An important open question in cosmology is the degree to which Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) solutions of Einstein's equations are able model large-scale behavior locally inhomogeneous observable universe. We investigate this problem by considering a range exact n-body constraint equations. These contain discrete masses, and so allow arbitrarily large density contrasts be modeled. restrict our study regularly arranged distributions masses topological 3-spheres. This has...

{\it Fermi Gamma-ray Space Telescope} observations of GRB110721A have revealed two emission components from the relativistic jet: photosphere, peaking at $\sim 100$ keV and a non-thermal component, which peaks 1000$ keV. We use photospheric component to calculate properties outflow. find strong evolution in flow properties: Lorentz factor decreases with time during bursts $\Gamma \sim 150$ (assuming redshift $z=2$; values are only weakly dependent on unknown efficiency parameters). Such...

We study the effects of inhomogeneities on evolution Universe, by considering a range cosmological models with discretized matter content. This is done using exact and fully relativistic methods that exploit symmetries in about submanifolds spacetimes themselves possess no continuous global symmetries. These allow us to follow our throughout their entire history, far beyond what has previously been possible. find while some space-like curves collapse anisotropic singularities finite time,...

Spacetimes admitting a four-dimensional transitive similarity group are studied. It is shown that when the homogeneous hypersurfaces (which necessarily exist) spacelike (spatially case) then connection components in Lorentz frame adapted to spatially slicing proportional t-1 where t proper time of slices. An analogous statement holds those slices timelike except coordinate case. The transitively self-similar universes correspond exactly exact power law solutions Wainwright's terminology,...

The search for solutions of Einstein's equations representing relativistic cosmological models with a discrete matter content has been remarkably fruitful in the last decade. In this review we discuss progress made study specific subclass cosmologies, black-hole lattice models. particular, illustrate techniques used construction these spacetimes, and examine their resulting physical properties. This includes large-scale dynamics, dressing mass due to interaction between individual black...

A framework is introduced which explains the existence and similarities of most exact solutions Einstein equations with a wide range sources for class hypersurface-homogeneous spacetimes admit Hamiltonian formulation. This includes spatially homogeneous cosmological models astrophysically interesting static spherically symmetric as well stationary cylindrically models. The involves methods finding exploiting hidden symmetries invariant submanifolds formulation field equations. It unifies,...

We examine the effect that magnetic part of Weyl tensor has on large-scale expansion space. This is done within context a class cosmological models contain regularly arranged discrete masses, rather than continuous perfect fluid. The natural set geodesic curves one should use to consider these requires existence non-zero tensor. include this object in evolution equations by performing Taylor series about hypersurface where it initially vanishes. At same time, measured as fraction age...

It is shown that the zilch conservation law arises as Noether current corresponding to a variational symmetry of duality-symmetric Maxwell Lagrangian. The action generator on Lagrangian, while non-vanishing, total divergence required by theory. nature was previously known only for some components tensor, notably optical chirality. By contrast, our analysis fully covariant and is, therefore, valid all tensor. presented here both real complex versions Lagrangians.

Two-dimensional Hamiltonian systems admitting second invariants which are quartic in the momenta investigated using Jacobi geometrization of dynamics. This approach allows for a unified treatment at both arbitrary and fixed energy. In differential geometric picture, invariant corresponds to existence fourth rank Killing tensor. Expressing metric terms Kahler potential, integrability condition tensor energy is non-linear equation involving potential. At energy, further conditions must be...

We investigate integrable two-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In case of a invariant, we provide some examples weakly systems. recover classical strongly Cartesian polar coordinates new parabolic elliptical coordinates.

We consider a family of cosmological models in which all mass is confined to regular lattice identical black holes. By exploiting the reflection symmetry about planes that bisect these lattices into halves, we are able evolution number geometrically distinguished surfaces exist within each them. find equations for symmetric can be written as simple set Friedmann-like equations, with source terms behave like interacting effective fluids. then show gravitational waves effectively trapped small...

The existence of Killing vectors and tensors in two-dimensional space-times is discussed. corresponding mechanical problem having a potential with two exponential terms analyzed detail number cases admitting are given. give rise to new exact solutions perfect fluid Bianchi Kantowski–Sachs cosmologies as well inflationary models scalar field source.

Spatially homogeneous and non-exceptional spatially self-similar spacetime metrics which are 'exact power law metrics' defined, explicitly parametrised shown to have fixed conformal 3-geometry in the natural slicing of by orbits symmetry group admit a homothetic Killing vector field not tangent that slicing. In fact exact exactly those spacelike homogeneity or self-similarity group. Such arise as 'singular point solutions' gravitational equations when formulated certain system first order...

Cubic invariants for two-dimensional Hamiltonian systems are investigated using the Jacobi geometrization procedure. This approach allows a unified treatment of at both fixed and arbitrary energy. In geometric picture invariant generally corresponds to third rank Killing tensor, whose existence energy value forces metric satisfy nonlinear integrability condition expressed in terms Kähler potential. Further conditions, leading system equations which is overdetermined except singular cases,...

In dynamic spacetimes in which asymmetric gravitational collapse/expansion is taking place, the timelike geodesic equation appears to exhibit an interesting property: Relative collapsing configuration, free test particles undergo ``acceleration'' and form a double-jet configuration parallel axis of collapse. We illustrate this aspect peculiar motion simple spatially homogeneous cosmological models such as Kasner spacetime. To estimate effect spatial inhomogeneities on cosmic jets, geodesics...

Invariants at arbitrary and fixed energy (strongly weakly conserved quantities) for two-dimensional Hamiltonian systems are treated in a unified way. This is achieved by utilizing the Jacobi metric geometrization of dynamics. Using Killing tensors we obtain an integrability condition quadratic invariants which involves analytic function S(z). For S(z) second-degree polynomial with real second derivative. The then reduces to Darboux's energy. four types classical positive-definite...

The reparametrization freedom in the choice of time variable dynamics spatially homogeneous cosmological models is used to reformulate field equations as a geodesic flow for "Jacobi geometry" particular gauge called Jacobi gauge. For diagonalizable this geometry conformally flat Lorentzian geometry. By choosing variables which are adapted symmetries geometry, considerable simplification achieved, and one can explain existence all known exact solutions terms analysis, well simplify study...

A new formulation of Carter's constant for geodesic motion in Kerr black holes is given. It shown that corresponds to the total angular momentum plus a precisely defined part which quadratic linear momenta. The characterization exact weak field limit obtained by letting gravitational go zero. suggested form can be useful current studies dynamics extreme mass ratio inspiral (EMRI) systems emitting radiation.

We discuss ultracompact stellar objects which have multiple necks in their optical geometry. There are fact physically reasonable equations of state for the number can be arbitrarily large. The proofs these statements rely on a recent regularized formulation field static spherically symmetric models due to Nilsson and Uggla. particular equation p=\rho-\rho_s plays central role this context.