Lorentz invariants

Problem:

In a certain reference frame a static uniform electric field E₀ is at an angle θ₀ ≠ π/2 to a static, uniform, magnetic induction B₀, (B₀ = E₀/c), in the SI system of units.
(a)  Determine the magnitude of E and B in a reference frame in which the angle between E and B is θ.
(b)  Is there a reference frame for which E and B are perpendicular to each other?  If so, what is it?

Solution:

• Concepts:
  The Lorentz invariance of the electromagnetic field-strength tensor
• Reasoning:
  In SI units E² - c²B² and (E∙B)² are invariant under a Lorentz transformation.
• Details of the calculation:
  (a)  E² - c²B² and (E∙B)² are the same in every reference frame.
  In K we have E₀∙B₀ = E₀B₀cosθ₀ = (E₀²/c)cosθ₀.
  In K' we have E'∙B' = E'B'cosθ.
  Since E₀² - c²B₀² = 0 in K, we have  E'² - c²B'² = 0 in K', therefore E' = cB'.
  We then have E'∙B' = (E'²/c)cosθ.
  E₀∙B₀ = E'∙B' -->  E'² = E₀²cosθ₀/cosθ.
  (b)  If E and B are perpendicular to each other, then E∙B = 0.  But E∙B is invariant and not zero in K.  There is no such frame.