Stationary untrapped boundary conditions in general relativity

A class of boundary conditions for canonical general relativity are proposed and studied at the quasi-local level. It is shown that for untrapped or marginal surfaces, fixing the area element on the 2-surface (rather than the induced 2-metric) and the angular momentum surface density is enough to have a functionally differentiable Hamiltonian, thus providing definition of conserved quantities for the quasi-local regions. If on the boundary the evolution vector normal to the 2-surface is chosen to be proportional to the dual expansion vector, we obtain a generalization of the Hawking energy associated with a generalized Kodama vector. This vector plays the role for the stationary untrapped boundary conditions which the stationary Killing vector plays for stationary black holes. When the dual expansion vector is null, the boundary conditions reduce to the ones given by the non-expanding horizons and the null trapping horizons.Comment: 11 pages, improved discussion section, a reference added, accepted for publication in Classical and Quantum Gravit

On a class of 2-surface observables in general relativity

The boundary conditions for canonical vacuum general relativity is investigated at the quasi-local level. It is shown that fixing the area element on the 2- surface S (rather than the induced 2-metric) is enough to have a well defined constraint algebra, and a well defined Poisson algebra of basic Hamiltonians parameterized by shifts that are tangent to and divergence-free on $. The evolution equations preserve these boundary conditions and the value of the basic Hamiltonian gives 2+2 covariant, gauge-invariant 2-surface observables. The meaning of these observables is also discussed.Comment: 11 pages, a discussion of the observables in stationary spacetimes is included, new references are added, typos correcte

Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has been added and several inaccuracies have been corrected. To appear in Journal of Theoretical Probabilit

Hunting for binary Cepheids using lucky imaging technique

Detecting companions to Cepheids is difficult. In most cases the bright pulsator overshines the fainter secondary. Since Cepheids play a key role in determining the cosmic distance scale it is crucial to find binaries among the calibrating stars of the period-luminosity relation. Here we present an ongoing observing project of searching for faint and close companions of Galactic Cepheids using lucky imaging technique.Comment: 4 pages, 2 figures, published in AN. Proceedings for the 6th Workshop of Young Researchers in Astronomy and Astrophysic

Total angular momentum from Dirac eigenspinors

The eigenvalue problem for Dirac operators, constructed from two connections on the spinor bundle over closed spacelike 2-surfaces, is investigated. A class of divergence free vector fields, built from the eigenspinors, are found, which, for the lowest eigenvalue, reproduce the rotation Killing vectors of metric spheres, and provide rotation BMS vector fields at future null infinity. This makes it possible to introduce a well defined, gauge invariant spatial angular momentum at null infinity, which reduces to the standard expression in stationary spacetimes. The general formula for the angular momentum flux carried away be the gravitational radiation is also derived.Comment: 34 pages, typos corrected, four references added, appearing in Class. Quantum Gra

Extremality conditions for isolated and dynamical horizons

A maximally rotating Kerr black hole is said to be extremal. In this paper we introduce the corresponding restrictions for isolated and dynamical horizons. These reduce to the standard notions for Kerr but in general do not require the horizon to be either stationary or rotationally symmetric. We consider physical implications and applications of these results. In particular we introduce a parameter e which characterizes how close a horizon is to extremality and should be calculable in numerical simulations.Comment: 13 pages, 4 figures, added reference; v3 appendix added with proof of result from section IIID, some discussion and references added. Version to appear in PR

On certain quasi-local spin-angular momentum expressions for small spheres

The Ludvigsen-Vickers and two recently suggested quasi-local spin-angular momentum expressions, based on holomorphic and anti-holomorphic spinor fields, are calculated for small spheres of radius $r$ about a point $o$. It is shown that, apart from the sign in the case of anti-holomorphic spinors in non-vacuum, the leading terms of all these expressions coincide. In non-vacuum spacetimes this common leading term is of order $r^4$, and it is the product of the contraction of the energy-momentum tensor and an average of the approximate boost-rotation Killing vector that vanishes at $o$ and of the 3-volume of the ball of radius $r$. In vacuum spacetimes the leading term is of order $r^6$, and the factor of proportionality is the contraction of the Bel-Robinson tensor and an other average of the same approximate boost-rotation Killing vector.Comment: 16 pages, Plain Te

