# Lectures on the Infrared Structure of Gravity and Gauge Theory [CL]

This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at https://goo.gl/3DJdOr. Please send typos or corrections to strominger@physics.harvard.edu.

A. Strominger
Fri, 17 Mar 17
18/50

# Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque [CL]

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

M. Bartuccelli, J. Deane and G. Gentile
Mon, 6 Mar 17
38/47

Comments: 32 pages, 8 figures, 5 tables

# Dirac-Bergmann Constraints in Physics: Singular Lagrangians, Hamiltonian Constraints and the Second Noether Theorem [CL]

There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac-Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed:
i) the connection of the rank and eigenvalues of the Hessian matrix in the Euler-Lagrange equations with the chains of first and second class constraints;
ii) the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints;
iii) the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations.
Review paper for a chapter of a future book.

L. Lusanna
Mon, 27 Feb 17
23/49

# Dirac-Bergmann Constraints in Relativistic Physics: Non-Inertial Frames, Point Particles, Fields and Gravity [CL]

There is a review of the physical theories needing Dirac-Bergmann theory of constraints at the Hamiltonian level due to the existence of gauge symmetries. It contains:
i) the treatment of systems of point particles in special relativity both in inertial and non-inertial frames with a Wigner-covariant way of eliminating relative times in relativistic bound states;
ii) the description of the electro-magnetic field in relativistic atomic physics and of Yang-Mills fields in absence of Gribov ambiguity in particle physics;
iii) the identification of the inertial gauge variables and of the physical variables in canonical ADM tetrad gravity in presence of the electro-magnetic field and of charged scalar point particles in asymptotically Minkowskian space-times without super-translations by means of a Shanmugadhasan canonical transformation to a York canonical basis adapted to ten of the 14 first-class constraints and the definition of the Hamiltonian Post-Minkowskian weak field limit.
Review paper for a chapter of a future book

L. Lusanna
Mon, 27 Feb 17
40/49

Comments: 30 pages. arXiv admin note: substantial text overlap with arXiv:1108.3224, arXiv:1205.2481

# Noether symmetries and stability of ideal gas solution in Galileon Cosmology [CL]

A class of generalized Galileon cosmological models, which can be described by a point-like Lagrangian, is considered in order to utilize Noether’s Theorem to determine conservation laws for the field equations. In the Friedmann-Lema\^{\i}tre-Robertson-Walker universe, the existence of a nontrivial conservation law indicates the integrability of the field equations. Due to the complexity of the latter, we apply the differential invariants approach in order to construct special power-law solutions and study their stability.

N. Dimakis, A. Giacomini, S. Jamal, et. al.
Tue, 7 Feb 17
31/64

# Inflationary $α$-attractor cosmology: A global dynamical systems perspective [CL]

We study flat FLRW $\alpha$-attractor $\mathrm{E}$- and $\mathrm{T}$-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of e-folds, is associated with a solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this inflationary attractor solution.’ A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.

A. Alho and C. Uggla
Thu, 2 Feb 17
10/52

# Exact collisional plasma fluid theories [CL]

An expansion of the velocity space distribution functions in terms of multi-index Hermite polynomials is carried out to derive a consistent set of collisional fluid equations for plasmas. The velocity-space moments of the often troublesome nonlinear Landau collision operator are evaluated exactly, and to all orders with respect to the expansion. The collisional moments are shown to be generated by applying gradients on two well-known functions, namely the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The expansion can be truncated at arbitrary order with quantifiable error, providing a consistent and systematic alternative to the Chapman-Enskog procedure which, in plasma physics, boils down to the famous Braginskii equations. To illustrate our approach, we provide the collisional ten-moment equations and prove explicitly that the exact, nonlinear expressions for the momentum- and energy-transfer rate satisfy the correct conservation properties.

D. Pfefferle, E. Hirvijoki and M. Lingam
Mon, 30 Jan 17
37/41

# Cosmological Evolution and Exact Solutions in a Fourth-order Theory of Gravity [CL]

A fourth-order theory of gravity is considered which in terms of dynamics has the same degrees of freedom and number of constraints as those of scalar-tensor theories. In addition it admits a canonical point-like Lagrangian description. We study the critical points of the theory and we show that it can describe the matter epoch of the universe and that two accelerated phases can be recovered one of which describes a de Sitter universe. Finally for some models exact solutions are presented.

A. Paliathanasis
Tue, 17 Jan 17
44/81

# The structure of invariant tori in a 3D galactic potential [CL]

We study in detail the structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type. We have used the color and rotation method to investigate the properties of the invariant tori in the 4D spaces of section. We compare our results with those of previous works and we describe the morphology of the rotational, as well as of the tube tori in the 4D space. We find sticky chaotic orbits in the immediate neighborhood of sets of invariant tori surrounding 3D stable periodic orbits. Particularly useful for galactic dynamics is the behavior of chaotic orbits trapped for long time between 4D invariant tori. We find that they support during this time the same structure as the quasi-periodic orbits around the stable periodic orbits, contributing however to a local increase of the dispersion of velocities. Finally we find that the tube tori do not appear in the 3D projections of the spaces of section in the axisymmetric Hamiltonian we examined.

