Mirta S. Iriondo

Null surfaces formulation in three dimensions

The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contain the same geometrical meaning as the 4-D formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constructed on asymptotically flat space times and some of their kinematical aspects discussed. A particular example, which nevertheless contains most of the generic features is explicitly constructed and analyzed, revealing the behavior predicted in the full theory.The null surface formulation of general relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contains the same geometrical meaning as the four-dimensional formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constructed on asymptotically flat space–times and some of their kinematical aspects are discussed. A particular example, which nevertheless contains most of the generic features, is explicitly constructed and analyzed, revealing the behavior predicted in the full theory.

Einstein's Equation in Ashtekar's Variables Constitutes a Symmetric Hyperbolic System

We show that the 3+1 vacuum Einstein field equations in Ashtekars variables constitutes a first order symmetric hyperbolic system for arbitrary but fixed lapse and shift fields, by suitable adding to the system terms proportional to the constraint equations.

Null cones from I+ and Legendre submanifolds

It is shown that the main variable Z of the null surface formulation of GR is the generating family of a constrained Lagrange submanifold that lives on the energy surface H=0 and that its level surfaces Z=const yield Legendre submanifolds on that energy surface. Thus, the singularity structure of past null cones with apex at I+ is obtained by studying the projection map of the Legendre submanifolds to the configuration space. The behavior of the coordinate system defined by the variable Z at the caustic points is analyzed. It is shown that a single function Z(xa,ζ,ζ) cannot generate the conformal structure of an asymptotically flat space–time that satisfies the generic and weak energy condition.

Understanding singularities in Cartan’s and null surface formulation geometric structures

In this work we establish a relationship between Cartan’s geometric approach to third-order ordinary differential equations and the three-dimensional null surface formulation. We then generalize both constructions to allow for caustics and singularities that necessarily arise in these formalisms.

Fast and Slow solutions in General Relativity: The Initialization Procedure

We apply recent results in the theory of PDE, specifically in problems with two different time scales, to Einstein’s equations near their Newtonian limit. The results imply a justification to post-Newtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned, the first post-Newtonian equations and their solutions in terms of Green’s functions are presented in order to indicate how to proceed in calculations with this approach.

Cartan's equivalence method and null coframes in General Relativity

Using Cartans equivalence method for point transformations, we obtain from first principles the conformal geometry associated with third-order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a family of Lorentzian metrics, the conformal group in three and four dimensions and the so-called normal metric connection. A special feature of this connection is that the non-vanishing components of its torsion depend on one relative invariant, the (generalized) Wunschmann invariant. We show that the above-mentioned construction naturally contains the null surface formulation of general relativity.

Null surfaces and the Bach equations

It is shown that the integrability conditions that arise in the null surface formulation (NSF) of general relativity (GR) impose a field equation on the local null surfaces which is equivalent to the vanishing of the Bach tensor. This field equation is written explicitly to second order in a perturbation expansion. The field equation is further simplified if asymptotic flatness is imposed on the underlying space–time. The resulting equation determines the global null surfaces of asymptotically flat, radiative space–times. It is also shown that the source term of this equation is constructed from the free Bondi data at I. Possible generalizations of this field equation are analyzed. In particular we include other field equations for surfaces that have already appeared in the literature which coincide with ours at a linear level. We find that the other equations do not yield null surfaces for GR.

Numerical Treatment of Interfaces for Second-Order Wave Equations

In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. We show that it is possible to implement an interface scheme of “penalty” type for the second-order wave equation, similar to the ones used for first-order hyperbolic and parabolic equations, and the second-order scheme used by Mattsson et al. 2008. These schemes, known as SAT schemes for finite difference approximations and penalties for spectral ones, and ours share similar properties but in our case one needs to pass at the interface a smaller amount of data than previously known schemes. This is important for multi-block parallelizations in several dimensions, for it implies that one obtains the same solution quality while sharing among different computational grids only a fraction of the data one would need for a comparable (in accuracy) SAT or Mattsson et al.’s scheme. The semi-discrete approximation used here preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge–Kutta method. This is crucial, since the explicit Runge–Kutta method would be impractical given the severe restrictions that arise from the stiff parts of the equations.

Integrating singular functions on the sphere

We obtain rigorous results concerning the evaluation of integrals on the two-sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure.