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In 1911, Jüttner proposed the generalization, for a relativistic gas, of the Maxwell–Boltzmann distribution of velocities. Here, we want to discuss, among others, the Jüttner probability density function (PDF). Both the velocity space and, consequently, the momentum space are not flat in special relativity. The velocity space corresponds to the Lobachevsky one, which has a negative curvature. This curvature induces a specific power for the Lorentz factor in the PDF, affecting the Jüttner normalization constant in one, two, and three dimensions. Furthermore, Jüttner distribution, written in terms of a more convenient variable, the rapidity, presents a curvature change at the origin at sufficiently high energy, which does not agree with our computational dynamics simulations of a relativistic gas. However, in one dimension, the rapidity satisfies a simple additivity law. This allows us to obtain, through the central limit theorem, a new, Lorentz-invariant, PDF whose curvature at the origin does not change for any energy value and which agrees with our computational dynamics simulations data. Also, we perform extensive first-principle simulations of a one-dimensional relativistic gas constituted by light and heavy particles.

The spacetime of a boosted Bondi-Sachs rotating black hole is considered as a proper background to examine electromagnetic configurations connected to analytic solutions of Maxwell equations. In our analysis, we first use the Bondi-Sachs transformations in order to bring the boosted rotating black hole metric into the Kerr-Schild form, from which zero angular momentum observers (ZAMOs) are constructed via the ADM formalism. In Kerr-Schild coordinates, we obtain the Killing fields as sources of Maxwell electrodynamics, and we fix a ZAMO in order to evaluate the components of the electric and magnetic fields, from which we obtain nonsingular patterns of an eventual momentum-energy emission of a boosted Kerr-Schild black hole. Distinct patterns are examined and discussed in the case of variations of the boost parameter $\ensuremath{\gamma}$. We extend our analysis by considering the nonsingular electromagnetic emission in the framework of a boosted Bondi-Sachs rotating black hole, as it moves at relativistic speeds. We also discuss possible mechanisms that may resemble magnetospheres of rotating boosted black holes and give rise to hydromagnetic flows from accretion discs and to the production of jets.

A solution of Einstein's vacuum field equation is derived that describes a general boosted Kerr black hole relative to a Lorentz frame at future null infinity. The metric contains five independent parameters -- mass $m$, rotation $\omega$, boost parameter $v/c$ and the boost direction defined by $(n_1,n_2,n_3)$ satisfying $(n_1)^2+(n_2)^2+(n_3)^2=1$ -- and reduces to the Kerr black hole when the boost parameter is zero and $n_1=1$. The solution describes the most general configuration that an astrophysical black hole must have. The black hole rotates about the $z$-axis with angular momentum proportional to $m \omega$ and the geometry has just one Killing vector $\partial/\partial{u}$, where $u$ is the retarded time coordinate. The boost turns the ergosphere asymmetric, with dominant lobes in the direction opposite to the boost. The event and Cauchy horizons, defined for the case $\omega < m$, are specified respectively by the radii $r_{\pm}=m \pm \sqrt{m^2-\omega^2}$. The horizons are topologically spherical and the singularity has the topology of a circle on planes that are orthogonal to the boost direction. We argue that this black hole geometry is the natural set to describe the remnants of the recently observed gravitational wave events $GW150914$, $GW151226$, $GW170814$ and $GW170817$\cite{gw1,gw2,gw3,gw4}. In the conclusions we discuss possible astrophysical processes in the asymmetric ergosphere and the electromagnetic dynamical effects that may result from the rotating black hole moving at relativistic speeds together with the precession of the boost axis about the rotation axis of the black hole., Comment: 7 pages, 8 figures. arXiv admin note: text overlap with arXiv:1608.01174; typing mistakes in eq.(16), eq.(19) and eq.(27) of previous version were corrected; new version with extended manuscript and new title; typing corrections; accepted for publication in Phys. Rev. D, April 8, 2019

We construct a FLRW universe considering an anisotropic scaling between space and time at extremely high and low energies only. In this context, Friedmann equations contain an additional term arising from spatial curvature which implements nonsingular bounces in the early Universe. The matter content of the model is a nonrelativistic pressureless perfect fluid and radiation. By breaking covariance diffeomorphism also at extreme large scales, an additional term furnishes late-time acceleration due to spatial curvature so that a cosmological constant is not needed. In order to probe the final fate of the universe we also introduce a lower order curvature term which dominates in deep IR. Given the observational parameters we obtain a concrete model in eternal recurrence in which the end of late-time acceleration takes place at a redshift $z \simeq -0.14$ and the universe recollapses at $z\simeq - 0.32$., 8 pages, 6 figures. arXiv admin note: text overlap with arXiv:1710.06911

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies in Ho\v{r}ava-Lifshitz gravity. The $6$-dim phase space presents two critical points, one asymptotic de Sitter attractor at infinity and a $2$-dim invariant plane. We identified four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces. The center with multiplicity two engenders in its neighborhood the topology of stable and unstable cylinders $R \times S^3$ of orbits. We show that the stable and unstable cylinders coalesce realizing a smooth homoclinic connection to the center manifold, a rare event of regular/non-chaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity two in the domain $B$ engenders a high instability in the dynamics so that the cylinders emerging from the center manifold towards the bounce have four distinct attractors: the center manifold itself, the de Sitter attractor at infinity and two further momentum-dominated attractors with infinite anisotropy. In the domain $C$ we examine the features of invariant manifolds of orbits about a saddle of multiplicity two. The presence of the saddle of multiplicity two engenders bifurcations of the invariant manifold as the energy $E_0$ of the system increases relative to the energy $E_{cr}$ providing structures that were not yet observed in the literature. The domain $D$ is not examined as most of its features are present already in the previous domains., Comment: Accepted for publication in Physical Review D

We show that gravitational wave radiative patterns from a point test particle falling radially into a Schwarzschild black hole, as derived by Davis, Ruffini, Press and Price [M. Davis et al., Phys. Rev. Lett. 27, 1466 (1971).], are present in the nonlinear regime of head-on mergers of black holes. We use the Bondi-Sachs characteristic formulation and express the gravitational wave luminosity and the net momentum flux in terms of the news functions. We then evaluate the ($\ensuremath{-}2$)-spin-weighted $\ensuremath{\ell}$-multipole decomposition of these quantities via exact expressions valid in the nonlinear regime and defined at future null infinity. Our treatment is made in the realm of Robinson-Trautman dynamics, with characteristic initial data corresponding to the head-on merger of two black holes. We consider mass ratios in the range $0.01\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1$. We obtain the exponential decay with $\ensuremath{\ell}$ of the total energy contributed by each multipole $\ensuremath{\ell}$, with an accurate linear correlation in the log-linear plot of the points up to $\ensuremath{\alpha}\ensuremath{\simeq}0.7$. Above this mass ratio the contribution of the odd modes to the energy decreases faster than that of the even modes, leading to the breaking of the linear correlation; for $\ensuremath{\alpha}=1$ the energy in all odd modes is zero. The dominant contribution to the total radiated energy comes from the quadrupole mode $\ensuremath{\ell}=2$ corresponding, for instance, to about $\ensuremath{\simeq}84%$ for small mass ratios up to $\ensuremath{\simeq}99.8%$ for the limit case $\ensuremath{\alpha}=1$. The total rescaled radiated energy ${E}_{W}^{\text{total}}/{m}_{0}{\ensuremath{\alpha}}^{2}$ decreases linearly with decreasing $\ensuremath{\alpha}$, yielding for the point particle limit $\ensuremath{\alpha}\ensuremath{\rightarrow}0$ the value $\ensuremath{\simeq}0.0484$, about 5 times larger than the result of Davis et al. [1]. The mode decomposition of the net momentum flux and of the associated gravitational wave impulses results in an adjacent-even-odd mode-mixing pattern. We obtain that the impulses contributed by each $(\ensuremath{\ell},\ensuremath{\ell}+1)$ mixed mode also accurately satisfy the exponential decay with $\ensuremath{\ell}$, for the whole mass ratio domain considered, $0.01\ensuremath{\le}\ensuremath{\alpha}l1$. The (2, 3) mode contributions to the total impulses are dominant. The mode-mixing effect can also be seen in the decomposition of the net kick velocity imparted to the system by the gravitational wave emission. The mixed mode impulses reach a maximum at $\ensuremath{\alpha}\ensuremath{\simeq}0.7$; for $\ensuremath{\alpha}g0.7$ the impulses decrease and are zero in the equal mass case, due to the decrease to zero of the odd modes of the news functions.

A solution of Einstein's vacuum field equations that describes a boosted Kerr black hole relative to an asymptotic Lorentz frame at the future null infinity is derived. The solution has three parameters (mass, rotation and boost) and corresponds to the most general configuration that an astrophysical black hole must have; it reduces to the Kerr solution when the boost parameter is zero. In this solution the ergosphere is north-south asymmetric, with dominant lobes in the direction opposite to the boost. However the event horizon, the Cauchy horizon and the ring singularity, which are the core of the black hole structure, do not alter, being independent of the boost parameter. Possible consequences for astrophysical processes connected with Penrose processes in the asymmetric ergosphere are discussed., Comment: 11 pages, 1figure

We examine numerically and analytically the problem of the relativistic velocity distribution in a 1-dim relativistic gas in thermal equilibrium. Our derivation is based on the special theory of relativity, the central limit theorem and the Lobachevsky structure of the velocity space of the theory, where the rapidity variable plays a crucial role. For v^2/c^2 << 1 and 1/��= k_B T/ m_0 c^2 << 1 the distribution tends to the Maxwell-Boltzmann distribution., 15 pages, 4 figures

We develop the canonical quantization of a midisuperspace model which contains, as a subspace, a minisuperspace constituted of a Friedman-Lema\^{\i}tre-Robertson-Walker Universe filled with homogeneous scalar and dust fields, where the sign of the intrinsic curvature of the spacelike hypersurfaces of homogeneity is not specified, allowing the study of topology change in these hypersurfaces. We solve the Wheeler-DeWitt equation of the midisuperspace model restricted to this minisuperspace subspace in the semi-classical approximation. Adopting the conditional probability interpretation, we find that some of the solutions present change of topology of the homogeneous hypersurfaces. However, this result depends crucially on the interpretation we adopt: using the usual probabilistic interpretation, we find selection rules which forbid some of these topology changes., Comment: 23 pages, LaTex file. We added in the conclusion some comments about path integral formalism and corrected litle misprintings