M. Katsanikas and P. Patsis
Mon, 9 Jan 17
21/52

Comments: 26 pages, 34 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential [CL]

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis and Zachilas 1994] in order to visualize them. As examples we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

M. Katsanikas, P. Patsis and A. Pinotsis
Mon, 9 Jan 17
26/52

Comments: 12 pages,22 figures, Accepted for publication in the International Journal of Bifurcation and Chaos

# The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation [CL]

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D “confined tori”. Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
42/52

Comments: 10 pages, 14 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Instabilities and stickiness in a 3D rotating galactic potential [CL]

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.

M. Katsanikas, P. Patsis and G. Contopoulos
Mon, 9 Jan 17
44/52

Comments: 29pages, 25 figures, accepted for publication in the International Journal of Bifurcation and Chaos

# Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows [CL]

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton’s principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. Conserved circulation laws, such as those of Kelvin, Alfv\’en and Bekenstein-Oron, emerge simply as special cases of the Poincar\’e-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin’s theorem to baroclinic (non-isentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.

C. Markakis, K. Uryu, E. Gourgoulhon, et. al.
Mon, 2 Jan 17
9/45

# Dynamical models and the onset of chaos in space debris [EPA]

The increasing threat raised by space debris led to the development of different mathematical models and approaches to investigate the dynamics of small particles orbiting around the Earth. Such models and methods strongly depend on the altitude of the objects above Earth’s surface, since the strength of the different forces acting on an Earth orbiting object (geopotential, atmospheric drag, lunar and solar attractions, solar radiation pressure, etc.) varies with the altitude of the debris.
In this review, our focus is on presenting different analytical and numerical approaches employed in modern studies of the space debris problem. We start by considering a model including the geopotential, solar and lunar gravitational forces and the solar radiation pressure. We summarize the equations of motion using different formalisms: Cartesian coordinates, Hamiltonian formulation using Delaunay and epicyclic variables, Milankovitch elements. Some of these methods lead in a straightforward way to the analysis of resonant motions. In particular, we review results found recently about the dynamics near tesseral, secular and semi-secular resonances.
As an application of the above methods, we proceed to analyze a timely subject namely the possible causes for the onset of chaos in space debris dynamics. Precisely, we discuss the phenomenon of overlapping of resonances, the effect of a large area-to-mass ratio, the influence of lunisolar secular resonances.
We conclude with a short discussion about the effect of the dissipation due to the atmospheric drag and we provide a list of minor effects, which could influence the dynamics of space debris.

A. Celletti, C. Efthymiopoulos, F. Gachet, et. al.
Fri, 30 Dec 16
39/64

# Number Density of Peaks in a Chi-Squared Field [CL]

We investigate the statistics of stationary points in the sum of squares of $N$ Gaussian random fields, which we call a “chi-squared” field. The behavior of such a field at a point is investigated, with particular attention paid to the formation of topological defects. An integral to compute the number density of stationary points at a given field amplitude is constructed. We compute exact expressions for the integral in various limits and provide code to evaluate it numerically in the general case. We investigate the dependence of the number density of stationary points on the field amplitude, number of fields, and power spectrum of the individual Gaussian random fields. This work parallels the work of Bardeen, Bond, Kaiser and Szalay, who investigated the statistics of peaks of Gaussian random fields. A number of results for integrating over matrices are presented in appendices.

J. Bloomfield, S. Face, A. Guth, et. al.
Tue, 13 Dec 16
49/77

Comments: 18 pages + 11 pages appendices, 3 figures

# Spacetime completeness of non-singular black holes in conformal gravity [CL]

We explicitly prove that the Weyl conformal symmetry solves the black hole singularity problem, otherwise unavoidable in a generally covariant local or non-local gravitational theory. Moreover, we yield explicit examples of local and non-local theories enjoying Weyl and diffeomorphism symmetry (in short co-covariant theories). Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free spherically symmetric and axi-symmetric exact solutions for black hole spacetimes conformally equivalent to the Schwarzschild or the Kerr spacetime. We first check the absence of divergences in the Kretschmann invariant for the rescaled metrics. Afterwords, we show that the new types of black holes are geodesically complete and linked by a Newman-Janis transformation just as in standard general relativity (based on Einstein-Hilbert action). Furthermore, we argue that no massive or massless particles can reach the former Schwarzschild singularity or touch the former Kerr ring singularity in a finite amount of their proper time or of their affine parameter. Finally, we discuss the Raychaudhuri equation in a co-covariant theory and we show that the expansion parameter for congruences of both types of geodesics (for massless and massive particles) never reaches minus infinity. Actually, the null geodesics become parallel at the r=0 point in the Schwarzschild spacetime (the origin) and the focusing of geodesics is avoided. The arguments of regularity of curvature invariants, geodesic completeness, and finiteness of geodesics’ expansion parameter ensure us that we are dealing with singularity-free and geodesically-complete black hole spacetimes.

