By Bak A. (ed.)
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Additional info for Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982
0 ji + N ∇ (55) The divergence term on the left side of (55) spoils the well-posed FOSH form we seek. However, if we use the identity Gij (g) + gij G0 0 (g) ≡ Rij (g) − gij Rk k (g) and the Einstein equations κ−1 Gαβ − Tαβ = 0, or κ−1 Rαβ = ραβ , we obtain (ε = −T 0 0 , ji = N T 0 i ) T j i − δ j i ε = (ρj i − δ j i ρk k ) ≡ S j i . (56) Then (54) and (55) obtain well-posed form (if S j i is assumed known), ¯ i ji = −2ji ∇ ¯ i N + 2N Kε + N K ij Sij , ∂¯0 ε + N ∇ (57) ¯ i ε = −2ε∇ ¯ i N + N Kji − ∇ ¯ j (N Sij ) .
Wheeler ed. Benjamin. 4. R. Geroch Domains of dependence J. Math. Phys. 111970 437-449. 5. Y. Choquet-Bruhat C ∞ solutions of non linear hyperbolic equations Gen. Rel. and Gravitation 2 1972 359-362. 6. Y. Nutku Harmonic maps in physics, Ann. Inst. Poincar´e, A 21, 1974 175-183. 7. C. W. Misner Harmonic maps as models of physical theories, Phys. Rev. D. 1978, 4510-4524. 8. Y. Choquet-Bruhat, D. Christodoulou and M. Francaviglia Cauchy data on a manifold, Ann. Inst. Poincar´e XXXI n◦ 4, 1979 399-414.
36 Arlen Anderson et al. where (˙) ≡ ∂¯0 ( ). A brief look at (29) shows that forming the combination Rij = Rij (g) − gij Rk k (g) leads to an equation of motion for the ADM canonical momentum π ij = g¯1/2 Kg ij − K ij (30) that contains no constraints. ) Indeed, using (28) and (29), we obtain the identity π˙ ij ≡ N g¯1/2 R(¯ g ) − N g¯−1/2 2π ik π j k − ππ ij g )g ij − Rij (¯ ¯ i∇ ¯ j N − g ij ∇ ¯ k∇ ¯ k N + N g¯1/2 Rij . +¯ g 1/2 ∇ (31) From the identity g˙ ij = −2N Kij , we have g˙ ij ≡ N g¯−1/2 (2πij − πgij ) .
Algebraic K-Theory, Number Theory, Geometry and Analysis: Proceedings of July 26-30, 1982 by Bak A. (ed.)
Categories: Geometry And Topology