Sitter Metric Research Articles

On non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics

We prove that the $TT$-gauge-fixed linearised Einstein operator is non-degenerate for Riemannian Kottler (Schwarzschild-anti de Sitter) metrics with dimension- and topology-dependent ranges of mass parameter. We provide evidence that this remains true for all such metrics except the spherical ones with a critical mass.

PARABOLIC CLASSICAL CURVATURE FLOWS

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Large scale magnetic fields from torsion modes and massive photon inflation

Previously, Barrow and Tsagas (2008 Phys. Rev. D 77 107302) showed that a slower decay of magnetic fields are present in open Friedmann universes, with traditional Maxwell equations. In their paper magnetic fields of the order of , which are far below the value required to seed galactic dynamos, were obtained. In this paper, galactic dynamo seeds of the order of are obtained from massive electrodynamics in an Einstein–Cartan–Proca expanding universe of de Sitter type. Slow decay of magnetic fields in photon–torsion coupling in quantum electrodynamics (Garcia de Andrade 2011 Phys. Lett. B 468 28) have been recently shown by the author Garcia de Andrade (2012 Phys. Lett. B 711 143) to also not be able to seed galactic dynamos. Torsion modes are constrained by the field equations. Spacetime torsion is shown to be explicitly responsible for the slow decay of a cosmic magnetic field. In the absence of massive photon torsion coupling the magnetic field decay is of the order , when torsion turns on . The pure massive-photon–torsion contribution amplifies the magnetic field by which characterizes an extremely slow magnetic dynamo action due to purely torsion gravitational effects. Recently Barrow, Tsagas and Yamamoto (2012 Phys. Rev. D 86 023535) have obtained superadiabatic amplification of B-fields in Friedmann open cosmology which lies within and which falls very comfortably within the limits to seed galactic dynamos. The are other simple solutions where a B-field decays as , a relatively weak photon–torsion coupling approximation. These solutions are obtained for de Sitter and Friedmann metrics. Numerical values as displayed in this new version of the paper specifically for GUT phases of inflation with and without massive photons; without photons we obtain the well known value of GR which is while for the values with massive photons one obtains . At the present time one obtains and with massive photons one obtains , which is unfortunately much stronger than the value obtained by astronomical observations of .

Astronomical time-of-flight photon speedometer.

A dual-band, fiber-optic, photon time-of-flight instrument was developed. Its design was optimized for measuring the velocity of visible photons emanating from relatively dim astronomical sources (apparent magnitude m>12), such as distant galaxies and quasars. We report the first direct photon group velocity measurements for extragalactic objects. The photon group velocity is found to be 3.00±0.03×108 ms-1 and is invariant, within experimental error, over the range of redshifts measured (0≤z≤1.33). This measurement provides additional validation of general relativity and is consistent with the Friedmann-Lemaître-Robertson-Walker and hyperbolic anti-de Sitter metrics but not with the elliptical de Sitter metric.

Yet another family of diagonal metrics for de Sitter and anti–de Sitter spacetimes

In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a 1+3 foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a 2+2 foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant Lambda and two discrete parameters +1,0,-1 related to the curvature of the 2-spaces. The class admits 3 distinct subcases for Lambda>0 and 8 subcases for Lambda<0. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of B-metrics as exact gravitational fields of a tachyon.

Five-dimensional spherical gravitational collapse of anisotropic fluid with cosmological constant

Our aim is to study five-dimensional spherically symmetric anisotropic collapse with a positive cosmological constant (PCC). For this purpose, five-dimensional spherically symmetric and Schwarzschild–de Sitter metrics are chosen in the interior and exterior regions respectively. A set of junction conditions is derived for the smooth matching of interior and exterior spacetimes. The apparent horizon is calculated and its physical significance is studied. It comes out that the whole collapsing process is influenced by the cosmological constant. The collapsing process under the influence of cosmological constant slows down and black hole size also reduced.

Nonequilibrium Dynamics of the σ-Model Modes on the de Sitter Space

The two-dimensional σ-model with the de Sitter target space is locally canonic in the north pole diamond of the Penrose diagram in the cosmological gauge. The left and right moving modes on the cylindrical base space are entangled among themselves and interact with the de Sitter metric. Firstly, we show that the untangled oscillators can be obtained from the entangled operators by applying a set of Bogoliubov transformations constrained by the requirement that the partial evolution generator be diagonal. Secondly, we determine the nonequilibrium dynamics of the untangled modes in the nonequilibrium thermofield dynamics formalism. The thermal modes are represented as thermal doublet oscillators that satisfy partial evolution equations of Heisenberg-type. From these we compute the local free one-body propagator of an arbitrary mode between two times. Thirdly, we discuss the field representation of the thermal modes. We show that there is a set of thermal doublet fields that satisfy the equal time canonical commutation relations, are solutions to the σ-model equations of motion, and can be decomposed in terms of thermal doublet oscillators. Finally, we construct a local partial evolution functional of Hamilton-like form for the thermal doublet fields.

