Higher-dimensional perfect fluids and empty singular boundaries

In order to find out whether empty singular boundaries can arise in higher dimensional Gravity, we study the solution of Einstein’s equations consisting in a (N + 2)-dimensional static and hyperplane symmetric perfect fluid satisfying the equation of state ρ = ηp, being η an arbitrary constant and N ≥ 2. We show that this spacetime has some weird properties. In particular, in the case η > −1, it has an empty (without matter) repulsive singular boundary. We also study the behavior of geodesics and the Cauchy problem for the propagation of massless scalar field in this spacetime. For η > 1, we find that only vertical null geodesics touch the boundary and bounce, and all of them start and finish at z = ∞; whereas non-vertical null as well as all time-like ones are bounded between two planes determined by initial conditions. We obtain that the Cauchy problem for the propagation of a massless scalar field is well-posed and waves are completely reflected at the singularity, if we only demand the waves to have finite energy, although no boundary condition is required.     

Affine gravity,Palatini formalism and charges

Affine gravity and the Palatini formalism contribute both to produce a simple and unique formula for calculating charges at spatial and null infinity for Lovelock type Lagrangians whose variational derivatives do not depend on second-order derivatives of the field components. The method is based on the covariant generalization due to Julia and Silva of the Regge–Teitelboim procedure that was used to define properly the mass in the classical formulation of Einstein’s theory of gravity. Numerous applications reproduce standard results obtained by other secure but mostly specialized method like in ADM energy for asymptotically flat spacetimes and in Abbot and Deser for asymptotically de Sitter and anti-de Sitter spacetimes, both at spatial infinity. As a novel application we calculate the Bondi energy loss in five dimensional gravity, based on the asymptotic solution given by Tanabe et al. and obtain, as expected, the same result. We also give the for Einstein–Gauss–Bonnet gravity and find the superpotential for Lovelock theories of gravity when the number of dimensions tends to infinity with maximally symmetrical boundaries. The paper is written in standard component formalism.     

Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment

We describe our explicit Lorentz-invariant solution of the Einstein and null geodesic equations for the deflection experiment of 2002 September 8 when a massive moving body, Jupiter, passed within 3.7’ of a line-of-sight to a distant quasar. We develop a general relativistic framework which shows that our measurement of the retarded position of a moving light-ray deflecting body (Jupiter) by making use of the gravitational time delay of quasar’s radio wave is equivalent to comparison of the relativistic laws of the Lorentz transformation for gravity and light. Because, according to Einstein, the Lorentz transformation of gravity field variables must depend on a fundamental speed c, its measurement through the retarded position of Jupiter in the gravitational time delay allows us to study the causal nature of gravity and to set an upper limit on the speed of propagation of gravity in the near zone of the solar system as contrasted to the speed of the radio waves. In particular, the v/c term beyond of the standard Einstein’s deflection, which we measured to 20% accuracy, is associated with the aberration of the null direction of the gravity force (“aberration of gravity”) caused by the Lorentz transformation of the Christoffel symbols from the static frame of Jupiter to the moving frame of observer. General relativistic formulation of the experiment identifies the aberration of gravity with the retardation of gravity because the speed of gravitational waves in Einstein’s theory is equal to the speed of propagation of the gravity force. We discuss the misconceptions which have inhibited the acceptance of this interpretation of the experiment. We also comment on other interpretations of this experiment by Asada, Will, Samuel, Pascual–Sánchez, and Carlip and show that their “speed of light” interpretations confuse the Lorentz transformation for gravity with that for light, and the fundamental speed of gravity with the physical speed of light from the quasar. For this reason, the “speed of light” interpretations are not entirely consistent with a retarded Liénard–Wiechert solution of the Einstein equations, and do not properly incorporate how the phase of the radio waves from the quasar is perturbed by the retarded gravitational field of Jupiter. Although all of the formulations predict the same deflection to the order of v/c, our formulation shows that the underlying cause of this deflection term is associated with the aberration of gravity and not of light, and that the interpretations predict different deflections at higher orders of v/c beyond the Shapiro delay, thus, making their measurement highly desirable for deeper testing of general relativity in future astrometric experiments like Gaia, SIM, and SKA.     