We examine the dynamics of scalar perturbations in closed Friedmann-Lema\^itre-Robertson- Walker (FLRW) universes in the framework of Brane World theory with a timelike extra dimension. In this scenario, the unperturbed Friedmann equations contain additional terms arising from the bulk-brane interaction that implement non-singular bounces in the models with a cosmological constant and non-interacting perfect fluids. The structure of the phase-space of the models allows for two basic configurations, namely, one bounce solutions or eternal universes. Assuming that the matter content of the model is given by dust and radiation, we derive the dynamical field equations for scalar hydrodynamical perturbations considering either a conformally flat (de Sitter) bulk or a perturbed bulk. We perform a numerical analysis which can shed some light on the study of cosmological scalar perturbations in bouncing brane world models. From a mathematical point of view we show that although the bounce enhances the amplitudes of scalar perturbations for one bounce models in the case of a de Sitter bulk, the amplitudes of the perturbations remain sufficiently small and bounded relative to the background values up to a certain scale. For one bounce models in the case of a perturbed bulk the amplitudes of all perturbations (apart from the Weyl fluid energy density) remain sufficiently small and bounded relative to the background values for any scale of the perturbations. We also discuss and compare the stability and bounded behaviour of the perturbations in the late accelerated phase of one bounce solutions. For eternal universes we argue that some of these features are maintained only for early times (typically of the order of the first bounce). In this sense we show that eternal solutions are highly unstable configurations considering the background model of this paper., Comment: 15 pages, 29 figures. Accepted for publication in Physical Review D

We examine the nonlinear dynamics of a closed Friedmann-Robertson-Walker universe in the framework of Brane World formalism with a timelike extra dimension. In this scenario, the Friedmann equations contain additional terms arising from the bulk-brane interaction which provide a concrete model for nonsingular bounces in the early phase of the Universe. We construct a nonsingular cosmological scenario sourced with dust, radiation and a cosmological constant. The structure of the phase space shows a nonsingular orbit with two accelerated phases, separated by a smooth transition corresponding to a decelerated expansion. Given observational parameters we connect such phases to a primordial accelerated phase, a soft transition to Friedmann (where the classical regime is valid), and a graceful exit to a de Sitter accelerated phase.

In the framework of braneworld formalism, we examine the dynamics of a Bianchi IX model with three scale factors on a 4-dim Lorentzian brane embedded in a 5-dim conformally flat empty bulk with a timelike extra dimension. The matter content is a pressureless perfect fluid restricted to the brane. In this scenario Einstein’s equations on the brane reduces to a 6-dim Hamiltonian dynamical system with additional terms – due to the bulk-brane interaction – that avoid the singularity and implement nonsingular bounces in the early phase of the universe. Due to an effective cosmological constant on the brane the phase space of the model presents two critical points (a saddle-center-center and a center-centercenter) in a finite region of phase space, and two asymptotic de Sitter critical points at infinity, one acting as an attractor to late-time acceleration dynamics. The critical points belong to a 2-dim invariant plane; together they organize the dynamics of the phase space. The center-center-center critical point corresponds to a stable Einstein universe configuration with perpetually oscillatory orbits in its neighborhood. On the other hand the saddle-center-center engenders in the phase space the topology of stable and unstable 4-dim cylinders R × S 3 , where R is a saddle direction and S 3 is the center manifold of unstable periodic orbits

We construct a spacetime whose only source of curvature is a Dirac magnetic monopole, and whose geometry inherits the structure of lines of singularities of the monopole electromagnetic potentials. The spacetime has the topology S³ × R, is stationary and asymptotically at but not asymptotically Minkowskian, with its at null infinity having the topology of S³. These mild pathologies, as acausality and string structure, allow the spacetime configuration to have a gravitational magnetic mass, which results proportional to the charge mu of the monopole. This suggests that the Dirac monopole may be the source of magnetic mass in gravitational configurations, which has no Newtonian analogue. Alsomu has the role of a NUT parameter in the metric of the spacetime, suggesting that the charge of the monopole can provide a physical realization of the NUT parameter.

We explicitly calculate the Green functions describing quantum changes of topology in Friedman-Lemaitre-Robertson-Walker Universes whose spacelike sections are compact but endowed with distinct topologies. The calculations are performed using the long wavelength approximation at second order in the gradient expansion. We argue that complex metrics are necessary in order to obtain a non-vanishing Green functions and interpret this fact as demonstrating that a quantum topology change can be viewed as a quantum tunneling effect. We demonstrate that quantum topological transitions between curved hypersurfaces are allowed whereas no transition to or from a flat section is possible, establishing thus a selection rule. We also show that the quantum topology changes in the direction of negatively curved hypersurfaces are strongly enhanced as time goes on, while transitions in the opposite direction are suppressed., 11 pages, 2 figures, reshaped version