Fuchsian PDE and their Role in Characterizing Singular Solutions to the Vacuum Einstein Equations

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Written by Ellery Ames

What are the dynamics of general singular solutions to the Einstein equations near singularities? Characterizing such behavior is one of the main outstanding problems in classical general relativity. In this talk I will present so called Fuchsian methods for obtaining solutions to singular symmetric hyperbolic PDE. These methods allow us to prove the existence of large families of singular solutions to the Einstein equations with asymptotic velocity dominated singular dynamics.

 

A binary neutron star coalescence forecasting algorithm for GRB observations

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Written by Gergely Debreczeni

Modern gravitational wave (GW) detectors are hunting for GWs originating from various sources. It is assumed that in some cases it is the coalescence (merging) of binary neutron stars (BNS) which is responsible for gamma-ray burst (GRB) - routinely observed by electromagnetic telescopes. Since also before the merging, during the inspiral phase the binary system emitts GWs , analysis groups of GW detectors are performing in-depth search for such events around the time-window of known, already detected GRBs. These joint analysis are very important in increasing the confidence of a possible GW detection. It is widely expected that the first direct detection of GWs will happen in the next few years, and what is a matter of fact that the sensitivity of the next generation of GW detectors will allow us to 'see' a few hundred seconds of inspiral of the binary system before the merge - for specific mass parameter range.
From the two above fact, it naturally follows, that one can (should!) turn around the logic and use the GWs emitted during the inspiral phase of a BNS to predict, in advance the time, sky location of a GRB and set up contraints on the physical parameters of the system. There exists no such prediction algorithm, as of today.
Despite the fact that it is not yet feasible to use this new method with the current GW detectors, it will be of utmost importance in the late-Advanced LIGO/Virgo era and definitely for Einstein Telescope.
The very goal of the research presented in this talk is to develop the above described zero-latency, binary NS coalescence 'forecasting' method and set up the associated alert system to be used by next generation of gravitational wave detectors and collaborating telescopes.

 

Null canonical formulation and integrability of cylindrical gravitational waves

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Written by Andreas Fuchs

We derive a Poisson bracket of the metric variables of free null initial data for cylindrically symmetric gravitational waves. The structure of this bracket is essentially identical to the bracket on the metric sector of the initial data for spacetimes without symmetries found by Michael Reisenberger [1]. Using the integrability of the dynamics of cylindrically symmetric gravitational waves an explicit transformation from metric data on a null hypersurface to so called monodromy data, a one parameter family of unimodular matrices, is obtained. The Poisson brackets of the monodromy data are then obtained from that of the null data. They are quite simple, and what is more, a unique preferred quantization is known [2]. It is also demonstrated that the transformation to monodromy data is invertible. The original results are joint work with Michael Reisenberger.

[1] Michael P. Reisenberger: The Poisson bracket on free null initial data for gravity Phys.Rev.Lett.101:211101, 2008, arXiv:gr-qc/0712.2541
[2] D. Korotkin, H. Samtleben Yangian Symmetry in Integrable Quantum Gravity Nucl.Phys. B527 (1998) 657-689, arXiv:hep-th/9710210v1

 

Chaotic heteroclinic structure for extreme gravity models

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Written by Juliette Hell

The Bianchi IX model is homogeneous but anisotropic. It is conjectured to show the relevant dynamics of GR on the local boundary, which is of BKL-chaotic type. We show how perturbing the first principles in the simplest interesting way put GR at a bifurcation where the dynamics changes dramatically. We consider the Hubble normalized system admitting the Kasner circle of equilibria, carrying a network of heteroclinics whose dynamics is described by the Kasner map. If the perturbation parameter is subcritical, Chaos of the Kasner map remains only on a nongeneric Cantor set of the Kasner circle. If the perturbation parameter is supercritical, the Kasner map is multivalued and loses continuity. The chaos is generic in the sense of iterated function systems, but the concept of BKL-era becomes meaningless. These ideas using symbolic dynamics can be generalized for FLRW models with scalar fields. Those models can be Hubble normalized and admit a circle of non proper equilibria. The Chaos does not takes place in the geometry but on the scalar field variables.

