GR19 Mexico July 5–9 2010

A2 session: Mathematical Relativity and

Other Progress in Classical Gravity Theory

Chair: Sergio Dain

June 30, 2010

Schedule

Monday

• 14:00 14:30 Abhay Ashtekar

• 14:30 15:00 Robert Wald

• 15:00 15:15 Sumio Yamada

• 15:15 15:30 Mu-Tao Wang

• 15:30 15:45 Gyula Fodor

• 15:45 16:00 Michael Reisenberger

• Break

• 18:45 19:00 Keiju Murata

• 19:00 19:15 Christian Luebbe

• 19:15 19:25 Marıa Eugenia Gabach Clement

• 19:25 19:35 Ernesto Nungesser

• 19:35 19:45 Luis Filipe Costa

Thursday

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• 14:00 14:30 Helmut Friedrich

• 14:30 14:45 Hakan Andreasson

• 14:45 15:00 Gustavo Dotti

• 15:00 15:15 Carlos Kozameh

• 15:15 15:30 Jose M M Senovilla

• 15:30 15:45 Alejandro Perez

• 15:45 16:00 Luca Lusanna

• Break

• 16:30 17:00 Piotr Chrusciel

• 17:00 17:15 Beverly Berger

• 17:15 17:30 Woei Chet Lim

• 17:30 17:45 Oscar Reula

• 17:45 18:00 Istvan Racz

• 18:00 18:15 Osvaldo Moreschi

• 18:15 18:30 Ram Gopal Vishwakarma

Friday

• 14:00 14:30 Jim Isenberg

• 14:30 14:45 Omar Eduardo Ortiz

• 14:45 15:00 Juan Antonio Valiente Kroon

• 15:00 15:15 Florian Beyer

• 15:15 15:30 Andres Acena

• 15:30 15:45 Alberto Chamorro

• 15:45 16:00 Ernesto Fabian Eiroa

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Titles and abstracts

Monday

1. Abhay Ashtekar: Evaporation of 2-d Black HolesThe Callen-Giddings-Harvey-Strominger black holes provide a simpleyet conceptually interesting setting for mathematical general relativity.One can study the gravitational collapse to black hole formation analyti-cally. The mean field approximation to full quantum equations providesinteresting PDEs. Although they were studied analytically and numer-ically a decade ago, more careful investigation has recently shown thatsome of the underlying concepts (such as the notion of Bondi energythat was used) were flawed and the prevailing intuition on what happensduring black hole evaporation has to be seriously corrected. While theprimary motivation of this investigation came from the issue of possi-ble information loss, this talk will focus on the analytical and numericalaspects of the PDEs. The unforeseen conclusions they lead to suggestthat it would be fruitful to analyze these PDEs rigorously. Various partsof this research were carried out in collaboration with Frans Pretorius,Fethi Ramazanoglu, Victor Taveras and Madhavan Varadarajan.

2. Robert Wald: Gravitational Self-ForceIt is well known that a sufficiently small “test body” in general relativitywill move on a geodesic of the background spacetime metric. It is ofconsiderable interest to determine the deviations from geodesic motionresulting from self-force effects. I describe work with Gralla that rig-orously derives perturbative corrections to geodesic motion due to thefinite size and mass of the body. The status of the MiSaTaQuWa equa-tion as a “self-consistent perturbative equation” is explained.

3. Sumio Yamada: Riemannian Penrose inequality revisitedWe re-examine the proof by Hugh Bray of the Riemannian Penrose in-equality using the conformal method. In particular in our new approach,we take a dual variational viewpoint of the inequality, fixing the ADMmass while changing the geometry of the threefold to maximize the areaof the outermost horizon. This is a joint work with Gilbert Weinstein.

4. Mu-Tao Wang: On the notion of quasilocal mass in generalrelativity

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There have been many attempts to define quasilocal mass for a spacelike2-surface in a spacetime by the Hamilton-Jacobi method. The essentialdifficulty in this approach is the subtle choice of the background configu-ration to be subtracted from the physical Hamiltonian. Quasilocal massshould be positive for general surfaces, but on the other hand shouldbe zero for surfaces in the flat spacetime. In this talk, I shall discribehow to use isometric embeddings into the Minkowski space to overcomethis difficulty and propose a new definition of gauge-independent quasi-local mass that has the desired properties, in addition to other naturalrequirements for a mass. This talk is based on a joint work with Shing-Tung Yau at Harvard.