C. Bambi, L. Modesto and L. Rachwal
Fri, 4 Nov 16
19/58

Comments: 22 pages, 21 pictures. arXiv admin note: text overlap with arXiv:1605.04173

# Initial conditions for cosmological perturbations [CL]

Penrose proposed that the big bang singularity should be constrained by requiring that the Weyl curvature vanishes there. The idea behind this past hypothesis is attractive because it constrains the initial conditions for the universe in geometric terms and is not confined to a specific early universe paradigm. However, the precise statement of Penrose’s hypothesis is tied to classical space-times and furthermore restricts only the gravitational degrees of freedom. These are encapsulated only in the tensor modes of the commonly used cosmological perturbation theory. Drawing inspiration from the underlying idea, we propose a quantum generalization of Penrose’s hypothesis using the Planck regime in place of the big bang, and simultaneously incorporating tensor as well as scalar modes. Initial conditions selected by this generalization constrain the universe to be as homogeneous and isotropic in the Planck regime \emph{as permitted by the Heisenberg uncertainty relations}.

A. Ashtekar and B. Gupt
Tue, 1 Nov 16
46/75

# Jacobi stability analysis of scalar field models with minimal coupling to gravity in a cosmological background [CL]

We perform the study of the stability of the cosmological scalar field models, by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. In the KCC approach we describe the time evolution of the scalar field cosmologies in geometric terms, by performing a “second geometrization”, by considering them as paths of a semispray. By introducing a non-linear connection and a Berwald type connection associated to the Friedmann and Klein-Gordon equations, five geometrical invariants can be constructed, with the second invariant giving the Jacobi stability of the cosmological model. We obtain all the relevant geometric quantities, and we formulate the condition of the Jacobi stability for scalar field cosmologies in the second order formalism. As an application of the developed methods we consider the Jacobi stability properties of the scalar fields with exponential and Higgs type potential. We find that the Universe dominated by a scalar field exponential potential is in Jacobi unstable state, while the cosmological evolution in the presence of Higgs fields has alternating stable and unstable phases. By using the standard first order formulation of the cosmological models as dynamical systems we have investigated the stability of the phantom quintessence and tachyonic scalar fields, by lifting the first order system to the tangent bundle. It turns out that in the presence of a power law potential both these models are Jacobi unstable during the entire cosmological evolution.

B. Danila, T. Harko, M. Mak, et. al.
Tue, 20 Sep 16
15/74

Comments: 24 pages, 14 figures, accepted for publication in Advances in High Energy Physics, special issue “Dark Physics in the Early Universe”

# Integrable Cosmological Potentials [CL]

The problem of classification of the Einstein–Friedman cosmological Hamiltonians $H$ with a single scalar inflaton field $\varphi$ that possess an additional integral of motion polynomial in momenta on the shell of the Friedman constraint $H=0$ is considered. Necessary and sufficient conditions for the existence of first, second, and third degree integrals are derived. These conditions have the form of ODEs for the cosmological potential $V(\varphi)$. In the case of linear and quadratic integrals we find general solutions of the ODEs and construct the corresponding integrals explicitly. A new wide class of Hamiltonians that possess a cubic integral is derived. The corresponding potentials are represented in a parametric form in terms of the associated Legendre functions. Six families of special elementary solutions are described and sporadic superintegrable cases are discussed.

V. Sokolov and A. Sorin
Wed, 31 Aug 16
47/61

Comments: 24 pages, LaTeX, 2 figures

# Electromagnetic 3D subsurface imaging with source sparsity for a synthetic object [IMA]

This paper concerns electromagnetic 3D subsurface imaging in connection with sparsity of signal sources. We explored an imaging approach that can be implemented in situations that allow obtaining a large amount of data over a surface or a set of orbits but at the same time require sparsity of the signal sources. Characteristic to such a tomography scenario is that it necessitates the inversion technique to be genuinely three-dimensional: For example, slicing is not possible due to the low number of sources. Here, we primarily focused on astrophysical subsurface exploration purposes. As an example target of our numerical experiments we used a synthetic small planetary object containing three inclusions, e.g. voids, of the size of the wavelength. A tetrahedral arrangement of source positions was used, it being the simplest symmetric point configuration in 3D. Our results suggest that somewhat reliable inversion results can be produced within the present a priori assumptions, if the data can be recorded at a specific resolution. This is valuable early-stage knowledge especially for design of future planetary missions in which the payload needs to be minimized, and potentially also for the development of other lightweight subsurface inspection systems.

S. Pursiainen and M. Kaasalainen
Thu, 25 Aug 16
13/43

Comments: 17 pages, 5 figures. This is an author-created, un-copyedited version of an article accepted for publication/published in Inverse Problems. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at this http URL

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# Ernst formulation of axisymmetric fields in $f(R)$ gravity: applications to neutron stars and gravitational waves [CL]

The Ernst formulation of the Einstein equations is generalised to accommodate $f(R)$ theories of gravity. It is shown that, as in general relativity, the axisymmetric $f(R)$ field equations for a vacuum spacetime that is either stationary or cylindrically symmetric reduce to a single, non-linear differential equation for a complex-valued scalar function. As a worked example, we apply the generalised Ernst equations to derive a $f(R)$ generalisation of the Zipoy-Voorhees metric, which may be used to describe the gravitational field outside of an ellipsoidal neutron star. We also apply the theory to investigate the phase speed of large-amplitude gravitational waves in $f(R)$ gravity in the context of soliton-like solutions that display shock-wave behaviour across the causal boundary.

