Stationary Metrics Research Articles

The Lorentzian scattering rigidity problem and rigidity of stationary metrics

The Lorentzian scattering rigidity problem and rigidity of stationary metrics

The molecular diversity of dissolved organic matter in forest streams across central Canadian boreal watersheds.

Small headwater streams can mobilize large amounts of terrestrially derived dissolved organic matter (DOM). While the molecular composition of DOM has important controls on biogeochemical cycles and carbon cycling, how stationary landscape metrics affect DOM composition is poorly understood, particularly in relation to non-stationary effects from hydrological changes across seasons. Here, we apply a combination of Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) and absorbance spectroscopy to characterize stream DOM from 13 diverse watersheds across the central Canadian boreal forests and statistically relate DOM compositional characteristics to landscape topography and hydrological metrics. We found that watershed runoff across different surface physiographies produced DOM with distinctly different chemical compositions related to runoff pH. Specifically, streams in sandy soil watersheds contained more abundant aromatic, nitrogenated and sulfurized fractions of DOM, likely due to a combination of lower soil capacity to absorb DOM than other soil types and high conifer forest coverage that generated acidic litterfall in more sandy watersheds. In contrast, streams with more neutral pH in watersheds with shallow soils had DOM resembling low oxidized phenolic molecules mainly due to increased brush/alder and deciduous vegetation coverage in relatively steeper watersheds. However, as precipitation and flows increased in the fall, the overall water chemistry of streams became more similar as runoff pH increased, the overall chemical diversity of DOM in streams decreased, and stream DOM resembled fresher, lower molecular weight lignin material likely originating from freshly produced leaf litter. Together, our findings show that during hydrologically disconnected periods, pH and landscape characteristics have important controls on the mobilization of aromatic DOM but that many landscape-specific characteristics in the Canadian boreal forest are less influential on DOM processing during wetter conditions where chemically similar, plant-derived DOM signatures are preferentially mobilized. These findings collectively help predict the composition of DOM across diverse watersheds in the Canadian boreal to inform microbial and contaminant biogeochemical processes in downstream ecosystems.

Математическая модель расчета и анализа характеристик оптического коммутатора в переходном режиме

The article considers a transient behavior of an optical switch described using a single-channel finite buffer queuing system with Poisson input stream and exponential service time distribution. To solve the Kolmogorov equations system and find the probabilities of states in the transition mode, the apparatus of the Laplace transform was used. Analytical expressions are found for determining the probability of losses and the throughput of an optical switch at a given point in time. In a numerical example, the performance metrics of an optical switch with a buffer size of two packets are analyzed for different network loads in transient and stationary modes. The stationary performance metrics obtained using the proposed approach coincided with the known results, which confirms the correctness of the conclusions. The optical switch transient behavior analyses at different network load showed that with a decreasing service rate the transition mode time increases.

The gravitational field outside a spatially compact stationary source in a generic fourth-order theory of gravity

By applying the symmetric and trace-free formalism in terms of the irreducible Cartesian tensors, the metric for the external gravitational field of a spatially compact stationary source is provided in F(X, Y, Z) gravity, a generic fourth-order theory of gravity, where X := R is Ricci scalar, Y := RμνRμν is Ricci square, and Z := RμνρσRμνρσ is Riemann square. A new type of gauge condition is proposed so that the linearized gravitational field equations of F(X, Y, Z) gravity are greatly simplified, and then, the stationary metric in the region exterior to the source is derived. In the process of applying the result, integrations are performed only over the domain occupied by the source. The multipole expansion of the metric potential in F(X, Y, Z) gravity for a spatially compact stationary source is also presented. In the expansion, the corrections of F(X, Y, Z) gravity to General Relativity are Yukawa-like ones, dependent on two characteristic lengths. Two additional sets of mass-type source multipole moments appear in the corrections and the salient feature characterizing them is that the integrations in their expressions are always modulated by a common radial factor related to the source distribution.

Relative velocity in pseudo-Riemannian spacetime

We give a coordinate independent definition of relative velocity of test particle in pseudo-Riemannian spacetime as measured along the observer’s line-of-sight in general and several instructive cases. In doing this, the test particle is considered as a luminous object, otherwise, if it is not, we assume that a light source is attached to it, which has neither mass nor volume. Then we utilize the general solution of independent definition of relative velocity of a luminous source in generic pseudo-Riemannian spacetime. As a corollary, we discuss the implications for the Minkowski metric, the test particle and observer at rest in an arbitrary stationary metric, the uniform gravitational field, the rotating reference frame, the Schwarzschild metric, the Kerr-type metrics, and the spatially homogeneous and isotropic Robertson-Walker (RW) spacetime of standard cosmological model. In the last case, it leads to cosmological consequence that the resulting, so-called, kinetic recession velocity of an astronomical object is always subluminal even for large redshifts of order one or more, so that it does not violate the fundamental physical principle of causality. We also calculate the measure of carrying away of a galaxy at redshift z by the expansion of space, which proves, in particular, that cosmological expansion of a flat 3D–space is fundamentally different from a kinematics of galaxies moving in a non-expanding flat 3D-space. So, it is impossible to mimic the true cosmological redshift by a Doppler effect caused by motion of galaxies in a non-expanding 3D-space, flat or curved. We also give a reappraisal of the `standard ́ kinematic interpretation of redshifts in RW spacetime as accumulated Doppler-shifts.