5. Gyula Fodor: Lifetime of gravitationally bound oscillatingscalar lumps (oscillatons)Apparently periodic spherically symmetric localized oscillating configu-rations (oscillatons) formed by a massive scalar field coupled to gravityhave been found by Seidel and Suen in 1991. By all practical numericalmethods oscillatons appear to be time-periodic. It was first pointed outby Don N. Page in 2004 that these states should necessarily lose energyby slowly emitting scalar radiation. I would like to present a method forthe calculation of the transcendentally small amplitude of the outgoingwaves in the limit when the central amplitude of the oscillaton is small.The method involves the study of the Fourier mode equations near apole in the complex plane and Borel summation. Substituting physi-cally plausible scalar field mass values into the results, it turns out thatthe time-scale on which the mass of the oscillaton changes is generallycomparable to the age of the universe.

6. Michael Reisenberger: Symplectic structure on free null ini-tial data for gravityFree initial data for general relativity on a pair of intersecting nullhypersurfaces are well known, but the lack of a Poisson bracket andconcerns about caustics have stymied the development of a constraintfree canonical theory. A way of neatly avoiding the problem of causticsand generator crossings will be explained and a Poisson bracket on freedata will be presented. On sufficiently regular functions of the solutionspacetime geometry this bracket matches the Peierls bracket defined onsuch functions by the Hilbert action. Some mysterious features of thePoisson bracket found will be discussed and explained.

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7. Keiju Murata: Non-equilibrium Condensation Process in aHolographic SuperconductorWe study the non-equilibrium condensation process in the holographicsuperconductor. When the temperature T is smaller than a critical tem-perature Tc, there are two black hole solutions, the Reissner-Nordstrom-AdS black hole and a black hole with a scalar hair. In the bound-ary theory, they can be regarded as the supercooled normal phase andthe superconducting phase, respectively. We consider perturbations onsupercooled Reissner-Nordstrom-AdS black holes and study their non-linear time evolution to know about physical phenomena associated withrapidly-cooled superconductors. We find that, for T < Tc, the initialperturbations grow exponentially and, eventually, spacetimes approachthe hairy black holes. We also clarify how the relaxation process from afar-from-equilibrium state proceeds in the boundary theory by observingthe time dependence of the superconducting order parameter. Finally,we study the time evolution of event and apparent horizons and dis-cuss their correspondence with the entropy of the boundary theory. Ourresult gives a first step toward the holographic understanding of thenon-equilibrium process in superconductors.

8. Christian Luebbe: Non-linear stability for radiative Einstein-Maxwell spacetimesWe present recent results on the stability of solutions to the Einstein-Maxwell equations. The analysis is carried out using the conformalfield equations. In particular we use a gauge based on conformal curvesthat leads to reduced field equations. Moreover this gauge choice givesus a priori knowledge of the location of the conformal boundary of theperturbed solution, which will be shown to be regular. The referencespacetimes employed in the stability analysis are derived using a resultby Simon. In detail, solutions of the electrostatic field equations areused to obtain initial data for the Einstein-Maxwell equations that leadto smooth solutions in a neighbourhood of timelike infinity. Our workuses hyperboloidal initial data for the perturbations and the results arehence semi-global in nature. A remarkable point about the overall anal-ysis is that very little explicit knowledge of the reference spacetimes isrequired. Instead the geometrical properties of the field equations andthe solutions are employed to obtain the result presented here.

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9. Marıa Eugenia Gabach Clement: Extreme black hole’s initialdata and its perturbationsWe discuss the existence of extreme black holes initial data for Einsteinequations. These initial data have similar properties to the extremeKerr and Reissner-Nordstrom data. In particular, we find that in thisextreme limit one of the asymptotic ends is cylindrical, and the totalmass is a minimum among the black hole family. We also treat pertur-bations of these extreme data and obtain the same asymptotic geometrywith the area of the cylindrical end preserved. The procedure employedcan be implemented in conformally flat initial data, and also in extremeKerr black hole.

10. Ernesto Nungesser: Isotropization of non-diagonal Bianchi I-symmetric spacetimes with collisionless matter at late timesassuming small dataAssuming that the space-time is close to isotropic in the sense thatthe shear parameter is small and that the maximal velocity of the par-ticles is bounded, we have been able to show that for non-diagonalBianchi I-symmetric spacetimes with collisionless matter the asymp-totic behaviour at late times is close to the special case of dust. Wealso have been able to show that all the Kasner exponents converge to 1

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and an asymptotic expression for the induced metric has been obtained.The key was a bootstrap argument.

