By Benz W.
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Additional info for (83, 64)-Konfigurationen in Laguerre-, Mobius-und weiteren Geometrien
11. W. (1995) Einstein and Yang-Mills theories in hyperbolic form without gauge ﬁxing. Phys Rev Lett 75:3377-3381. 12. W. (1996) A nonstrictly hyperbolic system for the Einstein equations with arbitrary lapse and shift. R. Acad Sci Paris S´erie IIb 323:835-841. 13. W. (1997) 3+1 general relativity in hyperbolic form. , Lasota J-P. ) Relativistic Astrophysics and Gravitational Radiation. North Holland, Amsterdam, 179-190. 14. W. (1997) Geometrical hyperbolic systems for general relativity and gauge theories.
We discuss several explicitly causal hyperbolic formulations of Einstein’s dynamical 3 + 1 equations in a coherent way, emphasizing throughout the fundamental role of the “slicing function,” α—the quantity that relates the lapse N to the determinant of the spatial metric g¯ through N = g¯1/2 α. The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations.
0 ji + N ∇ (58) By combining (57) plus (49), and (58) plus (50), we obtain expressions of gravity constraint evolution in the presence of matter, ¯ iC T = 2 C T ∇ ¯ j N + N KC T − N K ij (κ−1 Rij − Sij ) , ∂¯0 C T − N ∇ i j ¯ jCT = 2 CT ∇ ¯ i N (κ−1 Rij − Sij ) ¯ j N + 1 N KCjT − ∇ ∂¯0 CjT − N ∇ 2 (59) (60) where C T ≡ C + 2ε and CjT = Cj − 2jj . This is just the form we would anticipate on the basis of Hamiltonian dynamics and the form (49) and (50) of the vacuum constraints. Thus, for gravity plus a matter ﬁeld, we obtain results analogous to (44) and (45) for the total system.