 

Surgery and the Positive Mass Conjecture

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Written by Andreas Hermann

The Positive Mass Conjecture for asymptotically Euclidean manifolds has been proved in some special cases (e.g. for manifolds of dimension at most 7 or for spin manifolds) but the general case is still subject to current research. In this talk we present a surgery result obtained with Emmanuel Humbert which might help to give a proof in the general case.

 

Limits of spacetimes

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Written by Emma Jakobsson

When studying limits of spacetimes one is easily confused. The limit obtained when letting some free parameter approach a certain value is in general not unique, but depends on the choice of coordinates. This ambiguity led Geroch to formulate a definition for limits of a one-parameter family of spacetimes in 1969. The paper Limits of spacetimes, however, is vague when it comes to how to actually apply this definition. We have come up with an application of Geroch's definition, which makes it possible to see the limiting procedure in pictures. The general idea is to let the spacetime under consideration be represented by a 1+1-dimensional surface reflecting its essential causal structure, and embed this surface in 2+1-dimensional anti-de Sitter space. With the help of a conformally compactified picture of adS3 the result is reminiscent of a Penrose diagram, with the difference that the picture will change as we vary the parameter. In particular we have studied the limit e → m of the Reissner-Nordström solution.

 

The vector field method for Vlasov fields

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Written by Jeremie Joudioux

As for the wave equation, the Vlasov equation admits commutators arising from the geometry. This allows standard PDE techniques, such as the vector field method, to be applied to Vlasov fields. In this talk, the relevant geometric structures of the Vlasov equation will be explained, and exploited to apply the vector field method. The asymptotic behavior of Vlasov fields, with data in some weighted Sobolev spaces, on flat spacetime, can then be described using Klainerman-Sobolev inequalities.
This is a collaboration with D. Fajman (Vienna), and J. Smulevici (Orsay-Paris 11).

 

On the Einstein flow with positive cosmological constant

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Written by Klaus Kröncke

Let $(M,gamma)$ be a compact Riemannian Einstein metric with either positive or negative scalar curvature. Then, the metric $-dt^2+cosh^2(t)gamma$ (resp. $-dt^2+sinh^2(t)gamma$) is a solution of the Einstein equations with positive cosmological constant.
We show nonlinear stability of these solutions in the following sense: Let $(g,k)$ be an initial data set which is close enough to the initial data of the background solution and assume that $gamma$ has no Killing fields and does not admit the Laplacian eigenvalue $-2/n R(gamma)$.
Then the corresponding solution of the Einstein equation is geodesically complete in the expanding direction. Moreover, the solution can be globally foliated by CMC hypersurfaces which stay modulo scaling in a small neighbourhood of $gamma$.
This is joint work with David Fajman.

 

Conformal invariance without referring to Weyl rescalings

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Written by Andras Laszlo

Field theories invariant to conformal transformations are a very important class of models. Besides their theoretical significance due to their large symmetry group, they are important also from the practical point of view. For instance the kinematic part of the Standard Model Lagrangian also shows conformal invariance --the conformal symmetry is broken by the vacuum expectation value of the Higgs field in the Higgs self-interaction term. In the usual approach, a field theory is called conformal invariant whenever its field equations or its action is invariant to the conformal rescalings of the spacetime metric tensor along with corresponding transformation of the field quantities. For this, a group action of the conformal rescalings on the fields needs to be specified a priori, and conformal invariance only makes sense along with that. In this talk we introduce a simple new method of generating field theories in terms of their Lagrangian, without a priori specifying the group action of the conformal rescaling on the fields. The interesting aspect of this method is that it does not to refer to a spacetime metric tensor a priori, and therefore becomes particularly useful when searching for theories where the spacetime metric tensor is an emergent quantity, not a fundamental field.

