Niall Ó Murchadha

The physical gravitational degrees of freedom

When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving three-dimensional conformal Riemannian geometries obtained by imposing two general principles: (1) time is derived from change; (2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation, but also all the equations used in Yorks conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.

Geometry and Regularity of Moving Punctures

Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.

Scale-invariant gravity: Geometrodynamics

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyls idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t. scaling developed in the parallel particle dynamics paper by one of the authors. In spatially compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented. Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.

Wormholes and trumpets: Schwarzschild spacetime for the moving-puncture generation

We expand upon our previous analysis of numerical moving-puncture simulations of the Schwarzschild spacetime. We present a derivation of the family of analytic stationary 1 þ log foliations of the Schwarzschild solution, and outline a transformation to isotropic coordinates. We discuss in detail the numerical evolution of standard Schwarzschild puncture data, and the new time-independent 1 þ log data. Finally, we demonstrate that the moving-puncture method can locate the appropriate stationary geometry in a robust manner when a numerical code alternates between two forms of 1 þ log slicing during a simulation.

Total energy momentum in general relativity

The energy momentum of any asymptotically flat vacuum solution to the Einstein equations is a well‐defined, conserved, Lorentz‐covariant, timelike, future‐pointing vector. The only requirement is that one be given asymptotically flat initial data that satisfy very weak continuity and falloff conditions; the three‐metric must go flat faster than r−1/2. A large class of such data exists, consistent with the constraints, and the constraints play a key role in guaranteeing that the energy momentum is well behaved.

Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations

The assumption that a solution to the Einstein equations is static (or stationary) very strongly constrains the asymptotic behaviour of the metric. It is shown that one need only impose very weak differentiability and decay conditions a priori on the metric for the field equations to force the metric to be analytic near infinity and to have the standard Schwarzschildian fall-off.

Constant mean curvature slices in the extended Schwarzschild solution and the collapse of the lapse

An explicit CMC Schwarzschildean line element is derived near the critical point of the foliation, the lapse is shown to decay exponentially, and the coefficient of the exponent is calculated.

Trapped surfaces and the Penrose inequality in spherically symmetric geometries

We demonstrate that the Penrose inequality is valid for spherically symmetric geometries even when the horizon is immersed in matter. The matter field need not be at rest. The only restriction is that the source satisfies an energy condition outside the horizon. No restrictions are placed on the matter inside the horizon. The proof of the Penrose inequality gives a new necessary condition for the formation of trapped surfaces. This formulation may also be adapted to give a sufficient condition. We show that a modification of the Penrose inequality proposed by Gibbons for charged black holes can be broken in early stages of gravitational collapse. This investigation is based exclusively on the initial data formulation of general relativity.

Bowen-York trumpet data and black-hole simulations

The most popular method to construct initial data for black-hole-binary simulations is the puncture method, in which compactified wormholes are given linear and angular momentum via the Bowen-York extrinsic curvature. When these data are evolved, they quickly approach a trumpet topology, suggesting that it would be preferable to use data that are in trumpet form from the outset. To achieve this, we extend the puncture method to allow the construction of Bowen-York trumpets, including an outline of an existence and uniqueness proof of the solutions. We construct boosted, spinning and binary Bowen-York puncture trumpets using a single-domain pseudospectral elliptic solver, and evolve the binary data and compare with standard wormhole-data results. We also show that for boosted trumpets the black-hole mass can be prescribed a priori, without recourse to the iterative procedure that is necessary for wormhole data.

Non-CMC conformal data sets which do not produce solutions of the Einstein constraint equations

The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admits a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non-CMC data, much less is known. Here we exhibit the first class of non-CMC data for which we can prove that no solutions exist.