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, 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.
The source-free Maxwell's equations state that
formula_1
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
formula_2
We find that the curls must be zero, since,
formula_3
Moreover, we can substitute into the source-free Maxwell equations, to find that
formula_4
Simply dividing by the constant coefficients, we find that both the magnetic and electric field are static
formula_5
Derivation using Lie derivatives.
Defining the 1-form formula_6 and 2-form formula_7 in formula_8 as:
formula_9
Using the Hodge star operator, we can rewrite Maxwell's Equations with these forms as
formula_10.
The spherical symmetry condition requires that the Lie derivatives of formula_6 and formula_7 with respect to the vector field formula_13 that represents their rotations are zero
formula_14
By the definition of the Lie derivative as the directional derivative along formula_13
formula_16.
Therefore, formula_6 is equivalent to formula_18 under rotation and we can write for some function formula_19
formula_20.
Because the product of the components of the vector are just its length
formula_21.
And substituting back into our equation and rewriting for a function formula_22
formula_23.
Taking the exterior derivative of formula_6, we find by definition that,
formula_25.
And using our Maxwell equation that formula_26,
formula_27.
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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