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Niall Ó Murchadha

Publications and source records attributed to Niall Ó Murchadha.

At least 19 recordsLinked to original sources
arXiv2014

Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in 3, 4 and 6 dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in 3-dimensional spaces because of the link to General Relativity. In particular, all 3-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz-York method of finding initial data for Einstein equations.

gr-qc↗
arXiv2013

Constant Mean Curvature Slices of the Reissner-Nordström Spacetime

In order to specify a foliation of spacetime by spacelike hypersurfaces we need to place some restriction on the initial data and from this derive a way to calculate the lapse function $α$ which measures the proper time and interval between neighbouring hypersurfaces along the normal direction. Here we study a prescribed slicing known as the Constant Mean Curvature (CMC) slicing. This slicing will be applied to the Reissner-Nordström metric and the resultant slices will be investigated and then compared to those of the extended Schwarzschild solution.

gr-qc↗
arXiv2013

Potentials for transverse trace-free tensors

In the initial conditions of the $3 + 1$ formalism for numerical relativity, the transverse and trace-free (TT) part of the extrinsic curvature plays a key role. We know that TT tensors possess two degrees of freedom per space point. However, finding an expression for a TT tensor depending on only two scalar functions is a non-trivial task. Assuming either axial or translational symmetry, expressions depending on two scalar potentials alone are derived here for \emph{all} TT tensors in flat $3$-space. In a more general spatial slice, only one of these potentials is found, the same potential given in \cite{BakerPuzio} and \cite{Dain}, with the remaining equations reduced to a partial differential equation, depending on boundary conditions for a solution. As an exercise, we also derive the potentials which give the Bowen-York curvature tensor in flat space.

gr-qc↗
arXiv2012

Scaling up the extrinsic curvature in asymptotically flat gravitational initial data: Generating trapped surfaces

The existence of the initial value constraints means that specifying initial data for the Einstein equations is non-trivial. The standard method of constructing initial data in the asymptotically flat case is to choose an asymptotically flat 3-metric and a transverse-tracefree (TT) tensor on it. One can find a conformal transformation that maps these data into solutions of the constraints. In particular, the TT tensor becomes the extrinsic curvature of the 3-slice. We wish to understand how the physical solution changes as the free data is changed. In this paper we investigate an especially simple change: we multiply the TT tensor by a large constant. One might assume that this corresponds to pumping up the extrinsic curvature in the physical initial data. Unexpectedly, we show that, while the conformal factor monotonically increases, the physical extrinsic curvature decreases. The increase in the conformal factor however means that the physical volume increases in such a way that the ADM mass become unboundedly large. In turn, the blow-up of the mass combined with the control we have on the extrinsic curvature allows us to show that trapped surfaces, i.e., surfaces that are simultaneously future and past trapped, appear in the physical initial data.

gr-qc↗
arXiv2012

Scaling up the extrinsic curvature in gravitational initial data

Vacuum solutions to the Einstein equations can be viewed as the interplay between the geometry and the gravitational wave energy content. The constraints on initial data reflect this interaction. We assume we are looking at cosmological solutions to the Einstein equations so we assume that the 3-space is compact, without boundary. In this article we investigate, using both analytic and numerical techniques, what happens when the extrinsic curvature is increased while the background geometry is held fixed. This is equivalent to trying to magnify the local gravitational wave kinetic energy on an unchanged background. We find that the physical intrinsic curvature does not blow up. Rather the local volume of space expands to accommodate this attempt to increase the kinetic energy.

gr-qc↗
arXiv2012

Intrinsic time gravity and the Lichnerowicz-York equation

We investigate the effect on the Hamiltonian structure of general relativity of choosing an intrinsic time to fix the time slicing. 3-covariance with momentum constraint is maintained, but the Hamiltonian constraint is replaced by a dynamical equation for the trace of the momentum. This reveals a very simple structure with a local reduced Hamiltonian. The theory is easily generalised; in particular, the square of the Cotton-York tensor density can be added as an extra part of the potential while at the same time maintaining the classic 2 + 2 degrees of freedom. Initial data construction is simple in the extended theory; we get a generalised Lichnerowicz-York equation with nice existence and uniqueness properties. Adding standard matter fields is quite straightforward.