 

Post-Newtonian stationary disks: dynamic anti-dragging and general relativistic rotation laws

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Written by Patryk Mach

I will discuss recent developments in the theory of general-relativistic stationary gaseous disks. By means of post-Newtonian expansions we recovered the well-known geometric dragging, but we also obtained new (up to our knowledge) dynamic anti-dragging effects. I will discuss the physical requirements that allow one to specify, otherwise non-unique, post-Newtonian expansion terms. This gives rise to the discussion of general-relativistic rotation laws that in the Newtonian limit reduce to standard power-law relations. The presented results were obtained jointly with Piotr Jaranowski, Edward Malec, and Michał Piróg.
The talk is based on two preprints: arXiv:1501.04539 and arXiv:1410.8527 (to appear in Phys. Rev. D).

 

About stability islands of Anti-de Sitter space-studies of pure gravitational perturbation

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Written by Maciej Maliborski

To extend their studies of AdS stability problem to pure gravitational case Bizon and Rostworowski have considered cohomogenity-two biaxial Bianchi IX ansatz which also exhibits weakly turbulent behavior. In my talk I will briefly describe this simple 1+1 dimensional PDE system and present results of our studies. I will focus on time-periodic solutions of this system - a new class of globally regular solutions to the vacuum Einstein equations with negative cosmological constant - their structure and properties.

 

The characteristic initial value problem of colliding plane gravitational waves

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Written by Stefan Palenta

We consider the general initial value problem of colliding plane waves as search for a vacuum spacetime with 2 spacelike Killing vectors. This leads to the hyperbolic Ernst equation which can be treated by inverse scattering, showing many analogies to stationary axisymmetric spacetimes. With help of a corresponding linear Problem and a Riemann-Hilbert problem the solution to the Ernst equation can be expressed as an integral over jump functions. By expanding these jumps in Chebychev polynomials we get interesting approximative solutions.

 

Axisymmetric fully spectral code for hyperbolic equations

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Written by Rodrigo Panosso Macedo

We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for the time direction) collocation points. The method solves two issues of previous algorithms which were restricted to one spatial dimension, namely, (i) the inversion of a dense matrix and (ii) the acquisition of a sufficiently good initial-guess for non-linear systems of equations. For the first issue, we use the iterative bi-conjugate gradient stabilized method, which we equip with a pre-conditioner based on a singly diagonally implicit Runge-Kutta ("SDIRK"-) method. The SDIRK-method also supplies the code with a good initial-guess. The numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution. As an application, we solve general-relativistic wave equations on a black-hole space-time in so-called hyperboloidal slices and reproduce some recent results available in the literature.

 

Keplerian rotation curves in self-gravitating disks in the first post newtonian approximation

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Written by Michał Piróg

Purely newtonian results have shown that the self-gravitation of the disk in Keplerian accreting disk systems with polytropic gas speeds up its rotation. The rotational frequency is larger than that given by the well known strictly Keplerian formula which takes into account the central mass only. Thanks to this analysis we obtained important informations about relation between central mass and mass of the disk in such systems. Furthermore, we have shown that this method can be used to estimate mass of the disk in some class of astronomical objects. In our recent work we have shown how the analogous effect looks like in the first post-newtonian approximation. I would like to present the numerical results concerning this kind of systems in 1PN regime.

 

Robustness of the conformal constraint equations in a scalar-field setting

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Written by Bruno Premoselli

As is well known, the constraint equations arise in the initial-value formulation of the Einstein equations. The conformal method allows one to rewrite the constraint equations into a determined system of nonlinear, supercritical, elliptic PDEs. We will investigate in this talk stability properties for this elliptic system. The notion of stability considered here, defined as the continuous dependence of the set of solutions of the conformal constraint system in its coefficients, reformulates in terms of the relevance of the conformal method.
The analysis of the aforementioned stability properties involves blow-up techniques for the analysis of defects of compactness of supercritical nonlinear elliptic equations.
Joint work with Olivier Druet.