Pointwise decay for the wave equation on nonstationary spacetimes

This first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak form of the integrated local energy decay estimate and proves that solutions to the linear wave equation with variable coefficients in R1+3, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of these functions decay at spatial infinity. Roughly speaking, given metric perturbation terms and first-order terms whose vector fields decay at least as rapidly as the rate r−1−a, and potentials whose vector fields decay at least as rapidly as the rate r−2−a, our results imply that |ZJϕ(t,x)|≤Ct−2−a for bounded |x|, where Z denotes vector fields and J is a multi-index. We expect that this is a sharp decay rate. We prove results for both stationary and nonstationary metrics and coefficients. The proof uses integrated local energy decay to achieve an initial decay rate, and then uses an iteration involving the one-dimensional reduction (due to the positivity of the fundamental solution of the wave equation in three spatial dimensions) to achieve the full decay rate. For second-order perturbations of the Minkowski metric with coefficients that are not necessarily spherically symmetric, we analyze these coefficients' decay rates in a novel way to obtain a higher rate of pointwise decay for the solution than might be expected, assuming only weak pointwise decay on up to two time derivatives of the coefficients (and vector fields thereof); the latter weak decay is satisfied if, for example, the metric is stationary.

The charged Zipoy–Voorhees metric with astrophysical applications

Starting from an integral of the interaction region of colliding Einstein–Maxwell waves and by applying a coordinate transformation, we obtain the charged version of the static Zipoy–Voorhees (ZV) metric valid for all values of the distortion parameter gamma . In Schwarzschild coordinates, we investigate the effect of the charge in the newly found spacetime, stress the analogy with Reissner–Nordstrom metric and discuss some of its features. It is shown that from the expression of Weyl curvature, directional singularities become manifest. For astrophysical importance, we find lensing of null geodesics from the Gauss–Bonnet theorem in such non-spherically charged objects. To prepare the ground for our null, circular geodesics we consider the angular equation linearized about the symmetry plane theta =pi /2. This, in turn, suggests the distortion parameter (the ZV parameter) must be in the interval 1/2<gamma <1. It is found that the lensing angle is highly dependent on the distortion parameter, and becomes decisive on the effect of the charge. For a class of charged compact stars, we plot the deflection angle versus the ratio of impact parameter to the radius of the star. Plots have revealed that for perfectly spherical compact stars, it is hard to identify the effect of electric/magnetic charge, but for non-spherical compact stars the effect of electric charge becomes apparent. For comparison, the same lensing angle has also been found for the stationary ZV metric in the equatorial plane. Our analysis indicates that depending on the value of gamma , the stationary state induces counter effect on the bending angle and thus, when compared with the uncharged static ZV case, the bending angle decreases. The influence of the parameter gamma on the gravitational redshift is also displayed.

The Einstein equations and multipole moments at null infinity

We consider vacuum metrics admitting conformal compactification which is smooth up to the scri . We write metric in the Bondi–Sachs form and expand it into power series in the inverse affine distance 1/r. Like in the case of the luminosity distance, given the news tensor and initial data for a part of metric the Einstein equations define coefficients of the series in a recursive way. This is also true in the stationary case however now the news tensor vanishes and the role of initial data is taken by multipole moments which are equivalent to moments of Thorne. We find an approximate form of metric and show that in the case of vanishing mass the mass dipole may be different from zero. Then the known result about the Kerr like behaviour of a stationary metric is violated. Finally we find an approximate (up to the quadrupole moment) Bondi–Sachs form of the Kerr metric.

Test the growth models of black hole by jointing LIGO and Insight-HXMT observations

The growth model of black hole is still a controversial topic. In the stationary metric, all in-falling matter must be accumulated outside the event horizon of the black hole, as clocked by a distant external observer. In the time-dependent metric, all in-falling matter can fall into the event horizon of the final black hole within very short time. We test these two growth models by joining LIGO and Insight-HXMT observations. We find the stationary model is inconsistent with LIGO and Insight-HXMT observations.

Quantum probing of singularities at event horizons of black holes

It is proved that coordinate transformations of the Schwarzschild metric to new static and stationary metrics do not eliminate the mode of a particle “fall” to the event horizon of a black hole. This mode is unacceptable for the quantum mechanics of stationary states.

