We are interested in the perception of the electromagnetic phenomenon by observers that are more general than standard inertial observers in flat MINKOWSKI space-time. We model a relativistic observer as (principal) fibre bundle, where the fibres correspond to world lines of test particles and the base space defines the observer’s relative “space”. An EHRESMANN connection on the bundle splits local tangent spaces into “space”- and time subspaces. The time subspaces point in direction of the world lines, and the “space” subspaces are their orthogonal complements, according to the hypothesis of locality. A section map on the bundle defines a time synchronization. The curvature of the connection is a measure of the non-integrability of the local “spatial” subspaces. If the connection is flat, it allows for a section that is orthogonal to the world lines in all points, hence a global EINSTEIN time- synchronization. Time synchronization can be seen as gauge on the bundle, and the formalism of gauge theories provides the right tools. Gauge potential and curvature have a kinematic interpretation as velocity and vorticity fields. This approach encompasses all meaningful spatial-temporal decompositions of space-time. With this setting we split electromagnetic field entities accordingly and describe the effect of relative motion on measurable quantities. We will present the one plus three-dimensional form of MAXWELL’s equations and the constitutive relation and analyze SCHIFF’s treatment of his 1939 paradox. In our view, the coordinate- and frame-free approach brings to light the individual layers of mathematical structure and their physical significance, and generally favours structural considerations over technical ones. We emphasize, however, that our findings can be readily translated into classical RICCI calculus, whereby all textbook results can be recovered.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

Dynamics of the gravitational field will be presented in the Hamiltonian context. I will prove that the generator of this dynamics must be a quasi-local quantity and can be identified with gravitational energy. However, an ambiguity remains: different boundary conditions (which have to be imposed in order to define uniquely the field evolution) lead to different Hamiltonian frameworks and, consequently, to different definitions of the energy. I will relate various definitions existing in the literature (Hawking, Geroch, Bartnik, Brown-York, Liu-Yau and myself) with different way of controlling boundary data and will argue that only one of them represents the "true" field energy.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

After giving the most general formulation to date of the notion of integrability for axially symmetric harmonic maps from R 3 into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular we show that the problem of finding N- solitonic harmonic maps into a noncompact Grassmann manifold SU(p, q)/S(U(p) × U(q)) is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method by explicitly computing a 1-solitonic harmonic map for the two cases (p = 1, q = 1) and (p = 2, q = 1); and we show that the family of solutions obtained in each case contains, respectively, the Kerr family of solutions to the Einstein vacuum equations, and the Kerr-Newman family of solutions to the Einstein-Maxwell equations. Joint work with Shabnam Beheshti.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

The gravitational static 2-problem consists of showing that, under a suitable separation condition, two gravitating bodies can not be in static equilibrium. In Newtonian gravity such a separation condition is afforded by the assumption that the bodies in question are separated by a plane. We go on by considering a nonlinear generalization of Newtonian gravity due to Giulini and, finally, General Relativity. In both cases we prove that static 2-body configurations are ruled out under a condition which is in particular satisfied when there is a discrete symmetry interchanging the two bodies. This is work due to Rick Schoen and R.Beig.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

I show that for any finite number of point charges of arbitrary sign, magnitude, and at arbitrary positions in real three-dimensional space, there exists a unique electrostatic solution to the Maxwell-Born-Infeld field equations with finite energy. This solution is real analytic away from the point charges, and its electrostatic potential can be extended Lipschitz-continuously into the locations of these point charges. The electrostatic potential can be re-interpreted as the time function of an almost everywhere spacelike maximal hypersurface with lightcone defects. No struts between the point charges / lightcone defects occur. The results are discussed in the context of the N body problem in general relativity.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

In standard electrodynamics the field energy in a sphere around a point charge is infinite. This gives rise to an infinite self-force and to all the well-known pathologies associated with the (Abraham-)Lorentz-Dirac equation, such as pre-acceleration and run-away solutions. In the 1930s, Born and Infeld introduced a non-linear theory of electrodynamics; they were able to show that in their theory the field energy in a sphere around a static(!) point charge is finite. However, little is known about the Born-Infeld field of an accelerated(!) point charge, in particular about its regularity. In this talk I present a method of how to analyze this problem with the help of series expansions of the electromagnetic field in terms of light cone coordinates. The final goal is to rigorously prove that the self-force of an accelerated point charge in Born-Infeld theory is finite.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