 

Invariants at infinity of asymptotically hyperbolic metrics

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Written by Gicquaud Romain

The mass of an asymptotically Euclidean metric has played a very important role both in general relativity and in Riemannian geometry. A similar invariant can be defined for asymptotically hyperbolic metrics. This is no longuer a scalar but a vector in Minkowski space that transforms in a covariant way under the isometries of the hyperbolic space. In this talk I will classify the invariants having such a covariant property. The answer surprizingly involves polynomial solutions to the wave equation and to the linearized Einstein equation on Minkowski space.

 

Global hyperbolicity for spacetimes with continuous metrics

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Written by Clemens Saemann

Global hyperbolicity is the strongest commonly used causality condition in general relativity. It ensures well-posedness of the Cauchy problem for the wave equation, globally hyperbolic spacetimes are the class of spacetimes used in the initial value formulation of Einstein's equations and it plays an important role in the singularity theorems. These examples emphasize the importance of this notion in Lorentzian geometry and, in particular, in the theory of general relativity.
Classically (i.e., with smooth metric), there are four equivalent notions of global hyperbolicity. These are: compactness of the causal diamonds and (strong) causality, compactness of the space of causal curves connecting two points and causality, existence of a Cauchy hypersurface and the metric splitting of the spacetime.
We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining the first three equivalences above. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.

 

Numerical evolution of the axisymmetric vacuum Einstein equations in spherical coordinates: the linear case

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Written by Christian Schell

I discuss a new approach to solving the axisymmetric vacuum Einstein equations numerically. Spherical polar coordinates are best suited for situations such as gravitational collapse. Also they allow for a spectral approach using spherical harmonics. In this talk I consider the linearization of the equations about flat spacetime. I show why the considered situation requires a new gauge condition and derive an exact solution for our choice. After regularizing at the origin we present numerical evolutions and discuss their properties.

 

Axisymmetric constant mean curvature slices in the Kerr space-time

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Written by David Schinkel

Recently, there have been efforts to solve Einstein's equation in the context of a conformal compactification of space-time. Of particular importance in this regard are the so called CMC-foliations, characterized by spatial hyperboloidal hypersurfaces with a constant extrinsic mean curvature K. However, although of interest for general space-times, CMC-slices are known explicitly only for the spherically symmetric Schwarzschild metric. This work is devoted to numerically determining axisymmetric CMC-slices within the Kerr solution. We construct such slices outside the black hole horizon through an appropriate coordinate transformation in which an unknown auxiliary function A is involved. The condition K=const. throughout the slice leads to a nonlinear partial differential equation for the function A, which is solved with a pseudo-spectral method. The results exhibit exponential convergence, as is to be expected in a pseudo-spectral scheme for analytic solutions. As a by-product, we identify CMC-slices of the Schwarzschild solution which are not spherically symmetric.

 

Symmetry inheritance of scalar fields

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Written by Ivica Smolić

Physical fields don't necessarily have to have the same symmetries as the spacetime they live in. When this happens, we speak of the symmetry inheritance of fields. A recent discovery of (complex) scalar hair on rotating black hole by Herdeiro and Radu [2] has shown that symmetry noninheritance can be used to circumvent the Bekenstein's no-hair theorems [1]. In order to clarify formal background of this discovery we shall attempt to classify all possible obstructions of symmetry inheritance by scalar fields, both real and complex, and look more closely at special cases of stationary and axially symmetric spacetimes. Furthermore, we shall analyze the effects of such fields on black hole physics.

[1] J. Bekenstein: "Transcendence of the law of baryon-number conservation in black hole physics", Phys.Rev.Lett. 28 (1972) 452
[2] C.A.R. Herdeiro and E. Radu: "Kerr black holes with scalar hair", Phys.Rev.Lett. 112 (2014) 221101 [arXiv: 1403.2757]

 

Simple description of Maxwell field on Kerr black hole background

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Written by Tomasz Smolka

The Fackerell-Ipser equation describes electromagnetic wave propagation around Kerr black hole. It was derived in 1970s using Newman – Penrose formalism. I will present a geometric proof of the Fackerell-Ipser equation and discuss a generalization for special spacetimes equipped with conformal Yano-Killing tensors (CYK tensors).I will demonstrate non-standard properties of CYK tensors in Kerr spacetime which are useful in electrodynamics.