Photon spheres and sonic horizons in black holes from supergravity and other theories

We study closed photon orbits in spherically-symmetric static solutions of supergravity theories, a Horndeski theory, and a theory of quintessence. These orbits lie in what we shall call a photon sphere (anti-photon sphere) if the orbit is unstable (stable). We show that in all the asymptotically flat solutions we examine that admit a regular event horizon, and whose energy-momentum tensor satisfies the strong energy condition, there is one and only one photon sphere outside the event horizon. We give an example of a Horndeski theory black hole (whose energy-momentum tensor violates the strong energy condition) whose metric admits both a photon sphere and an anti-photon sphere. The uniqueness and non-existence also holds for asymptotically anti-de Sitter solutions in gauged supergravity. The latter also exhibit the projective symmetry that was first discovered for the Schwarzschild-de Sitter metrics: the unparameterised null geodesics are the same as when the cosmological or gauge coupling constant vanishes. We also study the closely related problem of accretion flows by perfect fluids in these metrics. For a radiation fluid, Bondi's sonic horizon coincides with the photon sphere. For a general polytropic equation of state this is not the case. Finally we exhibit counterexamples to a conjecture of Hod's.

Blowup of the solution to the Cauchy problem with arbitrary positive energy for a system of Klein–Gordon equations in the de Sitter metric

The ϕ4 model of a scalar (complex) field in the metric of an expanding universe, namely, in the de Sitter metric is considered. The initial energy of the system can have an arbitrarily high positive value. Sufficient conditions for solution blowup in a finite time are obtained. The existence of blowup is proved by applying H.A. Levine’s modified method is used.

Riemann curvature of a boosted spacetime geometry

The ultrarelativistic boosting procedure had been applied in the literature to map the metric of Schwarzschild–de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface. This paper evaluates the Riemann curvature tensor of the boosted Schwarzschild–de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Thus, for the first time in the literature, the singular limit of curvature, through Dirac’s [Formula: see text] distribution and its derivatives, is numerically evaluated for this class of spacetimes. Moreover, the analysis of the Kretschmann invariant and the geodesic equation shows that the spacetime possesses a “scalar curvature singularity” within a 3-sphere and it is possible to define what we here call “boosted horizon”, a sort of elastic wall where all particles are surprisingly pushed away, as numerical analysis demonstrates. This seems to suggest that such “boosted geometries” are ruled by a sort of “antigravity effect” since all geodesics seem to refuse to enter the “boosted horizon” and are “reflected” by it, even though their initial conditions are aimed at driving the particles toward the “boosted horizon” itself. Eventually, the equivalence with the coordinate shift method is invoked in order to demonstrate that all [Formula: see text] terms appearing in the Riemann curvature tensor give vanishing contribution in distributional sense.

Generalized Uncertainty Relations, Curved Phase-Spaces and Quantum Gravity

Modifications of the Weyl-Heisenberg algebra are proposed where the classical limit corresponds to a metric in (curved) momentum spaces. In the simplest scenario, the 2D de Sitter metric of constant curvature in momentum space furnishes a hierarchy of modified uncertainty relations leading to a minimum value for the position uncertainty . The first uncertainty relation of this hierarchy has the same functional form as the stringy modified uncertainty relation with a Planck scale minimum value for at . We proceed with a discussion of the most general curved phase space scenario (cotangent bundle of spacetime) and provide the noncommuting phase space coordinates algebra in terms of the symmetric and nonsymmetric metric components of a Hermitian complex metric , such . Yang’s noncommuting phase-space coordinates algebra, combined with the Schrodinger-Robertson inequalities involving angular momentum eigenstates, reveals how a quantized area operator in units of emerges like it occurs in Loop Quantum Gravity (LQG). Some final comments are made about Fedosov deformation quantization, Noncommutative and Nonassociative gravity.

Classification of minimum global embeddings for nonrotating black holes

We consider the problem of the existence of global embeddings of metrics of spherically symmetric black holes into an ambient space with the minimal possible dimension. We classify the possible types of embeddings by the type of realization of the metric symmetry by ambient space symmetries. For the Schwarzschild, Schwarzschild-de Sitter, and Reissner-Nordstrom black holes, we prove that the known global embeddings are the only ones. We obtain a new global embedding for the Reissner-Nordstrom-de Sitter metrics and prove that constructing such embeddings is impossible for the Schwarzschild-anti-de Sitter metric. We also discuss the possibility of constructing global embeddings of the Reissner-Nordstrom-anti-de Sitter metric.