11. Luis Filipe Costa: Spinning test particles in general relativity— exact gravito-electromagnetic analogiesWe compare the covariant equation describing the electromagnetic forceexerted on a magnetic dipole with Papapetrous equation for the grav-itational force exerted on a spinning test particle. We show that, ifPirani supplementary spin condition holds, there is an exact and fullygeneral analogy relating these two forces: both are determined by a con-traction of the spin 4-vector with a magnetic-type tidal tensor. Thesetidal tensors [PRD 78 024021, 2008] obey strikingly similar equations,which, in turn, are a covariant form of Maxwells equations and (someof) Einsteins field equations. We exemplify by considering gyroscopesin Schwarzschild and Kerr spacetimes, and comparing with the analo-gous situation of magnetic dipoles moving in the electromagnetic fieldof non-spinning and spinning charges. It is shown that, in the specialcase that the test particles center of mass is at rest and far from the sta-tionary source, the two forces are similar (which is in acco rdance withthe results known from linearized theory [PRD 6 406, 1972]); but that

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for generic dynamics key differences arise. These differences, which aretransparent in the symmetries of the tidal tensors, shed light on manyaspects of spin curvature coupling. In particular we show that: 1) inthe Kerr spacetime there are velocities for which no force is exerted onthe gyroscope, which in the electromagnetic analogue is forbidden bythe laws of electromagnetic induction; 2) that the electromagnetic forceon a dipole has a non-vanishing time projection, which is the powertransferred to it by Faradays induction, and is reflected in a varia-tion of its proper mass, whereas the fact that the force on a gyroscopeis spatial signals the absence of an analogous gravitational effect, ex-plaining the conservation of its proper mass; 3) whereas the total workdone on a magnetic dipole by a stationary magnetic field is zero, a sta-tionary gravitomagnetic field, by contrast, does work on mass currents,which is shown to quantitatively explain the Hawking-Wald Spin Inter-action Energy [PRL. 26 1344, 1971; PRD 6 406, 1972]. Central to theunderstanding of these forces is the issue of hidden momentum (e.g.[arXiv:1004.0679]), whose dynamical implications are also discussed.

Thursday

12. Helmut Friedrich: On radiative and static vacuum space-timesThe detailed understanding of asymptotically flat solution near the “crit-ical sets”, i.e. in the region where space-like infinity touches null in-finity, will allow us to relate analytically physical concepts defined onCauchy data to physical concepts defined on null infinity. Moreover,it offers possibilities to calculate numerically the entire development intime of Cauchy data, including its asymptotic structure and radiationfield, by solving finite Cauchy problems. In this talk we report on re-sults about the behaviour of gravitational fields near the critical sets.In the case of time reflection symmetric Cauchy data evidence is in-creasing that (finite) smoothness at null infinity is related to the databeing asymptotically static (up to a given order). This motivated and isconfirmed by studies of the existence and asymptotic behaviour of staticdata and their conformal classes. We will discuss the significance ofthese results for the analysis of gravitational fields near the critical sets.

13. Hakan Andreasson: Existence of axially symmetric static so-lutions of the Einstein-Vlasov system.We prove the existence of static, asymptotically flat non-vacuum space-times with axial symmetry where the matter is modeled as a collisionless

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gas. The axially symmetric solutions of the resulting Einstein-Vlasovsystem are obtained via the implicit function theorem by perturbing offa suitable spherically symmetric steady state of the Vlasov-Poisson sys-tem. This is a joint work with M. Kunze and G. Rein.

14. Gustavo Dotti: Cosmic censorship and linear stabilityThe current status of an ongoing program to study the linear stability ofthe most notable naked singularities: the negative mass Schwarzschildspacetime, the super-extreme Reissner-Nordstrom and the super-extremeKerr solutions, is reviewed. Theses spaces are found to be unstable, afact with implications on the weak cosmic censorship conjecture. Inconnection with strong cosmic censorship, the proof of the instability ofthe region beyond the Cauchy horizon of a Reissner-Nordstrom blackhole is given. The initial value problem for perturbations in the spheri-cal symmetric cases above present a number of technical difficulties thatwere overcome using intertwiners. The geometrical meaning of the in-tertwined perturbation fields is explained.

15. Carlos Kozameh: Equations of motion for Spin and Center ofmass in an asymptotically flat space timeWe show that from the knowledge of asymptotically shear-free null geodesiccongruences, i.e., congruences with shear that vanishes at future con-formal null infinity we can define the notion of the center-of-mass forasymptotically flata space times and its equations of motion. This in-cludes a kinematic meaning, in terms of the center of mass motion,for the Bondi three-momentum. In addition, we obtain insights intointrinsic spin and, in general, angular momentum, including an angu-lar momentum conservation law with well-defined flux terms. When aMaxwell field is present the asymptotically shear-free congruences allowus to determine/define at infinity a center-of-charge world-line and in-trinsic magnetic dipole moment.

