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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 Ein-stein 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)

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)

- 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)

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)

- Robert Bartnik
- 2008

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)

- Yong-Gao Chen, András Sárközy, +13 authors John Cossey
- 2008

On the monotonicity properties of additive representation functions.

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)

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

- Robert Bartnik, Paul Tod
- 2005

Conditions are given which, subject to a genericity condition on the Ricci tensor, are both necessary and sufficent for a 3-metric to arise from a static space-time metric. The vacuum field equations of General Relativity reduce for a static solution to a coupled system involving a (Riemannian) spatial metric g ij and a potential function V equal to the… (More)