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The Evolution of the Weyl and Maxwell Fields in Curved Space—Times
Author(s) -
de Vries Andreas
Publication year - 1996
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19961790103
Subject(s) - mathematics , covariant transformation , mathematical physics , spinor , maxwell's equations , hilbert space , hermitian matrix , order (exchange) , curved space , wave equation , field (mathematics) , physics , mathematical analysis , pure mathematics , finance , economics
The covariant Weyl (spin s = 1/2) and Maxwell ( s = 1) equations in certain local charts (u, φ) of a space‐time ( M, g ) are considered. It is shown that the condition g 00 ( x ) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ t φ = L (s) φ. Here L (s) is a linear first order differential operator on the pre—Hilbert space (C 0 ∞( U t , 2s+1 ). (…)), where U t ⊂ IR 3 is the image of the coordinate map of the spacelike hyper‐surface t = const, and (φ, C) = ƒU t ϕ *Q d (3) x with a suitable Hermitian n × n ‐ matrix Q = Q(t,x ). The total energy of the spinor field ϕ with respect to U t is then simply given by E = 〈ϕ,ϕ 〉. In this way inequalities for the energy change rate with respect to time, δ t |ϕ| 2 = 2Re (ϕ, L (s) ϕ) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.