2
$\begingroup$

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...".

quote from book

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?

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Would an elliptical current-carrying wire be a system in which the magnetic field can never be transformed away? $\endgroup$ – probably_someone Jan 31 at 1:02
  • $\begingroup$ That's troubling, because it suggests that everything that follows in the book may be untrustworthy! It's not obvious to me that the Lagrangian given there really is "the energy-density of the field in a local frame moving so that the magnetic field vanishes...". $\endgroup$ – S. McGrew Jan 31 at 1:06
  • $\begingroup$ that lagrangian density is also valid if the current is zero, that is, in free space, an assumption sometimes made $\endgroup$ – Wolphram jonny Jan 31 at 1:20
  • $\begingroup$ @probably_someone even an infinitely long straight, current-carrying wire (which generates a magnetic field, but whose electric field is zero) is a system in which the magnetic field cannot be transformed away. $\endgroup$ – Bruno De Souza Leão Jan 31 at 21:24
2
$\begingroup$

$E\cdot B$ 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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.