equations

If we give a Lorentz boost along $x_1$-direction, then in the boosted frame, electric and magnetic fields are given by
$$E_1' = E_1\\ E_2' = \gamma (E_2 - \beta B_3)\\ E_3' = \gamma (E_3 + \beta B_2)$$
And similar for components of B fields.

The Attempt at a Solution

I started with a frame in which the fields are parallel and see what kind of fields I can obtain after the transformation. The case where the boost is along the direction of E//B fields is trivial. Then I consider the case where I boost in the direction perpendicular to the E//B fields. By the equations I listed I find that I can produce E and B fields with some angle depending on $\beta$. But I am not seeing how I can go further from here. Am I in the right direction? Or should I try some other approach?

 

[phyzguy]

One way to approach problems like these is to construct invariants - i.e quantities that are the same in all frames. For example, given a four-vector A, the quantity $A^\mu A_\mu$ has the same value in all frames. It turns out there are two invariants that can be constructed from the electromagnetic field tensor $F^{\mu \nu}$, and hence from E and B. Try to discover what these are - this will help answer the question.

 

[Xavier1900]

One way to approach problems like these is to construct invariants - i.e quantities that are the same in all frames. For example, given a four-vector A, the quantity $A^\mu A_\mu$ has the same value in all frames. It turns out there are two invariants that can be constructed from the electromagnetic field tensor $F^{\mu \nu}$, and hence from E and B. Try to discover what these are - this will help answer the question.

Thanks for the hint. I realize that $F_{\rho \sigma}\tilde{F}^{\rho\sigma} = -4 \vec{B}\cdot \vec{E}$, which is a scalar. That means if E//B in one frame, $\vec{E}\cdot \vec{B} \neq 0$ in all frames. Thus I just need to find the right Lorentz boost for systems like that. I will try and see what I can find.