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- ROBERT BARTNIK, R. BARTNIK
- 2006

We show that the mass of an asymptotically flat n-manifold is a geometric invariant. The proof is based on harmonic coordinates and, to develop a suitable existence theory, results about elliptic operators with rough coefficients on weighted Sobolev spaces are summarised. Some relations between the mass. xalar curvature and harmonic maps are described and… (More)

Initial data for solutions of Einstein’s gravitational field equations cannot be chosen freely: the data must satisfy the four Einstein constraint equations. We first discuss the geometric origins of the Einstein constraints and the role the constraint equations play in generating solutions of the full system. We then discuss various ways of obtaining… (More)

- ROBERT BARTNIK
- 2009

Globally regular (ie. asymptotically flat and regular interior), spherically symmetric and localised (“particle-like”) solutions of the coupled Einstein Yang-Mills (EYM) equations with gauge group SU(2) have been known for more than 20 years, yet their properties are still not well understood. Spherically symmetric Yang–Mills fields are classified by a… (More)

- Robert Bartnik
- 2002

Physicists believe, with some justification, that there should be a correspondence between familiar properties of Newtonian gravity and properties of solutions of the Einstein equations. The Positive Mass Theorem (PMT), first proved over twenty years ago [45, 53], is a remarkable testament to this faith. However, fundamental mathematical questions… (More)

- Robert Bartnik, Andrew H. Norton
- SIAM J. Scientific Computing
- 2000

- Robert Bartnik
- 2004

A Hilbert manifold structure is described for the phase space F of asymptotically flat initial data for the Einstein equations. The space of solutions of the constraint equations forms a Hilbert submanifold C ⊂ F . The ADM energy-momentum defines a function which is smooth on this submanifold, but which is not defined in general on all of F . The ADM… (More)

- Robert Bartnik
- 1997

Properties of the Einstein equations in a coordinate gauge based on expanding null hypersurfaces foliated by metric 2-spheres are described. This null quasi-spherical (NQS) gauge leads to particularly simple analyses of the characteristic structure of the equations and of the propagation of gravitational shocks, and clarifies the geometry of timelike… (More)

We study spherically symmetric dynamical horizons (SSDH) in spherically symmetric Einstein/matter spacetimes. We first determine sufficient and necessary conditions for an initial data set for the gravitational and matter fields to satisfy the dynamical horizon condition in the spacetime development. The constraint equations reduce to a single second order… (More)

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp… (More)

We describe numerical techniques used in our construction of a 4th order in time evolution for the full Einstein equations, and assess the accuracy of some representative solutions. The scheme employs several novel geometric and numerical techniques, including a geometrically invariant coordinate gauge, which leads to a characteristic-transport formulation… (More)