16. Jose M M Senovilla: The boundary of the region with closedtrapped surfacesThe boundary of the region in spacetime containing closed trapped sur-faces is considered. In asymptotically flat black hole spacetimes, thisboundary will generally be strictly inside the event horizon. However,it will be outside any dynamical/trapping horizon associated to the eventhorizon. Actually, closed trapped surfaces can enter flat portions of the

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spacetime whose whole causal past is also flat. Explicit examples willbe presented, and open questions will be analyzed. (Co-authored withIngemar Bengtsson).

17. Alejandro Perez: Regular isolated black holesWe consider asymptotically flat spacetimes containing a black hole. Weshow that mild regularity conditions, in addition to the peeling condi-tions required by asymptotic flatness, severely restrict the fall-off be-havior of the radiation for asymptotically late observers at future nullinfinity (u—-¿Infinity). More precisely, we argue that at null infinity,the radiation must fall-off exponentially with an exponent proportionalto an integer times a characteristic coefficient which coincides with thesurface gravity for stationary black holes.

18. Luca Lusanna: ADM Gravity in the York Canonical Basis: IsDark Matter a Relativistic Inertial Effect?In special and general relativity the synchronization convention of dis-tant clocks may be simulated with a mathematical definition of globalnon-inertial frames (the only ones existing in general relativity dueto the equivalence principle) with well-defined instantaneous 3-spaces.For globally hyperbolic asymptotically Minkowskian spacetimes withoutsupertranslations this procedure can be used at the Hamiltonian level(ADM tetrad gravity ) in the recently found York canonical basis (di-agonalizing the York-Lichnerowicz approach), where it is possible forthe first time to disentangle tidal gravitational degrees of freedom fromgauge inertial ones. The most important inertial effect connected withclock synchronization (absent in Newtonian gravity) is the York time3K, the trace of the extrinsic curvature of 3-space. It is possible to makea Post-Minkowskian linearization of the Hamilton equations in the fam-ily of non-harmonic 3-orthogonal gauges parametrized by the York time.Post-Minkowskian gravitational waves with asymptotic background canbe defined: they propagate in non-Euclidean 3-spaces and enjoy of allthe standard properties. The Post-Newtonian limit of relativistic par-ticles coupled to ADM gravity shows the possibility to describe darkmatter as a relativistic inertial effect inside Einstein general relativityin a Post-Minkowskian reformulation of the Celestial Reference FrameICRS taking into account the York time, which has to be fitted to the ro-tation curves of galaxies. References: Gen.Rel.Grav. 39 (2007); arXiv0907.4087 and 1003.5143 (review in arXiv 0912.2935)

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19. Piotr Chrusciel: The Cauchy problem on the light coneI will present existence and uniqueness theorems for the Cauchy prob-lem for the Einstein equations on a light-cone.

20. Beverly Berger: Exploring the phenomenology of the BKLconjecture for spatially inhomogeneous cosmologiesLong ago, Belinskii, Khalatnikov, and Lifshitz (BKL) argued that theapproach to the singularity in generic gravitational collapse behavedlocally as a spatially homogeneous cosmology that was either velocitydominated (Kasner-like) or oscillatory (Mixmaster-like). This meansthat, operationally, in a numerical simulation of generic collapse, thePDEs of general relativity can be replaced at each spatial point by ODEsdescribing either the Kasner or Mixmaster cosmology. Numerical sim-ulations of collapse in spatially inhomogeneous cosmologies support thisargument. This suggests that, if one assumes this BKL conjecture to betrue, one could explore the phenomenology of generic collapse by evolv-ing, e.g., Mixmaster equations on a spatial grid with spatially dependent(smooth) initial conditions. The well known sensitivity to initial con-ditions would then be expected to yield an interesting, and potentiallyinformative, visualization of the approach to the singularit y. Whilethis BKL regime is reached at different (BKL) time for different spatialpoints, it is likely that sufficiently close to the singularity, almost all(i.e., except at a set of measure zero) spatial points are in this regime.An algorithm originally developed by Garfinkle will be used to generateeach local Mixmaster evolution. Numerical and analytic results in onespatial dimension will be presented along with comparison to genuinespatially inhomogeneous simulations.

21. Woei Chet Lim: Spike crossings in spacetimes with one Killingvector fieldTraditionally, dynamics near spacelike singularities was described bychaotic Mixmaster/BKL dynamics. Spacetimes with two commutingKilling vector fields exhibit a new phenomenon, namely spikes, whichare sub-horizon inhomogeneous structures whose dynamics differs fromBKL dynamics. I will report on the latest progress in the study of thecrossings of two spikes in the context of spacetimes with one Killingvector field.

