Celebratio Mathematica

A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Einstein and Yang–Mills theories in hyperbolic form without gauge fixing,” Phys. Rev. Lett. 75 : 19 (1995), pp. 3377–3381. MR 1356171 Zbl 1020.83503 article

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A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York: “Un système hyperbolique non strict pour les équations d’Einstein,” C. R. Acad. Sci., Paris, Sér. II, Fasc. b 323 : 12 (1996), pp. 835–841. Zbl 0916.35126 article

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A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J. W. York, Jr.: “Geometrical hyperbolic systems for general relativity and gauge theories,” Classical Quantum Gravity 14 : 1A (1997), pp. A9–A22. MR 1691883 Zbl 0866.58059 article

The evolution equations of Einstein’s theory and of Maxwell’s theory — the latter used as a simple model to illustrate the former — are written in gauge-covariant first-order symmetric hyperbolic form with only physically natural characteristic directions and speeds for the dynamical variables. Quantities representing gauge degrees of freedom (the spatial shift vector \( \beta^i(t, x^j) \) and the spatial scalar potential \( \phi(t, x^j) \), respectively) are not among the dynamical variables: the gauge and the physical quantities in the evolution equations are effectively decoupled. For example, the gauge quantities could be obtained as functions of \( (t, x^j) \) from subsidiary equations that are not part of the evolution equations. Propagation of certain (“radiative”) dynamical variables along the physical light cone is gauge invariant while the remaining dynamical variables are dragged along the axes orthogonal to the spacelike time slices by the propagating variables. We obtain these results by

taking a further time derivative of the equation of motion of the canonical momentum, and
adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier \( \beta^i \)) or of the Gauss law constraint of electromagnetism (Lagrange multiplier \( \phi \)).

General relativity also requires a harmonic time-slicing condition or a specific generalization of it that brings in the Hamiltonian constraint when we pass to first-order symmetric form. The dynamically propagating gravity fields straightforwardly determine the “electric” or “tidal” parts of the Riemann tensor

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