gr-qc↗
arXiv2011

Spin and orbital angular momentum of the tensor gauge field

Following the recent studies of the trickiness in spin and orbital angular momentum of the vector gauge fields, we perform here a parallel analysis for the tensor gauge field, which has certain relation to gravitation. Similarly to the vector case, we find a nice feature that after removing all gauge degrees of freedom the angular momentum of the tensor gauge field vanishes for a stationary system. This angular momentum also shows a one-parameter invariance over the infinitely many ways of complete gauge fixing for the tensor field. The tensor gauge coupling, however, does exhibit a critical difference from the vector gauge coupling that it may induce intrinsic interaction terms into the spatial translation and rotation generators, leaving none of the ten Poincaré generators interaction-free.

hep-th↗
arXiv2010

Conformal Superspace: the configuration space of general relativity

It has long been considered that conformal superspace is the natural configuration space for canonical general relativity. However, this was never definitively demonstrated. We have found that the standard conformal method of solving the Einstein constraints has an unexpected extra symmetry. This allows us to complete the project. We show that given a point and a velocity in conformal superspace, the Einstein equations generate a unique curve in conformal superspace.

gr-qc↗
arXiv2010

Quasilocal Energy in General Relativity

We show that the quasilocal mass defined by Wang and Yau is not well-defined at spatial infinity. It approaches neither the ADM mass nor the ADM energy. We suggest an alternative scheme which retains all the desirable characteristics of the Wang-Yau mass and, in addition, asymptotes to the ADM energy at infinity.

gr-qc↗
arXiv2009

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 {\it a priori}, without recourse to the iterative procedure that is necessary for wormhole data.

gr-qc↗
arXiv2009

Stationary hyperboloidal slicings with evolved gauge conditions

We analyze stationary slicings of the Schwarzschild spacetime defined by members of the Bona-Masso family of slicing conditions. Our main focus is on the influence of a non-vanishing offset to the extrinsic curvature, which forbids the existence of standard Cauchy foliations but at the same time allows gauge choices that are adapted to include null infinity (scri) in the evolution. These hyperboloidal slicings are especially interesting for observing outgoing gravitational waves. We show that the standard 1+log slicing condition admits no overall regular hyperboloidal slicing, but by appropriately combining with harmonic slicing, we construct a gauge condition that leads to a strongly singularity-avoiding hyperboloidal foliation that connects the black hole to scri.

gr-qc↗
arXiv2007

Where do moving punctures go?

Currently the most popular method to evolve black-hole binaries is the ``moving puncture'' method. It has recently been shown that when puncture initial data for a Schwarzschild black hole are evolved using this method, the numerical slices quickly lose contact with the second asymptotically flat end, and end instead on a cylinder of finite Schwarzschild coordinate radius. These slices are stationary, meaning that their geometry does not evolve further. We will describe these results in the context of maximal slices, and present time-independent puncture-like data for the Schwarzschild spacetime.

gr-qc↗
arXiv2006

The Einstein constraints: uniqueness and non-uniqueness in the conformal thin sandwich approach

We study the appearance of multiple solutions to certain decompositions of Einstein's constraint equations. Pfeiffer and York recently reported the existence of two branches of solutions for identical background data in the extended conformal thin-sandwich decomposition. We show that the Hamiltonian constraint alone, when expressed in a certain way, admits two branches of solutions with properties very similar to those found by Pfeiffer and York. We construct these two branches analytically for a constant-density star in spherical symmetry, but argue that this behavior is more general. In the case of the Hamiltonian constraint this non-uniqueness is well known to be related to the sign of one particular term, and we argue that the extended conformal thin-sandwich equations contain a similar term that causes the breakdown of uniqueness.

gr-qc↗
arXiv2005

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 3-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 York's 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.

gr-qc↗
arXiv2004

The Jang equation, apparent horizons, and the Penrose inequality

The Jang equation in the spherically symmetric case reduces to a first order equation. This permits an easy analysis of the role apparent horizons play in the (non)existence of solutions. We demonstrate that the proposed derivation of the Penrose inequality based on the Jang equation cannot work in the spherically symmetric case. Thus it is fruitless to apply this method, as it stands, to the general case. We show also that those analytic criteria for the formation of horizons that are based on the use of the Jang equation are of limited validity for the proof of the trapped surface conjecture.

gr-qc↗
arXiv2003

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) admit 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.

gr-qc↗
arXiv2003

Embedding spherical spacelike slices in a Schwarzschild solution

Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given Schwarzschild solution covers a significant subset of spherical superspace and these subsets form a sequence of nested domains as the Schwarzschild mass increases. This also demonstrates that spherical data offer an immediate counter example to the thick sandwich `theorem'.

gr-qc↗