ar X iv : g r - qc / 9 90 10 21 v 1 7 J an 1 99 9 Fixing Einstein ’ s equations

Einstein's theory of general relativity has not only proven to be physically accurate [1], it also sets a standard for mathematical beauty and elegance—geometrically. When viewed as a dynamical system of equations for evolving initial data, however, these equations have a serious flaw: they cannot be proven to be well-posed (except in special coordinates [2–4]). That is, they do not produce unique solutions that depend smoothly on the initial data. To remedy this failing, there has been widespread interest recently in reformulating Einstein's theory as a hyperbolic system of differential equations [5–20]. The physical and geometrical content of the original theory remain unchanged, but dynamical evolution is made sound. Here we present a new hyperbolic formulation that is strikingly close to the space-plus-time (" 3+1 ") form of Einstein's original equations. Indeed, the familiarity of its constituents make the existence of this formulation all the more unexpected. This is the most economical first-order symmetrizable hyperbolic formulation presently known to us that has only physical characteristic speeds, either zero or the speed of light, for all (non-matter) variables. It also serves as a foundation for unifying previous proposals. The source of the imperfection in Einstein's theory lies in the fact that it is a constrained theory. Physical initial data cannot be freely specified, yet even infinitesimally perturbed data that violate the physical constraints can lead to results so wildly divergent that they spoil the desired smooth dependence on initial data. This is particularly troublesome in numerical evolution where such violations are unavoidable. The lack of well-posedness is also a serious problem when addressing such a basic question as the global nonlinear stability of flat Minkowski spacetime. In fact, the proof of such stability [21] employs the hyperbolic wave equation of [7] which we discuss below. A well-posed formulation of Einstein's equations would also seem to be an essential starting point for the conventional approach to quantum gravity in which one first quantizes the (unconstrained) classical theory and then imposes the constraints. The desire to simulate numerically the full nonlinear evolution of Einstein's equations in three dimensions, such as in the collision of two black holes (cf., e.g., [22]), has motivated much of the recent effort on hyperbolic formulations: well-posed underlying equations make stable numerical evolution much more likely than otherwise would be the case, and formulations cast in first-order symmetrizable form are especially suited to numerical implementation. In addition, physical characteristic …