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22. Oscar Reula: On the geometrical uniqueness of the initial-boundary value problem in general relativity.In recent years considerable advances have been achieved in understand-ing the boundary value problem in general relativity. In particular,starting with the harmonic formulation of the evolution equations itwas found that there were large families of boundary data for which thesystem was strongly well posed, that is stable. Those conditions includemany which were consistent with constraint propagation, in the sensethat if initially the constraints were satisfied they would remain so forthe whole evolution, and among them some for which all the rest of thefields satisfied non-incoming radiation conditions. Thus they were allcandidates for being good conditions to represent radiating isolated sys-tems. But the gauge conditions are intermixed in all these conditionsand it is not clear which ones are better for such a representation. Oneway to explore this set of conditions is to ask for the following geo-metrical uniqueness questions: Can we specify instructions to imposeboundary conditions in such a way that two renditions of such instruc-tions, for the same initial data, would result in the same space-time?That is the resulting evolutions would be linked each other by a dif-feomorphism? In this work we analyze the problem in the linearizedversion and see that there is a set of preferred conditions which satisfycertain geometrical uniqueness, but also see that there remains an am-biguity.

23. Istvan Racz: On the topology of strictly stable surfacesIt is shown that strictly stable surfaces do possess exactly the same topo-logical properties as strictly stable MOTS in higher dimensional (n ≥ 4)spacetimes. In deriving this result no field equation is applied only ageneralized form of the dominant energy condition is required to hold.

24. Osvaldo Moreschi: Asymptotic global physical quantities innumerical relativityIt is very well known that difficulties arise in numerical representationsof spacetimes, when gravitational radiation, total momentum and angu-lar momentum are estimated at finite distances. We point out severalpossible problems that might arise from gauge and tetrad ambiguities.We indicate how to remove these freedom. We report on resent workon astrophysical systems involving these issues.

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25. Ram Gopal Vishwakarma: Gravitational Effect of Pressureand Dark EnergyOne of the most novel aspects of the theory of general relativity (GR)is its prediction that not only the energy density but the pressure ofmatter also gravitates. This is a purely relativistic effect resulting fromthe covariant character of the theory. However, this prediction hasnever been tested in any experiment so far. Here we examine this is-sue on the theoretical front and to our surprise, we find that the theoryseems to suffer from some fundamental inconsistencies. Especially, thestandard formulations of the energy-stress tensor seem to suffer fromparadoxes and inconsistencies in the presence of pressure. We recallthat the mysterious ‘dark energy’ (needed to explain the current cosmo-logical observations) poses a serious confrontation between fundamentalphysics and cosmology in view of its peculiar property- negative pres-sure. This crisis may be an outcome of the (so far untested) predictionof GR that the pressure of the matter source also gravitates.

Friday

26. Jim Isenberg: Initial Data for the Relativistic GraviationalN-Body ProblemIn general relativity, an initial data set for an isolated gravitationalsystem takes the form of a solution of the Einstein constraint equationswhich is asymptotically Euclidean on a specified end. Given a collec-tion of N such data sets with a subregion of interest chosen in each, weshow how to construct a family of new initial data sets, each of whichcontains isometric copies of each of the N chosen subregions, positionedin a chosen array in a single asymptotic end. These composite initialdata sets model isolated, relativistic gravitational systems containing Nchosen bodies in specified initial configurations. This work has been incollaboration with Piotr Chrusciel and Justin Corvino

27. Omar Eduardo Ortiz: On well-posedness, linear perturbationsand mass conservationfor axisymmetric Einstein equationsFor axially symmetric solutions of Einstein equations there exists agauge which has the remarkable property that the total mass can bewritten as a conserved, positive definite, integral on the spacelike slices.The mass integral provides a nonlinear control of the variables along thewhole evolution. In this gauge, Einstein equations reduce to a coupled

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hyperbolic-elliptic system which is formally singular at the axis. As afirst step in analyzing this system of equations we study linear pertur-bations on flat background. We prove that the linear equations reduceto a very simple system of equations which provide, thought the massformula, useful insight into the structure of the full system. However,the singular behavior of the coefficients at the axis makes the study ofthis linear system difficult from the analytical point of view. In order tounderstand the behavior of the solutions, we study the numerical evo-lution of them. We provide strong numerical evidence that the systemis well-posed and that its solutions have the expected behavior. Finally,this linear system allows us to formulate a model problem which is phys-ically interesting in itself, since it is connected with the linear stabilityof black hole solutions in axial symmetry. This model can contributesignificantly to solve the nonlinear problem and at the same time it ap-pears to be tractable.

28. Florian Beyer: Theory of second-order hyperbolic Fuchsianequations and applications to general relativityWe introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associatedinitial value problem when data are imposed on the singularity. Forthe proposed class of second-order hyperbolic Fuchsian systems, we es-tablish the existence of solutions with prescribed asymptotic behavioron the singularity. Our proof is based on a new scheme which is alsosuitable to design numerical approximations. Furthermore, we showthat the second-order Fuchsian framework is appropriate to handle Ein-stein’s field equations for Gowdy symmetric spacetimes and allows usto recover earlier results, while providing a direct approach leading toaccurate numerical solutions of the singular initial value problem. Asexamples we construct Gowdy solutions numerically with incomplete(i.e. non-compact) Cauchy horizons.

