Skew Symmetric Bundle Maps on Space-time

摘要: We study the \Lie Algebra" of group Gauge Transforma- tions Space-time. obtain topological invariants arising from this Lie Algebra. Our methods give us fresh mathematical points view on Lorentz Transformations, orientation conventions, Doppler shift, Pauli matrices, Electro-Magnetic Duality Rotation, Poynting vectors, and Energy Mo- mentum Tensor T. LetM be a space-time andT (M) its tangent bundle. ThusM is 4-dimensional manifold with nondegenerate inner producth ;i T index + ++. space bundle maps F : (M)! which are skew symmetric respect to metric, i.e. hFv;vi =0f or allv2 Tx(M )a nd allx2 M. A has invariant planes eigenvector lines in each Tx(M). necessary sucient conditions as when these plane systems line form subbundles Theorem 7.3. Also we determine those same underlying structure. This done by introducing map TF = FF 1 (tr 2 )I (M). Then rise homeomorphic Map(M;S ), M into circle S . (See 7.11.) also show that natural complexica- tion. (see Propositions 2.2 2.3) leads an equivalence between vector elds complexied (M)C. The several beautiful relations link our subject matter Cliord Algebras Quaternions. Corollaries 4.6 4.7 4.8.) naturally many contact Physics, especially classical electromag- netism. These considerations frequently govern choice notation. physical motivations remarks will explored Scholia; mo- tivations links found Remarks.