Birkhoff's theorem (electromagnetism)

In physics, in the context of electromagnetism, Birkhoff's theorem concerns spherically symmetric static solutions of Maxwell's field equations of electromagnetism.

The theorem is due to George D. Birkhoff. It states that any spherically symmetric solution of the source-free Maxwell equations is necessarily static. Pappas (1984) gives two proofs of this theorem,[1] using Maxwell's equations and Lie derivatives. It is a limiting case of Birkhoff's theorem (relativity) by taking the flat metric without backreaction.

Derivation from Maxwell's equations

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The source-free Maxwell's equations state that

 

Since the fields are spherically symmetric, they depend only on the radial distance in spherical coordinates. The field is purely radial as non-radial components cannot be invariant under rotation, which would be necessary for symmetry. Therefore, we can rewrite the fields as

 

We find that the curls must be zero, since,

 

Moreover, we can substitute into the source-free Maxwell equations, to find that

 

Simply dividing by the constant coefficients, we find that both the magnetic and electric field are static

 

Derivation using Lie derivatives

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Defining the 1-form   and 2-form   in   as:

 

Using the Hodge star operator, we can rewrite Maxwell's Equations with these forms[2] as

 .

The spherical symmetry condition requires that the Lie derivatives of   and   with respect to the vector field   that represents their rotations are zero

 

By the definition of the Lie derivative as the directional derivative along  

 .

Therefore,   is equivalent to   under rotation and we can write for some function  

 .

Because the product of the components of the vector are just its length

 .

And substituting back into our equation and rewriting for a function  

 .

Taking the exterior derivative of  , we find by definition that,

 .

And using our Maxwell equation that  ,

 .

Thus, we find that the magnetic field is static. Similarly, using the second rotational invariance equation, we can find that the electric field is static. Therefore, the solution must be static.

References

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  1. ^ Pappas, Richard C. (March 1984). "Proof of Birkhoff's theorem in electrodynamics". American Journal of Physics. 52 (3): 255–256. Bibcode:1984AmJPh..52..255P. doi:10.1119/1.13934. ISSN 0002-9505.
  2. ^ Flanders, Harley (1963). Differential Forms with Applications to the Physical Sciences. New York: Academic Press. pp. 46–47. ISBN 0-12-259650-1. OCLC 10441583.{{cite book}}: CS1 maint: date and year (link)