Constraints and evolution in cosmology

We review some old and new results about strict and non strict hyperbolic formulations of the Einstein equations.Comment: To appear in the proceedings of the first Aegean summer school in General Relativity, S. Cotsakis ed. Springer Lecture Notes in Physic

Geometrical Hyperbolic Systems for General Relativity and Gauge Theories

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 $(1)$ taking a further time derivative of the equation of motion of the canonical momentum, and $(2)$ adding a covariant spatial derivative of the momentum constraints of general relativity (Lagrange multiplier $\beta^{i}$) or of the Gauss's 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.Comment: 24 pages, latex, no figure

A Note on Positive Energy Theorem for Spaces with Asymptotic SUSY Compactification

We extend the positive mass theorem proved previously by the author to the Lorentzian setting. This includes the original higher dimensional positive energy theorem whose spinor proof was given by Witten in dimension four and by Xiao Zhang in dimension five

Einstein and Yang-Mills theories in hyperbolic form without gauge-fixing

The evolution of physical and gauge degrees of freedom in the Einstein and Yang-Mills theories are separated in a gauge-invariant manner. We show that the equations of motion of these theories can always be written in flux-conservative first-order symmetric hyperbolic form. This dynamical form is ideal for global analysis, analytic approximation methods such as gauge-invariant perturbation theory, and numerical solution.Comment: 12 pages, revtex3.0, no figure

Multiply Warped Products with Non-Smooth Metrics

In this article we study manifolds with $C^{0}$-metrics and properties of Lorentzian multiply warped products. We represent the interior Schwarzschild space-time as a multiply warped product space-time with warping functions and we also investigate the curvature of a multiply warped product with $C^0$-warping functions. We given the {\it{Ricci curvature}} in terms of $f_1$, $f_2$ for the multiply warped products of the form $M=(0,\ 2m)\times_{f_1}R^1\times_{f_2} S^2$.Comment: LaTeX, 7 page

Hamiltonian Time Evolution for General Relativity

Hamiltonian time evolution in terms of an explicit parameter time is derived for general relativity, even when the constraints are not satisfied, from the Arnowitt-Deser-Misner-Teitelboim-Ashtekar action in which the slicing density $\alpha(x,t)$ is freely specified while the lapse $N=\alpha g^{1/2}$ is not. The constraint ``algebra'' becomes a well-posed evolution system for the constraints; this system is the twice-contracted Bianchi identity when $R_{ij}=0$. The Hamiltonian constraint is an initial value constraint which determines $g^{1/2}$ and hence $N$, given $\alpha$.Comment: 4 pages, revtex, to appear in Phys. Rev. Let

Conformal ``thin sandwich'' data for the initial-value problem of general relativity

The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function.Comment: 7 pages, RevTe

The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves

The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.Comment: Version to appear in Class. Quantum Gra

Hyperbolic formulations and numerical relativity II: Asymptotically constrained systems of the Einstein equations

We study asymptotically constrained systems for numerical integration of the Einstein equations, which are intended to be robust against perturbative errors for the free evolution of the initial data. First, we examine the previously proposed "$\lambda$-system", which introduces artificial flows to constraint surfaces based on the symmetric hyperbolic formulation. We show that this system works as expected for the wave propagation problem in the Maxwell system and in general relativity using Ashtekar's connection formulation. Second, we propose a new mechanism to control the stability, which we call the ``adjusted system". This is simply obtained by adding constraint terms in the dynamical equations and adjusting its multipliers. We explain why a particular choice of multiplier reduces the numerical errors from non-positive or pure-imaginary eigenvalues of the adjusted constraint propagation equations. This ``adjusted system" is also tested in the Maxwell system and in the Ashtekar's system. This mechanism affects more than the system's symmetric hyperbolicity.Comment: 16 pages, RevTeX, 9 eps figures, added Appendix B and minor changes, to appear in Class. Quant. Gra

The constraint equations for the Einstein-scalar field system on compact manifolds

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum Gravit