Probabilistic Study of the Speed of Approach to Equilibrium for an Inelastic Kac Model

This paper deals with a one-dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter p>0. In particular, the paper provides bounds for certain distances—such as specific weighted χ-distances and the Kolmogorov distance—between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent α=2/(1+p). With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, close to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function. Research partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR grant 2006/134526).     

Non-Equilibrium Dynamics of Dyson’s Model with an Infinite Number of Particles

Dyson’s model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson’s model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in which every point of \mathbbZ{\mathbb{Z}} is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel.     

The surface layers dual to hydrodynamic boundaries

The AdS/hydrodynamics correspondence provides a 1–1 map between large wavelength features of AdS black branes and conformal fluid flows. In this note we consider boundaries between nonrelativistic flows, applying the usual boundary conditions for viscous fluids. We find that a naive application of the correspondence to these boundaries yields a surface layer in the gravity theory whose stress tensor is not equal to that given by the Israel matching conditions. In particular, while neither stress tensor satisfies the null energy condition and both have nonvanishing momentum, only Israel's tensor has stress. The disagreement arises entirely from corrections to the metric due to multiple derivatives of the flow velocity, which violate Israel's finiteness assumption in the thin wall limit.     

The volume inside a black hole

The horizon (the surface) of a black hole is a null surface, defined by those hypothetical “outgoing” light rays that just hover under the influence of the strong gravity at the surface. Because the light rays are orthogonal to the spatial two-dimensional surface at one instant of time, the surface area of the black hole is the same for all observers (i.e. the same for all coordinate definitions of “instant of time”). This value is 4π(2Gm/c ²)² for nonspinning black holes, with G = Newton’s constant, c = speed of light, and m = mass of the black hole. The three-dimensional spatial volume inside a black hole, in contrast, depends explicitly on the definition of time, and can even be time dependent, or zero. We give examples of the volume found inside a standard, nonspinning spherical black hole, for several different standard time-coordinate definitions. Elucidating these results for the volume provides a new pedagogical resource of facts already known in principle to the relativity community, but rarely worked out.     

Couplings of the rho meson in a holographic dual of QCD with Regge trajectories

The couplings g_ρHH of the ρ meson with a scalar meson H are calculated in a holographic dual of QCD in which the Regge trajectories for the mesons are manifest. In contrast to the conclusion in general AdS/QCD models, the resulting couplings grow linearly with the quantum number of excited H; thus they are far from universal. This non-universality seems to result from the disappearance of an explicit cutoff in the holographic dimension in this model. Correspondingly, the ρ-dominance for the electromagnetic form factors of H does not hold any more, even in an apparent sign-alternating manner. With these couplings at hand, the asymptotic behavior of the form factors can easily be calculated. The form factor exhibits the 1/Q^(2Δ-2) behavior, which is in accordance with the scaling behavior of the fixed-angle scattering amplitude based on the AdS/CFT correspondence. It is also pointed out that the asymptotic behavior can be matched to the results of perturbative QCD, if the conformal dimension Δ of the operator is replaced by the combination τ+L of the meson. PACS  11.25.Tq; 12.40.Vv; 14.40.Cs     

Existence and Uniqueness Theorems for Massless Fields on a Class of Spacetimes with Closed Timelike Curves

We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC’s), in which all future-directed CTC’s traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in L ₂ on ℐ ⁻ , we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite L ₂ -norm) and have the given initial data on ℐ ⁻ . A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC’s are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC’s. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture whose meaning is essentially that the Cauchy problem for free fields is well defined in the presence of CTC’s whenever the problem is well-posed in a geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region. Received: 28 November 1994/Accepted: 20 May 1996     

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.     

Dyadic Green’s function of a PEMC cylinder

The dyadic Green’s function of a PEMC cylinder is derived with the aid of the principle of scattering superposition and Ohm–Rayleigh method. The PEMC boundary conditions are presented in dyadic form and it shows that how the impedance parameter of PEMC and cross-polarized fields appear in the Green’s function. The asymptotic expansions of the dyadic function is calculated in order to attain a closed form for the electrical field.     

Strict Convexity of the Free Energy for a Class of Non-Convex Gradient Models

We consider a gradient interface model on the lattice with interaction potential which is a non-convex perturbation of a convex potential. We show using a one-step multiple scale analysis the strict convexity of the surface tension at high temperature. This is an extension of Funaki and Spohn’s result [8], where the strict convexity of potential was crucial in their proof. Supported by the DFG-Forschergruppe 718 ‘Analysis and stochastics in complex physical systems’.     