A. Suvorov and A. Melatos
Thu, 11 Aug 16
47/51

Comments: 12 pages, zero figures. Accepted for publication in PRD

# Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state [CL]

Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index $n$. The explicit form of the solution is presented for the polytropic index $n=1$, and for the indexes $n=1/2$ and $n=1/5$, respectively. The case of $n=3$ is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven terms only. The power series representations of the geometric and physical properties of the linear barotropic and polytropic stars are also obtained.

T. Harko and M. Mak
Tue, 26 Jul 16
75/75

Comments: 21 pages, 3 figures, accepted for publication in Astrophys. Space Science

# On dynamical systems approaches and methods in $f(R)$ cosmology [CL]

We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + \alpha R^2$, $\alpha > 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to $f(R)$ cosmology and discuss their advantages and disadvantages.

A. Alho, S. Carloni and C. Uggla
Wed, 20 Jul 16
4/66

# Hot dense magnetized ultrarelativistic spinor matter in a slab [CL]

Properties of hot dense ultrarelativistic spinor matter in a slab of finite width, placed in a transverse uniform magnetic field, are studied. The admissible set of boundary conditions is determined by the requirement that spinor matter be confined inside the slab. In thermal equilibrium, the chiral separation effect in the slab is shown to depend both on temperature and chemical potential; this is distinct from the unrealistic case of the magnetic field filling the unbounded (infinite) medium, when the effect is temperature-independent. In the realistic case of the slab, a stepwise behaviour of the axial current density at zero temperature is smoothed out as temperature increases, turning into a linear behaviour at infinitely large temperature. A choice of boundary conditions can facilitate either augmentation or attenuation of the chiral separation effect; in particular, the effect can persist even at zero chemical potential, if temperature is nonzero. Thus the boundary condition can serve as a source that is additional to the spinor matter density.

Y. Sitenko
Tue, 28 Jun 16
32/58

Comments: 27 pages, 5 figures. arXiv admin note: text overlap with arXiv:1603.09268

# Force-Free Foliations [CL]

Electromagnetic field configurations with vanishing Lorentz force density are known as force-free and appear in terrestrial, space, and astrophysical plasmas. We explore a general method for finding such configurations based on formulating equations for the field lines rather than the field itself. The basic object becomes a foliation of spacetime or, in the stationary axisymmetric case, of the half-plane. We use this approach to find some new stationary and axisymmetric solutions, one of which could represent a rotating plasma vortex near a magnetic null point.

G. Compere, S. Gralla and A. Lupsasca
Wed, 22 Jun 16
38/50

# Analytical theory for highly elliptical orbits including time-dependent perturbations [EPA]

Traditional analytical theories of celestial mechanics are not well-adapted when dealing with highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and so limited to quasi-circular orbits. On the other hand, the time-dependency due to the motion of the third body (e.g. Moon and Sun) is almost always neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the third-body disturbing function into a finite polynomial using Fourier series in multiple of the satellite’s eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, by combining the classical Brouwer-von Zeipel procedure and the time-dependent Lie-Deprit transforms, we have performed a normalization of the expanded Hamiltonian in order to eliminate all the periodic terms. One of the benefits is that the original Brouwer solution for J2 is not modified. The main difficulty lies in the fact that the generating functions of the transformation must be computed by solving a partial differential equation, involving derivatives with respect to the mean anomaly, which appears implicitly in the perturbation. We present a method to solve this equation by means of an iterative process. Finally we have obtained an analytical tool useful for the mission analysis, allowing to propagate the osculating motion of objects on highly elliptical orbits (e>0.6) over long periods efficiently with very high accuracy, or to determine initial elements or mean elements. Comparisons between the complete solution and the numerical simulations will be presented.

G. Lion and G. Metris
Tue, 14 Jun 16
9/67

# Towards an analytical theory of the third-body problem for highly elliptical orbits [EPA]

When dealing with satellites orbiting a central body on a highly elliptical orbit, it is necessary to consider the effect of gravitational perturbations due to external bodies. Indeed, these perturbations can become very important as soon as the altitude of the satellite becomes high, which is the case around the apocentre of this type of orbit. For several reasons, the traditional tools of celestial mechanics are not well adapted to the particular dynamic of highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and therefore limited to quasi-circular orbits [17, 25]. On the other hand, the time-dependency due to the motion of the third-body is often neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the disturbing function into a finite polynomial using Fourier expansions of elliptic motion functions in multiple of the satellite’s eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, we show how to perform a normalization of the expanded Hamiltonian by means of a time-dependent Lie transformation which aims to eliminate periodic terms. The difficulty lies in the fact that the generator of the transformation must be computed by solving a partial differential equation involving variables which are linear with time and the eccentric anomaly which is not time linear. We propose to solve this equation by means of an iterative process.