 

The differentiability of horizons

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Written by David Szeghy

Cauchy horizons and black hole horizons are well-known objects of general relativity. Recently, the differentiability properties of horizons have been intensively studied. It is known that a horizon is differentiable almost everywhere, but there are examples, where the horizon is non-differentiable on a dense set, R. J. Budzynski, W. Kondracki and A. Królak constructed an example of a compact Cauchy horizon that is not a differentiable manifold in [Bu-K-Kr]. Thus, the set of non-differentiable points can not be neglected. P. T. Chrusciel, J.H.G. Fu, G.J. Galloway and R. Howard in [C-F-G-H] examined the endpoint set of the generators. J. K. Beem and A. Królak in [B-Kr] proved that at the inner points of a generator the horizon is differentiable. Moreover, at the end point p of a generator, the horizon H is differentiable if and only if p is the end point of only 1 generator. An other proof of the above fact was given in a more general case by P. T. Chrusciel in [C].
We give a more detailed classification of the points of the generator. Let H be a horizon and γ:[(α,β)→H a past directed generator, where [(α,β) denotes the interval (α,β) or [α,β). We prove that along any generator γ the differentiability order can change only once, more precisely there are parameter values t0 in [α,β] and k≥1, for which H is exactly of class Ck at every point γ(t) if t is in (t0,β ), moreover, H is differentiable but not of class C1 at every γ(t) for which t is in (α,t0]. We also give a geometric characterization of γ(t0) as the first non-injectivity point along the geodesic γ of a suitable space-like submanifold R in H. Moreover, we show that the first cut point of R along γ is usually the endpoint of γ. Finally, we prove that if there is an inner point γ(t*) of a generator γ where the horizon H is smooth, then the horizon must be smooth at every point of γ.
References
[B-Kr] J. Beem and A. Królak, Cauchy horizon endpoints and differentiability, J. Math. Phys. 39, (1998) pp. 6001-6010
[Bu-K-Kr] R. J. Budzynski,W. Kondracki, A. Królak, On the differentiability of compact Cauchy horizons, Letters in Mathematical Physics, (2003), Volume 63, Issue 1, pp. 1-4
[C] P.T. Chrusciel, A remark on differentiability of Cauchy horizons, Classical Quantum Gravity 15, (1998) pp. 3845-3848
[C-F-G-H] P.T. Chrusciel, J.H.G. Fu, G.J. Galloway, R. Howard, On one differentiability properties of horizons and applications to Riemannian geometry, Journal of Geometry and Physics 41, Issues 1-2, (2002), pp. 1-12

 

Spherically Symmetric, Static Solutions of the Einstein-Vlasov System with Non-Vanishing Cosmological Constant

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Written by Maximilian Thaller

We consider the static Einstein-Vlasov system in spherical symmetry. Existence of different types of solutions to this system for zero cosmological constant has been shown for the isotropic and anisotropic case by Rein-Rendall, Rein, and Wolansky. In this talk I shortly review existing results on static solutions and describe a method to prove existence of static solutions to the Einstein-Vlasov system with positive cosmological constant. The energy density and the pressure of these solutions have compact support and outside a finite ball these solutions are identical to a Schwarzschild-deSitter spacetime. Moreover other classes of new non-vacuum solutions that we have construced will be presented like solutions containing a black hole surrounded by Vlasov matter for both negative and positive cosmological constants. Finally, the global structure of the constructed non-vacuum spacetimes will be discussed with the help of Penrose diagrams.
The results presented in the talk are joint work with H. Andréasson and D. Fajman.

 

Mass inequalities for axially symmetric, asymptotically flat initial data

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Written by Drazen Vrzan

About a decade ago it was proven that the ADM mass of axially symmetric, asymptotically flat initial data is greater or equal than the root of the angular momentum, and equality holds for extreme Kerr (only). We describe recent, stronger inequalities which also contain higher "momenta", focusing on the special case where the data are close to extreme Kerr in a suitable sense.