On the Orthocomplementation of State–Property–Systems of Contextual Systems

We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability of a state transition in general not only depends on the two states involved, but also on the measurement context which induces the state transition.     

A note on conserved charges of asymptotically flat and anti-de Sitter spaces in arbitrary dimensions

The calculation of conserved charges of black holes is a rich problem, for which many methods are known. Until recently, there was some controversy on the proper definition of conserved charges in asymptotically anti-de Sitter (AdS) spaces in arbitrary dimensions. This paper provides a systematic and explicit Hamiltonian derivation of the energy and the angular momenta of both asymptotically flat and asymptotically AdS spacetimes in any dimension D  ≥  4. This requires as a first step a precise determination of the asymptotic conditions of the metric and of its conjugate momentum. These conditions happen to be achieved in ellipsoidal coordinates adapted to the rotating solutions. The asymptotic symmetry algebra is found to be isomorphic either to the Poincaré algebra or to the so(D − 1,2) algebra, as expected. In the asymptotically flat case, the boundary conditions involve a generalization of the parity conditions, introduced by Regge and Teitelboim, which are necessary to make the angular momenta finite. The charges are explicitly computed for Kerr and Kerr–AdS black holes for arbitrary D and they are shown to be in agreement with thermodynamical arguments. The author is a FRIA-FNRS bursar (National Fund for Scientific Research, Belgium).     

Worldtube conservation laws for the null-timelike evolution problem

I treat the worldtube constraints which arise in the null-timelike initial-boundary value problem for the Bondi-Sachs formulation of Einstein’s equations. Boundary data on a worldtube and initial data on an outgoing null hypersurface determine the exterior spacetime by integration along the outgoing null geodsics. The worldtube constraints are a set of conservation laws which impose conditions on the integration constants. I show how these constraints lead to a well-posed initial value problem governing the extrinsic curvature of the worldtube, whose components are related to the integration constants. Possible applications to gravitational waveform extraction and to the well-posedness of the null-timelike initial-boundary value problem are discussed.     

Polarized angular distributions in the decays of the ψ’ charmonium state directly produced in unpolarized proton–antiproton collisions

Using the helicity formulism, we calculate the combined angular distribution function of the polarized gamma photons and electron in the cascade process pp̄→ψ’→χ_J+γ₁→ψ+γ₂+γ₁→e⁺+e^-+γ₂+γ₁ (J=0,1,2), when p̄ and p are unpolarized. We also present the partially integrated angular distribution functions in different cases. Our results show that by measuring the two-particle angular distribution of γ₁ and γ₂ and that of γ₂ and e^- with the polarization of either one of the two particles, one can determine the relative magnitudes as well as the relative phases of the helicity amplitudes in the radiative decay processes ψ’→χ_J+γ₁ and χ_J→ψ+γ₂. PACS  11.80.Cr; 13.20.Fc; 13.60.Le; 14.40.Gx     

Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] × Σ, where Σ is a compact manifold with smooth boundaries ∂Σ. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on ∂Σ. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell’s equations in the Lorentz gauge and Einstein’s gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.     

Zγ* production in e+e- interactions at ?s = 183-209 \textGeV\sqrt{s} = 183-209\,\text{GeV}

Measurements of Zγ^* production are presented using data collected by the DELPHI detector at centre-of-mass energies ranging from 183 to 209 GeV, corresponding to an integrated luminosity of about 667 pb^-1. The measurements cover a wide range of the possible final state four-fermion configurations: hadronic and leptonic (e⁺e^-qq̄, μ⁺μ^-qq̄, qq̄νν̄), fully leptonic (l⁺l^-l^’+l^’-) and fully hadronic final states (qq̄qq̄, with a low mass qq̄ pair). Measurements of the Zγ^* cross-section for the various final states have been compared with the Standard Model expectations and found to be consistent within the errors. In addition, a total cross-section measurement of the l⁺l^-l^’+l^’- cross-section is reported, and found to be in agreement with the prediction of the Standard Model.     

Mach’s principle and the notion of time

The role of time coordinate in the realization of March’s principle is highlighted. It is shown that Mach’s principle is linked to the definition of a ‘particle’. These results suggest a deep connection between quantum gravity and Mach’s principle.