G. Lion, G. Metris and F. Deleflie
Thu, 26 May 16
5/53

Comments: Proceedings of the International Symposium on Orbit Propagation and Determination – Challenges for Orbit Determination and the Dynamics of Artificial Celestial Bodies and Space Debris, Lille, France, 2011

# Finite Conformal Quantum Gravity and Nonsingular Spacetimes [CL]

We explicitly prove that a class of finite quantum gravitational theories (in odd as well as in even dimension) is actually a range of anomaly-free conformally invariant theories in the spontaneously broken phase of the conformal Weyl symmetry. At classical level we show how the Weyl conformal invariance is likely able to tame the spacetime singularities that plague not only Einstein gravity, but also local and weakly non-local higher derivative theories. This latter statement is rigorously proved by a singularity theorem that applies to a large class of weakly non-local theories. Following the seminal paper by Narlikar and Kembhavi, we provide an explicit construction of singularity-free black hole exact solutions conformally equivalent to the Schwarzschild metric. Furthermore, we show that the FRW cosmological solutions and the Belinski, Khalatnikov, Lifshitz (BKL) spacetimes, which exactly solve the classical equations of motion, are conformally equivalent to regular spacetimes. Finally, we prove that the Oppenheimer-Volkov gravitational collapse is a an exact (singularity-free) solution of the non-local conformally invariant theory compatible with the bounce paradigm.

L. Modesto and L. Rachwal
Mon, 16 May 16
4/48

# Symmetry Reduced Loop Quantum Gravity: A Bird's Eye View [CL]

This is a brief overview of the current status of symmetry reduced models in Loop Quantum Gravity. The goal is to provide an introduction to other more specialized and detailed reviews that follow. Since most of this work is motivated by the physics of the very early universe, I will focus primarily on Loop Quantum Cosmology and discuss quantum aspects of black holes only briefly.

A. Ashtekar
Tue, 10 May 16
6/85

# Supersymmetric Theory of Stochastic ABC Model: A Numerical Study [CL]

In this paper, we investigate numerically the stochastic ABC model, a toy model in the theory of astrophysical kinematic dynamos, within the recently proposed supersymmetric theory of stochastics (STS). STS characterises stochastic differential equations (SDEs) by the spectrum of the stochastic evolution operator (SEO) on elements of the exterior algebra or differentials forms over the system’s phase space, X. STS can thereby classify SDEs as chaotic or non-chaotic by identifying the phenomenon of stochastic chaos with the spontaneously broken topological supersymmetry that all SDEs possess. We demonstrate the following three properties of the SEO, deduced previously analytically and from physical arguments: the SEO spectra for zeroth and top degree forms never break topological supersymmetry, all SDEs possesses pseudo-time-reversal symmetry, and each de Rahm cohomology class provides one supersymmetric eigenstate. Our results also suggests that the SEO spectra for forms of complementary degrees, i.e., k and dim X -k, may be isospectral.

I. Ovchinnikov, Y. Sun, T. Ensslin, et. al.
Mon, 2 May 16
39/49

Comments: Revtex 4-1, 9 pages, 3 figures

# Stability and chaos in Kustaanheimo-Stiefel space induced by the Hopf fibration [CL]

The need for the extra dimension in Kustaanheimo-Stiefel (KS) regularization is explained by the topology of the Hopf fibration, which defines the geometry and structure of KS space. A trajectory in Cartesian space is represented by a four-dimensional manifold, called the fundamental manifold. Based on geometric and topological aspects classical concepts of stability are translated to KS language. The separation between manifolds of solutions generalizes the concept of Lyapunov stability. The dimension-raising nature of the fibration transforms fixed points, limit cycles, attractive sets, and Poincar\’e sections to higher-dimensional subspaces. From these concepts chaotic systems are studied. In strongly perturbed problems the numerical error can break the topological structure of KS space: points in a fiber are no longer transformed to the same point in Cartesian space. An observer in three dimensions will see orbits departing from the same initial conditions but diverging in time. This apparent randomness of the integration can only be understood in four dimensions. The concept of topological stability results in a simple method for estimating the time scale in which numerical simulations can be trusted. Ideally all trajectories departing from the same fiber should be KS transformed to a unique trajectory in three-dimensional space, because the fundamental manifold that they constitute is unique. By monitoring how trajectories departing from one fiber separate from the fundamental manifold a critical time, equivalent to the Lyapunov time, is estimated. These concepts are tested on N-body examples: the Pythagorean problem, and an example of field stars interacting with a binary.

J. Roa, H. Urrutxua and J. Pelaez
Mon, 25 Apr 16
6/40

Comments: Accepted in MNRAS. 12 pages, 9 figures

# Continuum Eigenmodes in Some Linear Stellar Models [CL]

We apply parallel approaches in the study of continuous spectra to adiabatic stellar models. We seek continuum eigenmodes for the LAWE formulated as both finite difference and linear differential equations. In particular, we apply methods of Jacobi matrices and methods of subordinancy theory in these respective formulations. We find certain pressure-density conditions which admit positive-measured sets of continuous oscillation spectra under plausible conditions on density and pressure. We arrive at results of unbounded oscillations and computational or, perhaps, dynamic instability.

