A useful exercise for future explorations
The mathematical context in which A. Lichnerowicz did write his formula has been clarified on the page presenting the invariants of any electromagnetic field. Let now calculate its value. It is obvious that the objective is to recover:
ψ = < F, F > = < H, H > - < E, E >
A. Lichnerowicz recommends calculating it using [01 ; § 8, p. 15, Equ.(8-1) and p. 16, Equ(8-4)].
I am trying to reproduce his calculation to obtain the result [01 ; § 8, p. 17 Eq.(8-5)]. In a first attempt, I naively interpret the notation [01 ; § 8, p. 17] as a simple scalar product. This approach yields:
2. Ψ
=
This interpretation of the product < F, F > leads to an expression for the volume density of energy carried by the electromagnetic field.
This is clearly not the expected result.
The difference in sign distinguishing it from the expected expression cannot stem from a difference in the convention adopted for the signature of the metric.
Indeed, in [02 ; p. XIII (1992)], as in [01, § 8, first line below Eq.(8-1) (1955)], the signature is of the (+ - - -) type and the calculation of the expression [02 ; § 25, p. 76, Eq.(25,1)] leads to [02 ; § 25, p. 76, Eq.(25,3)] which is identical to [01 ; § 8, p. 17 Eq.(8-5)].
Furthermore, reference [02 ; § 23, p.73] proposes a matrix representation of electromagnetic fields completely consistent with current practices.
In conclusion: the notation < F, F > must not be interpreted as a simple scalar product.
Should we then "requalify" the previous calculation by taking into account the fact that the physical context is related to a Minkowski geometry?
In other words, should this expression be interpreted as follows:
ψ= < F, F >[η]
… an expression in which the diagonal matrix (+ - - -) [η] of M(4, R) precisely represents this geometry?
The answer is equally negative.
Proof: if this track were acceptable, the calculation would have to be reiterated as follows:
Ψ
=
∑α ∑ (ηαβ. Fαβ. Fαβ)
=
(η00. F00. F00) + (η01. F01. F01) + etc.
As a matter of facts, the components of the metric tensor systematically vanish as soon as the two indices differ.
α ≠ β ⇒ ηαβ = 0, Fαβ ≠ 0
Simultaneously, the only non-zero components of the electromagnetic field tensor are those for which the two indices differ.
As a result, this interpretation of the notation used by A. Lichnerowicz would automatically yield a zero result.
This is not the case.
c.q.f.d.
The exact result is only obtained by actually working with alternating forms, indicator tensors, and adjoint tensors.
A half-century leap forward and a look at courses devoted to tensor calculus later, notably those of rank two, allow us to rediscover the historical result; indeed, in any metric:
Thus, by specifying the writing that appeared in [01 ; § 8, p. 17] using the language associated with tensor calculus... the classical result [01 ; § 8, p. 17 Equ.(8-5)] is recovered exactly (with the correct signs) :
Ψ
=
< F, F >
=
½. ∑α ∑β Fαβ. Fαβ
=
< H, H > - < E, E >
c.q.f.d.
[01] Lichnerowicz, A.. : Théories relativistes de la gravitation et de l'électromagnétisme, Relativité générale et théories unitaires, Masson et Cie, Éditeurs 1955, 289 pages.
[02] Landau, L. D. et Lifschitz, E. M.: Lehrbuch der theoretischen Physik, II : Klassische Feldtheorie; Akademie Verlag GmbH, Berlin 1992 ISBN 3-05-501550-9, 480 S.