29. Juan Antonio Valiente Kroon: A rigidity property for asymp-totically simple developments of time symmetric dataGiven an analytic, time symmetric, asymptotically Euclidean initialdata set for the vacuum Einstein equations, we analyse the implica-tions of its development being asymptotically simple. To this end weconsider a class of initial data which includes static solutions. It isshown that the development of this data is smooth at the sets where

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null infinity touches spatial infinity if and only if the data is static ina neighbourhood of infinity [1]. This result generalises a similar anal-ysis for conformally flat data [2]. The present result brings further inevidence the privileged role that asymptotically static data plays amongthe class of time symmetric data with an asymptotically simple devel-opments —as conjectured in e.g. [3,4]. The analysis makes extensiveuse of the conformal field equations and of Friedrichs framework of thecylinder at spatial infinity [5,6]. References [1] J A Valiente Kroon.[2] J A Valiente Kroon. A rigidity property of of asymptotically simplespacetimes arising from conformally flat data. Comm. Math. Phys.(in press) [3] J A Valiente Kroon. A new class of obstructions to thesmoothness of null infinity. Comm. Math. Phys. 244, 133 (2004) [4] JA Valiente Kroon. Does asymptotic simplicity allow for radiation nearspatial infinity? Comm. Math. Phys. 252, 211 (2004) [5] H Friedrich.Gravitational fields near space-like and null infinity. J. Geom. Phys.24, 83 (1998). [6] H Friedrich. Smoothness at null infinity and thestructure of initial data. In The Einstein Equations and the large scalebehaviour of gravitational fields, P T Chrusciel and H Friedrich eds,Birkhauser (2004).

30. Andres Acena: Minimal data at a point for solutions to cer-tain geometric systemsIn this talk recent results in the characterization of solutions to cer-tain geometrical system of equations in a three dimensional Rieman-nian manifold will be presented. The system of equations has beenconstructed as to include several physically interesting systems of equa-tions, such as the stationary Einstein vacuum field equations or har-monic maps coupled to gravity in three dimensions. A characterizationof its solutions in a neighbourhood of a given point through sequencesof symmetric trace free tensors is given and necessary and sufficientconditions on the data for the existence of the solution are presented,thus providing a complete characterization of all the solutions aroundthe given point.

31. Alberto Chamorro: Physical Content of the Principle of Gen-eral CovarianceThe issue of whether or not the Principle of General Covariance (GCP)has physical content has been matter of debate and confusion since theinception of General Relativity. In our view the physical meaning of co-

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ordinates is related to the question of the possible physical significanceof that principle. We believe that the latter may be taken as an appro-priate generalized principle of relativity with physical content. With thepurpose of throwing light over the subject, after presenting our versionof the GCP, we define and construct quasi-Minkowskian coordinates as-sociated to the word-line of an observer who transports an orthonormaltetrad (QMCCR). We view the QMCCR as the coordinates that wouldbe obtained by that observer by applying operational protocols valid inflat space-time to get the standard Lorentzian coordinates of an event.The set of all the QMCCR is in general an infinite family all of whosemembers collapse to the usual Lorentzian coordinates when the observeris in free fall, his or her space triad does not rotate (R = 0) and thecurvature of space-time vanishes. This implements the idea that theset of all the operational protocols which are equivalent -in the senseof assigning the same numerical values- to obtain the Lorentzian co-ordinates of events in flat space-time split into inequivalent subsets ofoperational prescriptions under the presence of a gravitational field orwhen the observer is not inertial. Something similar must happen withall the physical quantities. Other considerations will be presented

32. Ernesto Fabian Eiroa: Recent progress in thin shell worm-holesIn this talk, some recent advances in the geometrical construction ofthin shell wormholes are presented. Configurations with spherical andcylindrical symmetry are both considered. The stability under perturba-tions preserving the symmetry and the presence of matter violating theenergy conditions are discussed.

