Session L10: Mathematical and Post-Newtonian Relativity

Black hole in a post-Newtonian tidal field

In this talk I describe an ongoing project that aims to determine the tidal distortion of a nonrotating black hole when it is placed in the presence of other nearby bodies. The context of this work is very general, but the focus here will be on post-Newtonian tidal environments. The first part of the project consists of calculating the black-hole metric in terms of arbitrary tidal fields that characterize the tidal environment. In the second part the tidal fields are determined by inserting the black hole within a global spacetime that contains the other bodies. Here the global spacetime contains an arbitrary number of bodies that move slowly under their weak mutual gravity, and its metric is described by post-Newtonian theory. We calculate the tidal fields acting on the black hole and express them in terms of the post-Newtonian potentials. Because the post-Newtonian metric of the N-body system is not valid in the strongly-gravitating environment of the black hole, we work in a buffer region around the black hole where the tidal metric and the post-Newtonian metric are both valid. The tidal fields are determined by matching the two metrics in this buffer region.   

Metric of a Nonrotating Black Hole in a Tidal Environment

We consider the perturbed field of a Schwarzschild black hole with a mass $M$ much smaller than the local radius of curvature $R$ generated by the external Universe. We discuss light-cone coordinates used to represent the combined metric of the black hole and external matter. The metric of the external Universe is found as an expansion in STF harmonics, then the Einstein equations are solved for the full metric at the first nonlinear order $O((M/R)^4)$. We discuss the gauge freedom and analize the solution.   

Three-body equations of motion in successive post-Newtonian approximations

There are periodic solutions to the equal-mass three-body (and N-body) problem in Newtonian gravity. The figure-eight solution is one of them. In this paper, we discuss its solution in the first and second post-Newtonian approximations to General Relativity. To do so we derive the canonical equations of motion in the ADM gauge from the three-body Hamiltonian. We then integrate those equations numerically, showing that quantities such as the energy, linear and angular momenta are conserved down to numerical error. We also study the scaling of the initial parameters with the physical size of the triple system. In this way we can assess when general relativistic results are important and we determine that this occur for distances of the order of 100M, with M the total mass of the system. For distances much closer than those, presumably the system would completely collapse due to gravitational radiation. This sets up a natural cut-off to Newtonian N-body simulations. The method can also be used to dynamically provide initial parameters for subsequent full nonlinear numerical simulations.   

Angular momentum at null infinity

I will describe a definition of angular momentum at null infinity which appears to be satisfactory. It is natural, resolves the supertranslation problem, allows a computation of fluxes, and gives physically plausible characterizations of spin and center of mass. It is a development of Penrose's twistor-based ideas, but I will recast it in conventional (non-twistor) terms. The supertranslation problem prevents a consistent treatment of gravitational angular momentum in special-relativistic terms. The resolution of this difficulty turns out to be that the angular momentum is not a pure $j=1$ quantity $M_{ab}$, but acquires higher-$j$ terms as well. These higher-$j$ terms fit into the theory just so as to give geometrically natural definitions of spin and center of mass. Remarkably, too, they correspond precisely to the Bondi shear. So shear and angular momentum should be regarded as different elements of a single unified concept. While this definition reproduces conventional results in weak-field slow-motion limits, it has novel features in more general situations. Systems which are asymmetric and highly dynamical may radiate angular momentum (including ``conventional,'' $j=1$, angular momentum) at {\em first order} in the gravitational wave strength. Astrophysical systems might have measurable ``hops'' and spin-changes associated with such emissions of center-of-mass or spin angular momentum. \newline See gr-qc/0709.1078; to appear in Gen. Rel. Grav.   

Beyond Discrete Vacuum Spacetimes

In applications to pre-geometric models of quantum gravity, one expects matter to play an important role in the geometry of the spacetime. Such models often posit that the matter fields play a crucial role in the determination of the spacetime geometry. However, it is not well understood at a fundamental level how one couples matter into the Regge geometry. In order to better understand the nature of such theories that rely on Regge Calculus, we must first gain a better understanding of the role of matter in a lattice spacetime. We investigate consistent methods of incorporating matter into spacetime, and particularly focus on the role of spinors in Regge Calculus. Since spinors are fundamental to fermionic fields, this investigation is crucial in understanding fermionic coupling to discrete spacetime. Our focus is primarily on the geometric interpretation of the fields on the lattice geometry with a goal on understanding the dynamic coupling between the fields and the geometry.   

Measuring the Scalar Curvature with Clocks and Photons: Voronoi-Delaunay Lattices in Regge Calculus

The Riemann scalar curvature plays a central role in Einstein's geometric theory of gravity. We describe a new geometric construction of this scalar curvature invariant at an event (vertex) in a discrete spacetime geometry. This allows one to constructively measure the scalar curvature using only clocks and photons. Given recent interest in discrete pre-geometric models of quantum gravity, we believe it is ever so important to reconstruct the curvature scalar with respect to a finite number of communicating observers. This derivation makes use of a fundamental lattice cell built from elements inherited from both the original simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding hinge-based expression in Regge Calculus (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted average of deficits per weighted average of dual areas.   

Geometrization of Electromagnetism in Purely Affine and Metric-Affine Gravity

The purely affine Lagrangian for linear electrodynamics, that has the form of the Maxwell Lagrangian in which the metric tensor is replaced by the symmetrized Ricci tensor and the electromagnetic field tensor by the tensor of homothetic curvature, is dynamically equivalent to the Einstein-Maxwell equations in the metric-affine and metric formulation. We apply to a purely affine Lagrangian the Legendre transformation with respect to the tensor of homothetic curvature to show that the corresponding Legendre term and the new Hamiltonian density are related to the metric-affine Maxwell-Palatini Lagrangian for the electromagnetic field. Therefore the purely affine picture, in addition to generating the gravitational Lagrangian that is linear in the curvature, justifies why the electromagnetic Lagrangian is quadratic in the electromagnetic field.   

A modification of Einstein-Schrodinger theory which closely approximates Einstein-Weinberg-Salam theory

The Lambda-Renormalized Einstein-Schrodinger theory is a modification of the original Einstein-Schrodinger theory in which a cosmological constant term is added to the Lagrangian, and this theory has been shown to closely approximate Einstein-Maxwell theory. Here we generalize this theory to non-Abelian fields by allowing the fields to be composed of 2x2 Hermitian matrices, and we consider the case where the symmetric part of the fields are multiples of the identity matrix. The resulting theory incorporates the U(1) and SU(2) gauge terms of the Weinberg-Salam Lagrangian, and when the rest of the Weinberg-Salam Lagrangian is included, we get a close approximation to Einstein-Weinberg-Salam theory. In particular, the field equations match those of Einstein-Weinberg-Salam theory except for additional terms which are $<10^{-13}$ of the usual terms for worst-case field strengths and rates of change accessible to measurement. The Lagrangian density is invariant under U(1) and SU(2) gauge transformations.