C. Winfield
Tue, 8 Mar 16
7/83

# Scalar-tensorial equivalence for higher order $f\left( R,\nabla_μ R,\nabla_{μ_{1}}\nabla_{μ_{2}}R,…,\nabla_{μ_{1}}…\nabla_{μ_{n} }R\right)$ theories of gravity [CL]

The equivalence between theories depending on the derivatives of $R$, i.e. $f\left( R,\nabla R,…,\nabla^{n}R\right)$, and scalar-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that $f\left( R,\nabla R,…,\nabla^{n}R\right)$ theories are equivalents to Brans-Dicke theories with kinetic terms $\omega_{0}=0$ and $\omega_{0}= – \frac{3}{2}$ for metric and Palatini formalisms respectively. This result is analogous to what happens for $f(R)$ theories. Furthermore, sufficient conditions are established for $f\left( R,\nabla R,…,\nabla^{n}R\right)$ theories to be written as scalar-tensorial theories. Finally, some examples are studied and the comparison of $f\left( R,\nabla R,…,\nabla^{n}R\right)$ theories to $f\left( R,\Box R,…\Box^{n}R\right)$ theories are performed.

R. Cuzinatto, C. Melo, L. Medeiros, et. al.
Tue, 8 Mar 16
31/83

# Electromagnetic Field in Lyra Manifold: A First Order Approach [CL]

We discuss the coupling of the electromagnetic field with a curved and torsioned Lyra manifold using the Duffin-Kemmer-Petiau theory. We will show how to obtain the equations of motion and energy-momentum and spin density tensors by means of the Schwinger Variational Principle.

R. Casana, C. Melo and B. Pimentel
Thu, 3 Mar 16
22/75

Comments: Matches version published ten years ago celebrating 100 years of Relativity. arXiv admin note: substantial text overlap with arXiv:gr-qc/0509117

# Oscillating solutions of the Vlasov-Poisson system — A numerical investigation [GA]

Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov-Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along one-parameter families of polytropic steady states we establish an Eddington-Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.

T. Ramming and G. Rein
Fri, 26 Feb 16
68/68

# On The Big Bang Singularity in $k=0$ FLRW Cosmologies [CL]

In this brief paper, we consider the dynamics of a spatially flat FLRW spacetime with a positive cosmological constant and matter obeying a barotropic equation of state. By performing a change of variables on the Raychaudhuri equation, we are able to compactify the big bang singularity to a finite point. We then use Chetaev’s instability theorem to prove that such a model is always past asymptotic to a big bang singularity assuming only the weak energy condition, which is more general than the strong energy condition used in the classical singularity theorems of cosmology.

I. Kohli
Tue, 9 Feb 16
23/63

# Geometric dark energy traversable wormholes constrained by astrophysical observations [CL]

In this letter, we introduce the astrophysical observations into the wormhole research, which is not meant to general parameters constraints for the dark energy models, in order to understand more about in which stage of the universe evolutions wormholes may exist through the investigation of the evolution behavior of the cosmic equation of state parameter. As a concrete instance, we investigate the Ricci dark energy (RDE) traversable wormholes constrained by astrophysical data-sets. Particularly, we can discover from Fig. \ref{fig5} of the present work, when the effective equation of state parameter $\omega_X<-1$, namely, the Null Energy conditions (NEC) is violated clearly, the wormholes will appear (open). Subsequently, six specific solutions of static and spherically symmetric traversable wormhole supported by the RDE are obtained. Except for the case of constant redshift function, in which the solution is not only asymptotically flat but also traversable, the remaining five solutions are all not asymptotically flat, therefore, the exotic matter from the RDE fluids is spatially distributed in the vicinity of the throat. Furthermore, we analyze the physical characteristics and properties of the RDE traversable wormholes. It is worth noting that, through the astrophysical observations, we get constraints on the parameters of RDE model, explore the type of exotic RDE fluids in different stages of the universe changing, limit the number of available models for wormhole research, reduce the number of the wormholes corresponding to different parameters for RDE model and provide a more apparent picture for wormhole investigations from the new perspective of observational cosmology background

D. Wang and X. Meng
Tue, 2 Feb 16
32/68

# Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics [CL]

The paper describes the unique geometric properties of ideal magnetohydrodynamics (MHD), and demonstrates how such features are inherited by extended MHD models, which incorporate two-fluid effects. The helicities and other geometric expressions for these models are presented in a topological context, emphasizing their universal features. Some of the results presented include: the generalized Kelvin circulation theorems, the existence of two Lie-dragged 2-forms, and two concomitant helicities (which can be studied via the Jones polynomial from Chern-Simons theory). The ensuing commonality is traced to the existence of an underlying Hamiltonian structure for all the extended MHD models, exemplified by the presence of a unique noncanonical Poisson bracket, and its associated energy.