This is a review of some results on the Born-Infeld model: i) existence of a convex “entropy” (in the sense of the mathematical theory of hyper- bolic conservation laws) for a suitably extended version of the system; ii) Galilean invariance of the extended system; iii) conjectures on the weak completion of the model and the induced interaction between field and matter; iv) linearization of the model by gauging the ambient metric.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

In the first part of the talk a Schwarzschild black hole is considered. We assume that light sources are distributed on a (big) sphere of radius R that emit, at an instant of time, photons isotropically. We calculate the resulting photon distribution and find that in the long-time limit the density becomes infinitely large near the photon sphere at r = 3 m . This suggests that every Schwarzschild black hole in Nature should be surrounded by a shell of very high photon density which could be detrimental to the health of any observer who comes close to this region. In the second part we discuss how the situation changes if a Kerr black hole is considered. The first part is based on the Bachelor Thesis of Dennis Philipp and the second part is ongoing work with Arne Grenzebach.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

Electromagnetic waves in vacuum are described by the dispersion relation which is a rem- iniscence of the Lorentzian geometry. I will discuss a system of integral equations for two differential 2-forms.This is a most general expression of the Maxwell electromagnetism in media and in present of gravity field. A covariant jump condition on an arbitrary smooth hypersurface represents a characteristic equation of the system. I derive a dispersion rela- tion for the system with a linear local constitutive relation from the jump condition. This dispersion relation is an expression of the Finsler type geometry which is a basis of the birefrigence phenomena. The axion modification of the electrodynamics in the framework of the Carroll-Field-Jackiw (CFJ) model will be discussed. I show that in order to rep- resent the axion contribution to the wave propagation it is necessary to go beyond the geometric approximation. The axion field modifies the global topological structure of light cones surfaces. In CFJ-electrodynamics, such a modification results in violation of causal- ity. In addition, the optical metrics in axion electrodynamics are not pseudo-Riemannian. In fact, for all types of the axion field, they are even non-Finslerian.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the temporal frequency ω, the spatial wave (co)vector k, and the polarisation (co)vector a. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating ω and k. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is represented by two light-cones. Kummer, whilst analysing quadratic line complexes as models for light rays in an optical apparatus, discovered in the framework of projective geometry a quartic surface that is linked to the Fresnel one. Given an arbitrary dispersionless linear (meta)material or vacuum, we aim to establish whether the resulting Fresnel surface is equivalent to, or is more general than, a Kummer surface.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

After a brief introduction to general relativity and spinors in curved space-time, a spa- tially homogeneous and isotropic cosmological model is introduced where Dirac spinors are coupled to classical gravity. If the scale function is large, the universe behaves just like the classical Friedmann dust solution. If however the scale function is small, quantum effects lead to oscillations of the energy-momentum tensor. After explaining the basic mechanisms, it is shown numerically and analytically that these quantum oscillations can prevent the formation of a big bang or big crunch singularity. We sketch the existence proof for time-periodic solutions, where the universe goes through an infinite number of expansion and contraction cycles. This is joint work with Christian Hainzl.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

I will present a class of diagrams, which we call “projection diagrams”, which provide a useful tool to visualise the global structure of spacetimes. Various examples will be discussed in detail, including cosmological models, and the Kerr spacetimes with or without a cosmological constant. Based on joint work with Christa Oelz and Sebastian Szybka.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Electromagnetic Spacetimes" (2012)

Introducing a Dirac Potential instead of the standard Poisson potential in the Vlasov equation change it into a very singular problem at a cross road of very subjects. In particular Penrose's method differenciates between linearly well posed and ill posed problems. It leads to a reuslt of ill posedness for the full non linear problem. On the other hand when restricted to kinetic hydrodynamic it gives rises to a well posed problem which locally in times is related to the semi classical limit of the Non Linear Schrodinger Equation. Moreover since the Non linear Schrodinger equation is integrable by inverse scattering some type of integrability with an infinite set of conserved quantities shows up in the present equation. In fact this equation can also be used as a model for water waves (in some convenient scaling). Then it carries the name of Benney equation and it is in this case that properties related to integrability were observed by Benney, Miura Novikov and others.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

