We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.Comment: 22 pages, 6 figures, to appear in Phys. Rev.

We find a one-parameter family of variables which recast the 3+1 Einstein equations into firstorder symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.

Laplacian in the hyperbolic space H n and linearization stability of the Einstein equation for Robertson-Walker modelsWe introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, nonlinear gravity. © 1999 American Institute of Physics. ͓S0022-2488͑99͒03002-9͔ implying that solutions to the complete set of equations are insensitive to properties of the evolution field in the vicinity of the constraint surface. In discrete approximations, on the other hand, a͒ Electronic mail: simo@mayu.physics.duq.edu JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 2 FEBRUARY 1999 909 0022-2488/99/40(2)/909/15/$15.00

The well posedness of the evolution of the constraints in standard 3ϩ1 general relativity is established by means of the Bianchi identities. Other related nonstandard 3ϩ1 cases which can be analyzed along similar lines are considered as well. The well posedness of the propagation of the constraints is relevant to the problem of unconstrained numerical evolution. ͓S0556-2821͑97͒01910-3͔