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Posters

1. Maria Florencia Parisi: Multigrid Numerical Simulation ofthe Ricci Flow equation in S3

In this work we develop a numerical method to evolve the Ricci Flowequation in 3-dimensions in a spherical background geometry, whichuses a multigrid formulation where the computational domain (S3) iscovered by 8 identical grids or patches. This allows both to study nontrivial topologies, as well as parallelize the computations, where eachpatch is evolved separately by a processor.This brings the problem ofensuring the proper transmission of data between grids, by means ofimposing suitable conditions at the interfaces without loosing precisionin the results. In order to achieve this, we obtained the SAT (stand-ing for Simultaneous Approximation Term) or penaltyoperators thatadequate to the formalism, and successfully implemented them in a nu-merical code based on MPI. This work is an ongoing project, where thementioned numerical code is in a test and optimization stage. We do,however, have some preliminary results that seem to point out its cor-rect and stable operation, faithfully reproducing the expected behaviorof some known solutions.

2. Eoin Condron: Self-similar collapse of the self-interacting cylin-drical scalar fieldWe investigate self-similar scalar field solutions to the Einstein equa-tions in whole cylinder symmetry. Imposing self similarity on the space-time gives rise to a set of single variable functions describing the metric.Furthermore, it is shown that the scalar field is dependent on a singleunknown function of the same variable and that the scalar field poten-tial has exponential form. The Einstein equations then appear as a setof ODEs. We discuss the number of degrees of freedom at an arbitrarypoint. Existence and uniqueness of solutions is discussed where initialdata is taken to be along the axis of symmetry. This is the first step inaddressing the question of cosmic censorship in this class of spacetimes.This is based on joint work with Dr Brien Nolan.

3. Oscar Pablo Zandron: Path-integral quantization in topolog-ically (2+1) massive supergravity theory. Diagrammatic toone loop structure.The path integral quantization for higher derivative Chern-Simons the-ories in (2+1) topologically massive supergravity is treated. The di-agrammatic to one loop and the Feynman rules are constructed and

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later on, the regularization and renormalization of this higher deriva-tive model is analysed in the framework of the perturbation theory.

4. Alberto Carrasco Ferreira: A uniqueness theorem for staticquasi-local black holesMarginally outer trapped surfaces (MOTS) are widely considered asgood quasi-local replacements for black holes. In this work we investi-gate sufficient conditions under which a static Killing initial data setpossessing a MOTS satisfies the hypotheses required for the classicaluniqueness theorems for black holes.

5. Nestor Ortiz: Conformal diagrams for the gravitational col-lapse of a spherically symmetric dust cloudWe present an algorithm to construct conformal diagrams describingthe causal structure in the interior of a relativistic, collapsing mattercloud in spherical symmetry. This algorithm is based on a careful studyof the light rays in the vicinity of the singularity and on the numericalintegration of radial null geodesics. We apply this technique to the col-lapse of a spherical dust cloud, and analyze the local and global visibilityof the resulting singularity.

6. Cesar S. Lopez-Monsalvo: Relativistic thermal dynamicsIn this work we present the covariant dynamics of a fully coupled two-fluid system whose species are particles and entropy. This is a general-ization of Carter’s original work (Carter, 1989), which on the groundsof simplicity, ignored entrainment effects which are generic to mostmulti-fluid systems. This is also an extension to general relativity of theNewtonian work performed by Andersson and Comer. The key result ofthe formalism is the relativistic generalization of the Maxwell-Cattaneoequation, which governs the dynamics of heat. In this construction,the relaxation time of temperature disturbances on a medium dependscrucially on the entrainment between matter and entropy and is fullydetermined once the equation of state is set. Therefore no additionalparameters are needed, in contrast with previous results where the re-laxation time has to be directly measured or obtained from relativistickinetic theory. We use the relativistic two-stream analysis to assess sta-bility and to verify that the matter models one can propose are causallywell behaved. These requirements are verified directly on the masterfunction (Lagrangian density) or, equivalently, at the level of the equa-tion of state one would like to use. We present a toy model equation ofstate of matter and radiation slightly out-of-equilibrium to illustrate thefeatures and constraints which are imposed by our stability analysis.

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7. Gabriel Abreu: Kodama timeIn a general time-dependent (3+1)-dimensional spherically symmetricspacetime, the so-called Kodama vector is a naturally defined geometricquantity that is timelike outside the evolving horizon and so defines apreferred class of fiducial observers. However the Kodama vector doesnot by itself define any preferred notion of time. We demonstrate thata preferred time coordinate - which we shall call Kodama time - can beintroduced by taking the additional step of applying the Clebsch decom-position theorem to the Kodama vector. We thus construct a geomet-rically preferred coordinate system for any time-dependent sphericallysymmetric spacetime, and explore its properties. In particular we usethis formalism to construct a general class of conservation laws, gen-eralizing Kodamas energy flux. We study the geometrically preferredfiducial observers, and demonstrate that it is possible to define and cal-culate a generalized notion of surface gravity that is valid throughoutthe e ntire evolving spacetime. Furthermore, by building and suitablynormalizing set of radial null geodesics, we can show that this general-ized surface gravity passes several consistency tests and has a physicallyappropriate static limit. http://arXiv.org/pdf/1004.1456v2