M. Lingam, G. Miloshevich and P. Morrison
Tue, 2 Feb 16
57/68

# Causal Nature and Dynamics of Trapping Horizons in Black Hole Collapse and Cosmology [CL]

In calculations of gravitational collapse to form black holes, trapping horizons (foliated by marginally trapped surfaces) make their first appearance either within the collapsing matter or where it joins on to a vacuum exterior. Those which then move outwards with respect to the matter have been proposed for use in defining black holes, replacing the global concept of an “event horizon” which has some serious drawbacks for practical applications. We focus here on studying the properties of trapping horizons within spherical symmetry (which gives some simplifications while retaining the most essential general features). Their locations are then given by exactly the same condition ($R=2M$) as for the event horizon in the vacuum Schwarzschild metric, and the same condition also applies for cosmological trapping horizons. We have investigated the causal nature of these horizons (i.e. whether they are spacelike, timelike or null), making contact with the Misner-Sharp formalism, which has often been used for numerical calculations of spherical collapse. We follow two different approaches, one using a geometrical quantity $\alpha$ and the other using the horizon velocity measured with respect to the collapsing (or expanding) matter. Simple expressions are found for each of these in terms of local fluid parameters, and the connection between them allows a full description of the possible behaviours, depending on the initial density profile and the equation of state. After revisiting the FLRW universe model and the pressureless Oppenheimer-Snyder collapse model in the light of this, we have carried out numerical simulations for stellar collapse with non-zero pressure, making contact with pioneering calculations from the 1960s where some features of the emergence and subsequent behaviour of trapping horizons could already be seen.

A. Helou, I. Musco and J. Miller
Mon, 25 Jan 16
6/56

Comments: 29 pages, 11 figures, to be submitted to Physical Review D

# On the astrodynamics applications of Weierstrass elliptic and related functions [EPA]

Weierstrass elliptic and related functions have been recently shown to enable analytical explicit solutions to classical problems in astrodynamics. These include the constant radial acceleration problem, the Stark problem and the two-fixed center (or Euler’s) problem. In this paper we review the basic technique that allows for these results and we discuss the limits and merits of the approach. Applications to interplanetary trajectory design are then discussed including low-thrust planetary fly-bys and the motion of an artificial satellite under the influence of an oblate primary including $J_2$ and $J_3$ harmonics.

D. Biscani
Wed, 20 Jan 16
30/58

Comments: Presented at the AAS/AIAA Space Flight Mechanics Meeting, Napa, CA in February 14, 2016

# Evolution and Dynamics of a Matter creation model [CL]

In the flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) geometry, we consider the expansion of the universe powered by the gravitationally induced adiabatic’ matter creation. To demonstrate how matter creation works well with the expanding universe, we have considered a general creation rate and analyzed this rate in the framework of dynamical analysis. The dynamical analysis hints the presence of a non-singular universe (without the big bang singularity) with two successive accelerated phases, one at the very early phase of the universe (i.e., inflation), and the other one describes the current accelerating universe, where this early, late accelerated phases are associated with an unstable fixed point (i.e., repeller) and a stable fixed (attractor) points, respectively. We have described this phenomena by analytic solutions of the Hubble function and the scale factor of the FLRW universe. Using Jacobi Last multiplier method, we have found a Lagrangian for this matter creation rate describing this scenario of the universe. To match with our early physics results, we introduce an equivalent dynamics driven by a single scalar field and discussed the associated observable parameters compared them with the latest PLANCK data sets. Then introducing the teleparallel modified gravity, we have established an equivalent gravitational theory in the framework of matter creation. Further, introducing an equivalence between matter creation and decaying vacuum, we have found an equivalent decaying vacuum model. Finally, we have discussed a model independent test, cosmography, for the present matter creation model.

S. Pan, J. Haro, A. Paliathanasis, et. al.
Mon, 18 Jan 16
13/50

# Scale dynamical origin of modification or addition of potential in mechanics. A possible framework for the MOND theory and the dark matter [CL]

Using our mathematical framework developed in \cite{cresson-pierret_scale} called \emph{scale dynamics}, we propose in this paper a new way of interpreting the problem of adding or modifying potentials in mechanics and specifically in galactic dynamics. An application is done for the two-body problem with a Keplerian potential showing that the velocity of the orbiting body is constant. This would explain the observed phenomenon in the flat rotation curves of galaxies without adding \emph{dark matter} or modifying Newton’s law of dynamics.

F. Pierret
Thu, 7 Jan 16
35/36

# On the Hojman conservation quantities in Cosmology [CL]

We discuss the application of the Hojmans Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojman’s method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noether’s Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like $f(R)$ gravity, the application of Hojman’s method provide us with the same results with that of Noether’s theorem. Moreover we study the special Ansatz. $\phi\left( t\right) =\phi\left( a\left( t\right) \right)$, which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of the Noether conservation law of momentum for the free particle. On the other hand, for $f(T)$ teleparallel gravity, it is not the existence of Hojman’s conservation laws which provide us with the special function form of $f(T)$ functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.

A. Paliathanasis, P. Leach and S. Capozziello
Tue, 5 Jan 16
64/104

# Closed-form solutions of the Wheeler-DeWitt equation in a scalar-vector field cosmological model by Lie symmetries [CL]

We apply as selection rule to determine the unknown functions of a cosmological model the existence of Lie point symmetries for the Wheeler-DeWitt equation of quantum gravity. Our cosmological setting consists of a flat Friedmann-Robertson-Walker metric having the scale factor $a(t)$, a scalar field with potential function $V(\phi)$ minimally coupled to gravity and a vector field of its kinetic energy is coupled with the scalar field by a coupling function $f(\phi)$. Then, the Lie symmetries of this dynamical system are investigated by utilizing the behavior of the corresponding minisuperspace under the infinitesimal generator of the desired symmetries. It is shown that by applying the Lie symmetry condition the form of the coupling function and also the scalar field potential function may be explicitly determined so that we are able to solve the Wheeler-DeWitt equation. Finally, we show how we can use the Lie symmetries in order to construct conservation laws and exact solutions for the field equations.