Photon regions and shadows of accelerated black holes

In an earlier paper, we have analytically determined the photon regions and the shadows of black holes of the Plebański class of metrics which are also known as the Kerr–Newman–NUT–(anti-)de Sitter metrics. These metrics are characterized by six parameters: Mass, spin, electric and magnetic charges, gravitomagnetic NUT charge and the cosmological constant. Here, we extend this analysis to the Plebański–Demiański class of metrics which contains, in addition to these six parameters, the so-called acceleration parameter. All these metrics are axially symmetric and stationary type D solutions to the Einstein–Maxwell equations with a cosmological constant. We derive analytical formulas for the photon regions (i.e. for the regions that contain spherical lightlike geodesics) and for the boundary curve of the shadow as it is seen by an observer at Boyer–Lindquist coordinates (rO, ϑO) in the domain of outer communication. Whereas all relevant formulas are derived for the whole Plebański–Demiański class, we concentrate on the accelerated Kerr metric (i.e. only mass, spin and acceleration parameter are different from zero) when discussing the influence of the acceleration parameter on the photon region and on the shadow in terms of pictures. The accelerated Kerr metric is also known as the rotating C-metric. We discuss how our analytical formulas can be used for calculating the horizontal and vertical angular diameters of the shadow and we estimate these values for the black holes at the center of our Galaxy and at the center of M87.

Energy problem in the Einstein–Cartan theory

The Einstein–Cartan–Kibble–Sciama equations are presented in the following form: the codifferential of the differential of a tetrad is the tetrad current with the ordinary conservation law. This form is an analogue of the form for non-Abelian quantum field theory. A canonical gauge tetrad is proposed for six two-directions with extreme sectional Riemannian curvature. A nonlocal conservation law for the contracted Bianchi identities is presented. The physical and geometric meaning of the Gibbons-Hawking Lagrangian and surface term are proposed. Application of the tetrad current concept is illustrated for the Schwarzschild and de Sitter metrics.

Spin field equations and Heun’s equations

The Kerr-Newman-(anti) de Sitter metric is the most general stationary black hole solution to the Einstein-Maxwell equation with a cosmological constant. We study the separability of the equations of the massless scalar (spin s=0), neutrino (s=1/2), electromagnetic (s=1), Rarita-Schwinger (s=3/2), and gravitational (s=2) fields propagating on this background. We obtain the angular and radial master equations, and show that the master equations are transformed to Heun’s equation. Meanwhile, we give the condition of existence of event horizons for Kerr-Newman-(anti) de Sitter spacetime by using Sturm theorem.

On Perturbations of the Schwarzschild Anti-De Sitter Spaces of Positive Mass

In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.

Asymptotics of Linear Waves and Resonances with Applications to Black Holes

We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|<M and M\Lambda a << 1. We prove that if the initial data is localized at frequencies \lambda >> 1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t < C log\lambda, where \nu_min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\infty)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.

Spinning massive test particles in cosmological and general static spherically symmetric spacetimes

A Lagrangian formalism is used to study the motion of a spinning massive particle in Friedmann–Robertson–Walker and Gödel spacetimes, as well as in a general Schwarzschild-like spacetime and in static spherically symmetric conformally flat spacetimes. Exact solutions for the motion of the particle and general exact expressions for the momenta and velocities are displayed for different cases. In particular, the solution for the motion in spherically symmetric metrics is presented in the equatorial plane. The exact solutions are found using constants of motion of the particle, namely its mass, its spin, its angular momentum, and a fourth constant, which is its energy when the metric is time-independent, and a different constant otherwise. These constants are associated to Killing vectors. In the case of the motion on the Friedmann–Robertson–Walker metric, a new constant of motion is found. This is the fourth constant which generalizes previously known results obtained for spinless particles. In the case of general Schwarzschild-like spacetimes, our results allow for the exploration of the case of the Reissner–Nordstrom–(Anti)de Sitter metric. Finally, for the case of the conformally flat spacetimes, the solution is explicitly evaluated for different metric tensors associated to a universe filled with static perfect fluids and electromagnetic radiation. For some combination of the values of the constants of motion the particle trajectories may exhibit spacelike velocity vectors in portions of the trajectories.

Dissipative fields in de Sitter and black hole spacetimes: Quantum entanglement due to pair production and dissipation

For free fields, pair creation in expanding universes is associated with the building up of correlations that lead to nonseparable states, i.e., quantum mechanically entangled ones. For dissipative fields, i.e., fields coupled to an environment, there is a competition between the squeezing of the state and the coupling to the external bath. We compute the final coherence level for dissipative fields that propagate in a two-dimensional de Sitter space, and we characterize the domain in parameter space where the state remains nonseparable. We then apply our analysis to (analogue) Hawking radiation by exploiting the close relationship between Lorentz violating theories propagating in de Sitter and black hole metrics. We establish the robustness of the spectrum and find that the entanglement among Hawking pairs is generally much stronger than that among pairs of quanta with opposite momenta.

HIGH ENERGY PARTICLE COLLISIONS NEAR THE BIFURCATION SURFACE

We consider generic nonextremal stationary dirty black holes. It is shown that in the vicinity of any bifurcation surface the energy of collision of two particles in the center of mass frame can grow unbound. This is a generic property that, in particular, includes collisions near the inner black hole horizon analyzed earlier by different methods. The similar results are also valid for cosmological horizons. The case of the de Sitter metric is discussed.