# Can the magnetic field *always* be transformed away?

In the book, "Einstein's General Theory of Relativity..." by Øyvind Grøn and Sigbjorn Hervik, the following statement is made: "The Lagrangian density of an electromagnetic field is the energy-scalar representing the energy-density of the field in a local frame moving so that the magnetic field vanishes...".

However, it is my understanding that often it is not possible to make the magnetic field vanish by transforming to a moving frame -- for example, when the electric and magnetic fields are parallel in some frame, or when the fields are perpendicular but |E| < |B| in some frame. Was this an error on the author's part, or is my understanding wrong?

Written in terms of the electric and magnetic fields, the Lagrangian density for the electromagnetic field becomes $$\dfrac{1}{2}(E^2 - B^2)$$. So if there is a frame in which the magnetic field vanishes, the Lagrangian is simply $$\dfrac{1}{2}E^2$$ in that frame, and this, in turn, coincides with the energy density. However, your objection is right: you cannot in general guarantee the existence of a frame in which the magnetic field is zero, so the author's choice of words was indeed unfortunate and misleading.
$$E\cdot B$$ is an invariant, up to sign, under Lorentz transformations. Hence, if in a reference frame it does not vanish (and you can construct many field configurations of this type in a given reference frame), then $$B$$ cannot vanish in every reference frame. You are right. There are situations where the density Lagrangian coincides with the density energy in no reference frames.