In the first part of the talk we focus on the dynamics of a many-particle self-gravitating system, described with either classical or relativistic kinetic theory. As a way to gain insight into the relativistic models, we will review what is known about dispersive behavior in the classical setting. Then we shall present some preliminary results concerning dispersive behavior for the relativistic models, together with necessary conditions for the existence of steady states as well. The second part of the talk is devoted to discuss several macroscopic limits of kinetic models, paying special attention to the relativistic BGK equation.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

A solution of the relativistic Vlasov-Maxwell system is said to be isolated from incoming radiation if it is not hit by electromagnetic energy coming from the pass null infinity of Minkowski space. In this talk I will review the results available concerning the existence and the properties of isolated solutions to the relativistic Vlasov-Maxwell system.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

The Nordström-Vlasov system describes the evolution of a population of self-gravitating collisionless particles. We study the existence and uniqueness of mild solution for the Cauchy problem in one dimension. This approach does not require any derivative for the initial particle density. For any initial particle density uniformly bounded with respect to the space variable by some function with finite kinetic energy and any initial smooth data for the field equation we construct a global solution, preserving the total energy. Moreover the solution propagates with finite speed, which coincides with the light speed.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

In this talk, I will discuss two recent results on the Cauchy problem for the relativistic Vlasov-Darwin (RVD) system: the existence and uniqueness of global in time classical solutions with a small initial datum, and the uniqueness of weak solutions with compact support. These results rely on the formulation of the RVD system in terms of the generalized phase space variables, and the scalar and vector potentials. Existence of classical solutions is proved by constructive arguments. Improvements are made on the time decay estimates previously known for the electromagnetic field and its space derivatives. Uniqueness of weak solutions is proved with techniques derived from optimal transportation. This work is in collaboration with Martial Agueh and Reinhard Illner.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

I will present the theorem obtained with C. Villani proving the Landau damping in the nonlinear perturbative regime for the classical Vlasov-Poisson equation, and discuss the method and some further developements in progress.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

The Vlasov-Maxwell system is a fundamental kinetic model of plasma dynamics. When one considers relativistic velocities and includes effects due to collisions with a fixed background of particles, the result is the relativistic Vlasov-Maxwell-Fokker-Planck system. The first Lorentz-invariant model of this type was recently derived by Calogero and Felix in 2010. Hence, we discuss the first well-posedness results for global-in-time classical solutions of this system posed in a lower-dimensional setting. Our methods utilize a gain in regularity stemming from the Fokker-Planck term to arrive at smooth solutions even from weak initial data.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

This talk will deal with non-linear stability of spherical steady states to Vlasov systems. In the case of the classical gravitational Vlasov-Poisson system, we have proved recently that all spherical stationary states which are deceasing functions of their microscopic energy are non linearly stable under general perturbations. The proof is based on two main steps: A quantitative control of the potential field via a Poincaré-Antonov like inequality, and a compactness argument leading to the stability of the whole distribution function. The first aim of this talk is to present a new functional inequality which makes fully quantitative the compactness part of this proof as well. Applications of this inequality to other contexts will also be discussed. The second aim of the talk will be to explore the extension of this strategy to relativistic contexts: the relativistic Vlasov-Poisson (RVP) and the Vlasov-Manev (VM) systems. We show that the strategy extends to the RVP although the potential field is less regular than in the classical case, while the theory for VM faces a serious difficulty : A Poincaré-Antonov inequality with a fractional Laplacian is not available. Nevertheless, we show how a standard variational approach in the case of the VM system may be used to prove the non-linear stability of the minimizers of the Hamiltonian, although the uniqueness of these minimizers is not guaranteed (here, the Euler-Lagrange equation is a fractional Laplacian and not a Poisson equation). Finally, in the case of the so-called "Pure Manev system", we exhibit a continuous family of self-similar blow-up solutions whose profiles are close to the steady states.

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)

• Thematic program: Relativity, Electromagnetism, Gravitation and Singularities (2012)
• Event: Workshop on "Relativistic Vlasov Theory" (2012)