We review the theory of stationary black hole solutions of vacuum Einstein equations.

We introduce and study the mechanical system which describes the dynamics and statics of rigid bodies of constant density floating in a calm incompressible fluid. Since much of the standard equilibrium theory, starting with Archimedes, allows bodies with vertices and edges, we assume the bodies to be convex and take care not to assume more regularity than that implied by convexity. One main result is the (Lyapunov) stability of equilibria satisfying a condition equivalent to the standard ‘metacentric’ criterion.

We show how to parameterise solutions of the general relativistic vector constraint equation on Einstein manifolds by unconstrained potentials. We provide a similar construction for the trace-free part of tensors satisfying the linearised scalar constraint. Previous work of ours has provided similar different constructions for solutions of the linearized constraints in the case where the cosmological constant Λ is zero. We use our new potentials to show that one can shield linearised gravitational fields using linearised gravitational fields without imposing the TT gauge (as done in previous work), for any value of Λ ∈ R .

We construct compact initial data of constant mean curvature for Einstein’s 4d vacuum equations with positive, where is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer (2004 Class. Quantum Grav. 21 73). We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are —symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential ‘symmetry breaking’ by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the ‘round donut’ of topology . In the absence of symmetry breaking, the Lichnerowicz equation for a symmetric momentum on reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

We construct compact initial data of constant mean curvature for Einstein’s 4d vacuum equations with positive, where is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer (2004 Class. Quantum Grav. 21 73). We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are —symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential ‘symmetry breaking’ by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the ‘round donut’ of topology . In the absence of symmetry breaking, the Lichnerowicz equation for a symmetric momentum on reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

We construct compact initial data of constant mean curvature (K) over tilde for Einstein's 4d vacuum equations with (Lambda) over cap = Lambda - ((K) over tilde2/3) positive, where Lambda is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer (2004 Class. Quantum Grav. 21 73). We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are U(1) x U(1)-symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential 'symmetry breaking' by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the 'round donut' of topology S-2 x S. In the absence of symmetry breaking, the Lichnerowicz equation for a U(1) x U(1) symmetric momentum on S-3 reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

We analyse the effect of post-Newtonian gravitational fields on propagation of light in a cylindrical waveguide in both a straight configuration and a spool configuration. We derive an equation for the dependence of the wave vector upon the vertical location of the waveguide. It is shown that the gravitational field produces a small shift in the wave vector, which we determine, while the spooling creates additional modes which could perhaps be measurable in future accurate experiments.

We present an elementary argument that one can shield linearized gravitational fields using linearized gravitational fields. This is done by using third-order potentials for the metric, which avoids the need to solve singular equations in shielding or gluing constructions for the linearized metric.

We present an elementary argument that one can shield linearised gravitational fields using linearised gravitational fields. This is done by using third-order potentials for the metric, which avoids the need to solve singular equations in shielding or gluing constructions for the linearised metric.

We propose a new model which describes relativistic hydrodynamics and generalizes the standard Euler system of isentropic perfect fluids. Remarkably, our system admits a convex extension which allows us to transform it to a symmetric hyperbolic form. This result sheds new light even on the relativistic Euler system.

We prove existence of solutions for an elastic body interacting with itself through its Newtonian gravitational field. Our construction works for configurations near one given by a self-gravitating ball of perfect fluid. We use an implicit function argument. In so doing we have to revisit some classical work in the astrophysical literature concerning linear stability of perfect fluid stars. The results presented here extend previous work by the authors, which was restricted to the astrophysically insignificant situation of configurations near one of vanishing stress. In particular, “mountains on neutron stars”, which are made possible by the presence of an elastic crust in neutron stars, can be treated using the techniques developed here.

We prove the existence of initial data sets which possess an asymptotically flat and an asymptotically cylindrical end. Such geometries are known as trumpets in the community of numerical relativists.

In this talk I describe recent joint works with R.Schoen and with G.Gibbons and R.Schoen which prove the non-existence of certain asymptotically flat, stationary solutions of the Einstein equations with more than one body. The basic restriction is for example satisfied when spacetime has an isometry reversing the sign of the timelike Killing vector and fixing a hypersurface in the space of Killing trajectories which is disjoint from the bodies. I also give a detailed treatment of the Newtonian situation.

We prove that, given a stress‐free, axially symmetric elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary, axisymmetric solution to the Einstein equations coupled to the equations of relativistic elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency. © 2009 Wiley Periodicals, Inc.

The static n-body problem of general relativity states that there are, under a reasonable energy condition, no static n-body configurations for n > 1, provided the configuration of the bodies satisfies a suitable separation condition. In this paper we solve this problem in the case that there exists a closed, noncompact, totally geodesic surface disjoint from the bodies. This covers the situation where the configuration has a reflection symmetry across a noncompact surface disjoint from the bodies.

Generalizing previous work by two of the authors, we prove the non-existence of certain stationary configurations in general relativity having a spatial reflection symmetry across a non-compact surface disjoint from the matter region. Our results cover cases such as that of two symmetrically arranged rotating bodies with anti-aligned spins in n + 1 (n ⩾ 3) dimensions, or two symmetrically arranged static bodies with opposite charges in (3 + 1) dimensions. They also cover certain symmetric configurations in (3 + 1)-dimensional gravity coupled to a collection of scalars and Abelian vector fields, such as those that arise in supergravity and Kaluza–Klein models. We also treat the bosonic sector of simple supergravity in 4 + 1 dimensions.

We construct solutions, for small values of G and angular frequency Ω, of special relativistic scalar gravity coupled to ideally elastic matter which have helical but no stationary or axial symmetry. They correspond to a body without any symmetries in steady rotation around one of its axes of inertia, or two bodies moving on a circle around their center of gravity. Our construction is rigorous, but modulo an unproved conjecture on the differentiability of a certain functional.

We study Killing vector elds in asymptotically at space{times. We prove the following result, implicitly assumed in the uniqueness theory of stationary black holes. If the conditions of the rigidity part of the positive energy theorem are met, then in such space{times there are no asymptotically null Killing vector elds except if the initial data set can be embedded in Minkowski space{time. We also give a proof of the non{ existence of non{singular (in an appropriate sense) asymptotically at space{times which satisfy an energy condition and which have a null ADM four{momentum, under conditions weaker than previously considered.