Quasi-local energy-momentum and two-surface characterization of the pp-wave spacetimes

In the present paper the determination of the {\it pp}-wave metric form the geometry of certain spacelike two-surfaces is considered. It has been shown that the vanishing of the Dougan--Mason quasi-local mass $m_{\$}$, associated with the smooth boundary $\$:=\partial\Sigma\approx S^2$ of a spacelike hypersurface $\Sigma$, is equivalent to the statement that the Cauchy development $D(\Sigma)$ is of a {\it pp}-wave type geometry with pure radiation, provided the ingoing null normals are not diverging on $\$$ and the dominant energy condition holds on $D(\Sigma)$. The metric on $D(\Sigma)$ itself, however, has not been determined. Here, assuming that the matter is a zero-rest-mass-field, it is shown that both the matter field and the {\it pp}-wave metric of $D(\Sigma)$ are completely determined by the value of the zero-rest-mass-field on $\$$ and the two dimensional Sen--geometry of $\$$ provided a convexity condition, slightly stronger than above, holds. Thus the {\it pp}-waves can be characterized not only by the usual Cauchy data on a {\it three} dimensional $\Sigma$ but by data on its {\it two} dimensional boundary $\$$ too. In addition, it is shown that the Ludvigsen--Vickers quasi-local angular momentum of axially symmetric {\it pp}-wave geometries has the familiar properties known for pure (matter) radiation.Comment: 15 pages, Plain Tex, no figure

On the roots of the Poincare structure of asymptotically flat spacetimes

The analysis of vacuum general relativity by R. Beig and N. O Murchadha (Ann. Phys. vol 174, 463 (1987)) is extended in numerous ways. The weakest possible power-type fall-off conditions for the energy-momentum tensor, the metric, the extrinsic curvature, the lapse and the shift are determined, which, together with the parity conditions, are preserved by the energy-momentum conservation and the evolution equations. The algebra of the asymptotic Killing vectors, defined with respect to a foliation of the spacetime, is shown to be the Lorentz Lie algebra for slow fall-off of the metric, but it is the Poincare algebra for 1/r or faster fall-off. It is shown that the applicability of the symplectic formalism already requires the 1/r (or faster) fall-off of the metric. The connection between the Poisson algebra of the Beig-O Murchadha Hamiltonians and the asymptotic Killing vectors is clarified. The value H[K^a] of their Hamiltonian is shown to be conserved in time if K^a is an asymptotic Killing vector defined with respect to the constant time slices. The angular momentum and centre-of-mass, defined by the value of H[K^a] for asymptotic rotation-boost Killing vectors K^a, are shown to be finite only for 1/r or faster fall-off of the metric. Our center-of-mass expression is the difference of that of Beig and O Murchadha and the spatial momentum times the coordinate time. The spatial angular momentum and this centre-of-mass form a Lorentz tensor, which transforms in the correct way under Poincare transformations.Comment: 34 pages, plain TEX, misleading notations changed, discussion improved and corrected, appearing in Class. Quantum Gra

On quasi-local charges and Newman--Penrose type quantities in Yang--Mills theories

We generalize the notion of quasi-local charges, introduced by P. Tod for Yang--Mills fields with unitary groups, to non-Abelian gauge theories with arbitrary gauge group, and calculate its small sphere and large sphere limits both at spatial and null infinity. We show that for semisimple gauge groups no reasonable definition yield conserved total charges and Newman--Penrose (NP) type quantities at null infinity in generic, radiative configurations. The conditions of their conservation, both in terms of the field configurations and the structure of the gauge group, are clarified. We also calculate the NP quantities for stationary, asymptotic solutions of the field equations with vanishing magnetic charges, and illustrate these by explicit solutions with various gauge groups.Comment: 22 pages, typos corrected, appearing in Classical and Quantum Gravit