Past Relativity Seminar

This talk will be based on arxiv:1106.5254.

This talk will be based on arxiv:1106.5254.

There have been significant progress in the calculation of scattering amplitudes in N=4 SYM. In this talk I will consider `form factors', which are defined not only with on-shell asymptotic states but also with one off-shell operator inserted. Such quantities are in some sense the hybrid of on-shell quantities (such

as scattering amplitudes) and off-shell quantities (such as correlation functions). We will see that form factors inherit many nice properties of scattering amplitudes, in particular we will consider their supersymmetrization and the dual picture.

In field theory simple forms of certain scattering amplitudes in four dimensional theories with massless particles are known. This has been shown to be closely related to underlying (super)symmetries and has been a source of inspiration for much development in the last years. Away from four dimensions much less is known with some concrete development only in six dimensions. I will show how to construct promising on-shell superspaces in eight and ten dimensions which permit suggestively simple forms of supersymmetric four point scattering amplitudes with massless particles. Supersymmetric on-shell recursion relations which allow one to compute in principle any amplitude are constructed, as well as the three point `seed' amplitudes to make these work. In the three point case I will also present some classes of supersymmetric amplitudes with a massive particle for the type IIB superstring in a flat background.

Consider the space M of parabolas y=ax^2+bx+c, with (a, b, c) as coordinates on M. Two parabolas generically intersect at two (possibly complex) points, and we can define a conformal structure on M by declaring two points to be null separated iff the corresponding parabolas are tangent. A simple calculation of discriminant shows that this conformal structure is flat.

In this talk (based on joint works with Godlinski and Sokolov) I shall show how replacing parabolas by rational plane curves of higher degree allows constructing curved conformal structures in any odd dimension. In dimension seven one can use this "twistor" construction to find G_2 structures in a conformal class.

A celebrated result in mathematical general relativity is the uniqueness of the Kerr(-Newman) black-holes as regular solutions to the stationary and axially-symmetric Einstein(-Maxwell) equations. The axial symmetry can be removed if one invokes Hawking's rigidity theorem. Hawking's theorem requires, however, real analyticity of the solution. A recent program of A. Ionescu and S. Klainerman seeks to remove the analyticity requirement in the vacuum case. They were able to show that any smooth extension of "Kerr data" prescribed on the horizon, satisfying the Einstein vacuum equations, must be Kerr, using a characterization of Kerr metric due to M. Mars. In this talk I will give a characterization for the Kerr-Newman metric, and extend the rigidity result to cover the electrovacuum case.

I will outline two areas currently under study by myself and my co-workers, particularly Jonathan Holland: one concerns the relation between the exceptional Lie group G_2 and Einstein's gravity; the second will introduce and apply the concept of a causal geometry.

We consider Einstein-scalar field Lichnerowicz equations in the positive case in compact Riemannian manifolds. We discuss existence and stability issues for these equations

Canonical quantization of gravitational field will beconsidered. Examples for which the procedure can be completed (without reducingthe degrees of freedom) will be presented and discussed. The frameworks appliedwill be: Loop Quantum Gravity, relational construction of the Dirac observablesand deparametrization.

I will present existence and uniqueness results for theCauchy problem as in the title.

Extremal black holes are of interest as they are expected have simpler quantum descriptions than their non-extremal counterparts.  Any extremal black hole solution admits a well defined notion of a near horizon geometry which solves the same field equations. I will describe recent progress on the general understanding of such near horizon geometries in four and higher dimensions. This will include the proof of near-horizon symmetry enhancement and the explicit classification of near-horizon geometries (in a variety of settings). I will also discuss how one can use such results to prove classification/uniqueness theorems for asymptotically flat extremal vacuum black holes in four and five dimensions.

Algebraic classification of the Weyl tensor is an important tool for solving the Einstein equation. I shall review the classification for spacetimes of dimension greater than four, and recent progress in using it to construct new exact solutions. The higher-dimensional generalization of the Goldberg-Sachs theorem will be discussed.

Two classes of solutions of the Einstein equations with symmetry which

are frequently studied are the Bianchi and Gowdy models. The aim of this

talk is to explain certain relations between these two classes of

spacetimes which can provide insights into the dynamics of both. In

particular it is explained that the special case of the Gowdy models known as circular loop spacetimes are Bianchi models in disguise. Generalizations of Gowdy spacetimes which can be thought of as inhomogeneous perturbations of some of the Bianchi models are introduced.

Results concerning their dynamics are presented.