Invariant Wave Maps from Gödel’s Universe
نویسنده
چکیده
Gödel’s metric is a solution g↵ to the Einstein field equations, with cosmological constant, in the presence of an incoherent matter distribution. Gödel’s universe G4 ↵ = R4 , g↵ is the total space of a principal bundle R ! G4 ↵ ! M3 over a 3dimensional nondegenerate CR manifold M3 = G4 ↵/K got as the space of orbits of a null Killing vector field K on G4 ↵. Invariant wave maps : G4 ↵ ! N are precisely the vertical lifts of subelliptic harmonic maps : M3 ! N . For every such we solve the L2 Dirichlet problem for the (degenerate elliptic) Jacobi operator J b and prove that J b has a discrete spectrum. 1. Why wave maps from G ↵? We start with a few motivational remarks bringing into the picture wave maps from Gödel’s universe. Gödel’s metric g↵ = ⇣ dx + e 1 dx 2 + (dx) + 1 2 e 1 (dx) + (dx) is a solution to Einstein’s field equations for an incoherent matter distribution at rest Rμ⌫ + ⇤gμ⌫ 1 2 R(g) gμ⌫ = 8⇡ c2 Tμ⌫ , T μ⌫ = ⇢ vv , (v) ⌘ (1, 0, 0, 0), ⇤ = ↵ 2 2 , ↵ ⇢ = 8⇡ c2 , and the field equations are the Euler-Lagrange equations of the variational principle S⌦(g) = 0 where S⌦(g) = Z
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تاریخ انتشار 2015