A. Paliathanasis and B. Vakili
Fri, 18 Dec 15
11/70

Comments: 14 pages, 2 figures; to appear in Gen. Rel. Grav

# Testing modified gravity and no-hair relations for the Kerr-Newman metric through quasi-periodic oscillations of galactic microquasars [CL]

We construct multipole moments for stationary, asymptotically flat, spacetime solutions to higher-order curvature theories of gravity. The moments are defined using $3+1$ techniques involving timelike Killing vector constructions as in the classic papers by Geroch and Hansen. Using the fact that the Kerr-Newman metric is a vacuum solution to a particular class of $f(R)$ theories of gravity, we compute all its moments, and find that they admit recurrence relations similar to those for the Kerr solution in general relativity. It has been proposed previously that modelling the measured frequencies of quasi-periodic oscillations from galactic microquasars enables experimental tests of the no-hair theorem. We explore the possibility that, even if the no-hair relation is found to break down in the context of general relativity, there may be an $f(R)$ counterpart that is preserved. We apply the results to the microquasars GRS $1915$+$105$ and GRO J$1655$-$40$ using the diskoseismology and kinematic resonance models, and constrain the spins and `charges’ [which are not really electric charges in the $f(R)$ context] of their black holes.

A. Suvorov and A. Melatos
Wed, 9 Dec 15
33/63

Comments: 14 pages, 5 figures; Accepted for publication in PRD

# Non-chaotic evolution of triangular configuration due to gravitational radiation reaction in the three-body problem [CL]

Continuing work initiated in an earlier publication [H. Asada, Phys. Rev. D {\bf 80}, 064021 (2009)], the gravitational radiation reaction to Lagrange’s equilateral triangular solution of the three-body problem is investigated in an analytic method. The previous work is based on the energy balance argument, which is sufficient for a two-body system because the number of degrees of freedom (the semi-major axis and the eccentricity in quasi-Keplerian cases for instance) equals to that of the constants of motion such as the total energy and the orbital angular momentum. In a system with three (or more) bodies, however, the number of degrees of freedom is more than that of the constants of motion. Therefore, the present paper discusses the evolution of the triangular system by directly treating the gravitational radiation reaction force to each body. The perturbed equations of motion are solved by using the Laplace transform technique. It is found that the triangular configuration is adiabatically shrinking and keeps to be in equilibrium with increasing the orbital frequency due to the radiation reaction if the mass ratios satisfy the Newtonian stability condition. Long-term stability involving the first post-Newtonian corrections is also discussed.

Fri, 4 Dec 15
45/64

# Derivation of the Hall and Extended Magnetohydrodynamics Brackets [CL]

There are several plasma models intermediate in complexity between ideal magnetohydrodynamics (MHD) and two-fluid theory, with Hall and Extended MHD being two important examples. In this paper we investigate several aspects of these theories, with the ultimate goal of deriving the noncanonical Poisson brackets used in their Hamiltonian formulations. We present fully Lagrangian actions for each, as opposed to the fully Eulerian, or mixed Eulerian-Lagrangian, actions that have appeared previously. As an important step in this process we exhibit each theory’s two advected fluxes (in analogy to ideal MHD’s advected magnetic flux), discovering also that with the correct choice of gauge they have corresponding Lie-dragged potentials resembling the electromagnetic vector potential, and associated conserved helicities. Finally, using the Euler-Lagrange map, we show how to derive the noncanonical Eulerian brackets from canonical Lagrangian ones.

E. DAvignon, P. Morrison and M. Lingam
Fri, 4 Dec 15
62/64

# Bifurcation sequences in the symmetric 1:1 Hamiltonian resonance [CL]

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under $Z_2 \times Z_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.

A. Marchesiello and G. Pucacco
Thu, 3 Dec 15
65/65

Comments: 36 pages, 10 figures, accepted on International Journal of Bifurcation and Chaos. arXiv admin note: substantial text overlap with arXiv:1401.2855

# Quasi-local approach to general universal horizons [CL]

Theories of gravity with a preferred foliation usually display arbitrarily fast signal propagation, changing the black hole definition. A new inescapable barrier, the universal horizon, has been defined and many static and spherically symmetric examples have been studied in the literature. Here, we translate the usual definition of universal horizon in terms of an optical scalar built with the preferred flow defined by the preferred spacetime foliation. The new expression have the advantage of being of quasi-local nature and not depend on specific spacetime symmetries to be well defined. Therefore, we propose it as a definition for general quasi-local universal horizons. We also to give a general (peeling) surface gravity definition for general spacetimes. Using the new formalism we show that there are no universal analog of cosmological horizons for FLRW models, for any scale factor function and we also state that quasi-local universal horizons are restricted to trapped regions of the spacetime. We analyze the evolution of the universal horizon area under simplifying assumptions and we conclude with our view on the next steps for the understanding of black holes in non relativistic gravity theories.

A. Maciel
Mon, 30 Nov 15
75/78