8. Jaykov Foukzon: Relativistic length expansion in general ac-celerated system revisitedThe aim of the present article is to give an exact and correct repre-sentation of the essentially important part of modern special relativitytheory that touches upon the behavior of the proper length of acceler-ated moving bodies. In particular we pointed out that standard solutionof the Bell’s problem [3]-[4]revision needed. Classical solution of therelativistic length expansion in general accelerated system completelyrevisited.Instant proper length measurement between J.S.Bell’s rocketsalso is considered successfully.http://arxiv.org/ftp/arxiv/papers/0910/0910.2298.pdf

9. Xianghui Luo Luo: Power law inflation with electromagnetismIn the talk, I will talk about how we proved future global stability ofpower law expanding cosmological models with perturbations that con-tain electromagnetic field. Future global stability means the questionthat given a background solution to Einstein equations coupled to somematter fields, of which every causal geodesic is complete to the future,whether a small perturbation results in a solution that also has thisproperty. In other words, whether the property that all freely fallingobjects have infinite future is stable against small perturbations.

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10. Emily Duffy: Cauchy Horizon Stability in the Self SimilarLTB SpacetimeWe undertake a rigorous study of the linear stability of the Cauchyhorizon in the self-similar Lemaitre-Tolman-Bondi spacetime. We usea combination of energy methods and asymptotic analysis to analyticallydetermine the growth and asymptotic behaviour of the perturbations asthey evolve through the spacetime. We first show that the Lp norm, for1

leqp <

infty of a particular average of the perturbation generically divergesat the Cauchy horizon. We then use this to determine the asymptoticbehaviour of the perturbation as it evolves towards the Cauchy horizonand discuss the implications for cosmic censorship. This is based onjoint work with Dr Brien Nolan.

11. Matteo Smerlak: Thermal time and the Tolman-Ehrenfest ef-fectThe thermal time hypothesis has been introduced as a possible basis fora fully general-relativistic thermodynamics. Here I use the notion ofthermal time to study thermal equilibrium on stationary spacetimes.Notably, I show that the Tolman-Ehrenfest effect (the variation of tem-perature in space so that T =

√g00 remains constant) can be reappraised

as a manifestation of this fact: at thermal equilibrium, temperature islocally the rate of flow of thermal time with respect to proper time -pictorially, the speed of (thermal) time. This derivation of the Tolman-Ehrenfest effect makes no reference to the physical mechanisms underly-ing thermalization, thus illustrating the import of the notion of thermaltime. Joint work with Carlo Rovelli.

12. Marcos Ariel Ramirez: Splitting thin shells of counter rotat-ing particles and their thick Einstein-Vlasov counterpartsIn this work we study the dynamics of self gravitating spherically sym-metric thin shells made of counter rotating particles. We consider allpossible velocity distributions for the particles, and show that the equa-tions of motion by themselves do not constrain this distribution. Wetherefore consider the dynamical stability of the resulting configurationsunder several possible processes. This includes the stability of staticconfigurations as a whole, where we find a lower bound for the com-pactness of the shell. We analyse also the stability of the single particleorbits and find conditions for “single particle evaporation”. In the caseof a shell with particles whose angular momentums are restricted to two

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values, we consider the conditions for stability under splitting into twoseparate shells. This analysis leads to the conclusion that under certainconditions, which are given explicitly, an evolving shell may split intotwo or more separate shells. We provide e xplicit examples to illustratethis phenomenon. We also include a derivation of the thick to thin shelllimit for an Einstein shell that shows that the limiting distribution ofangular momentum is unique, covering continuously a finite range ofvalues. Finally we deal with Einstein-Vlasov systems which are static,spherically symmetric and whose particles can only have a discrete setof values for their angular momentum. We prove some general proper-ties which hold for a wide class of these shells and compare with previousresults. We also develop a concrete family of shells and for these wedemonstrate the existence of a thin shell limit and show that this limitis in accordance with the thin shells that we have analysed before.

13. Alejandro Gallardo Lozada: The Boundary Field Theory In-duced by the Chern-Simons TheoryThe Chern-Simons theory defined on a 3-dimensional manifold withboundary is written as a 2-dimensional field theory defined only on theboundary of the three-manifold. The resulting theory is, essentially, thepull-back to the boundary of a symplectic structure defined on the spaceof auxiliary fields in terms of which the connection one-form of theChern-Simons theory is expressed when solving the condition of van-ishing curvature. The counting of the physical degrees of freedom livingin the boundary associated to the odel is performed using Diracs canoni-cal analysis for the particular case of the gauge group SU(2). The resultis that the specific model has one physical local degree of freedom.

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