Normally hyperbolic trapping on asymptotically stationary spacetimes

We prove microlocal estimates at the trapped set of asymptotically Kerr spacetimes: these are spacetimes whose metrics decay inverse polynomially in time to a stationary subextremal Kerr metric. This combines two independent results. The first one is purely dynamical: we show that the stable and unstable manifolds of a decaying perturbation of a time-translation-invariant dynamical system with normally hyperbolic trapping are smooth and decay to their stationary counterparts. The second, independent, result provides microlocal estimates for operators whose null-bicharacteristic flow has a normally hyperbolic invariant manifold, under suitable non-degeneracy conditions on the stable and unstable manifolds; this includes operators on closed manifolds, as well as operators on spacetimes for which the invariant manifold lies at future infinity.

Invariant solutions of Einstein field equations in pure radiation fields

The exact static accelerating solutions of Einstein’s equations in the non-comoving pure radiation fields, with an indefinite non-degenerate stationary metric in cylindrical coordinates, are obtained. The considered space-time is of Petrov type I and is not conformally flat. Moreover, the classical symmetry method is used for symmetry reduction and finding the solutions in terms of hyperbolic functions. The conservation laws are found by using the multiplier approach.

Travel Time Tomography in Stationary Spacetimes

In this paper, we consider the boundary rigidity problem on a cylindrical domain in \({\mathbb {R}}^{1+n}\), \(n\ge 2\), equipped with a stationary (time-invariant) Lorentzian metric. We show that the time separation function between pairs of points on the boundary of the cylindrical domain determines the stationary spacetime, up to some time-invariant diffeomorphism, assuming that the metric satisfies some a-priori conditions.

Surface Tension and $$\varGamma $$-Convergence of Van der Waals–Cahn–Hilliard Phase Transitions in Stationary Ergodic Media

We study the large scale equilibrium behavior of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic media. Specifically, we are interested in free energy functionals of the following form \begin{equation*} \mathcal{F}^{\omega}(u) = \int_{\mathbb{R}^{d}} \left(\frac{1}{2} \varphi^{\omega}(x,Du(x))^{2} + W(u(x)) \right) \, dx, \end{equation*} where $W$ is a double-well potential and $\varphi^{\omega}(x,\cdot)$ is a stationary ergodic Finsler metric. We show that, at large scales, the random energy $\mathcal{F}^{\omega}$ can be approximated by the anisotropic perimeter associated with a deterministic Finsler norm $\tilde{\varphi}$. To find $\tilde{\varphi}$, we build on existing work of Alberti, Bellettini, and Presutti, showing, in particular, that there is a natural sub-additive quantity in this context.

On testing CDM and geometry-driven Milky Way rotation curve models with Gaia DR2

ABSTRACT Flat rotation curves (RCs) in disc galaxies provide the main observational support to the hypothesis of surrounding dark matter (DM). Despite of the difficulty in identifying the DM contribution to the total mass density in our Galaxy, stellar kinematics, as tracer of gravitational potential, is the most reliable observable for gauging different matter components. From the Gaia second data release catalogue, we extracted parallaxes, proper motions, and line-of-sight velocities of unprecedented accuracy for a carefully selected sample of disc stars. This is the angular momentum supported population of the Milky Way (MW) that better traces its observed RC. We fitted such data to both a classical, i.e. including a DM halo, velocity profile model, and a general relativistic one derived from a stationary axisymmetric galaxy-scale metric. The general relativistic MW RC results statistically indistinguishable from its state-of-the-art DM analogue. This supports the ansatz that a weak gravitational contribution due to the off-diagonal term of the metric, by explaining the observed flatness of MW’s RC, could fill the gap in a baryons-only MW, thus rendering the Newtonian-origin DM a general relativity-like effect. In the context of Local Cosmology, our findings are suggestive of the Galaxy’s phase space as the exterior gravitational field in equilibrium far from a Kerr-like inner source, possibly with no need for extra matter to account for the disc kinematics.

Maxwell equations in curved space-time: non-vanishing magnetic field in pure electrostatic systems

Solutions of the Maxwell equations for electrostatic systems with manifestly vanishing electric currents in the curved space-time for stationary metrics are shown to exhibit a non-vanishing magnetic field of pure geometric origin. In contrast to the conventional magnetic field of the Earth it can not be screened away by a magnetic shielding. As an example of practical significance we treat electrostatic systems at rest on the rotating Earth and derive the relevant geometric magnetic field. We comment on its impact on the ultimate precision searches of the electric dipole moments of ultracold neutrons and of protons in all electric storage rings.

Non-inertial effects on a generalized DKP oscillator in a cosmic string space-time

We examine the relativistic generalized DKP oscillator for a spin-zero field in a cosmic-string background space-time characterized by a stationary cylindrical symmetric metric. We solve the radial part of the wave function for linear, Coulomb (and shifted Coulomb) and Cornell functions. Thus we obtain the energy eigenvalues and eigenfunctions of the generalized DKP oscillator.

Rotating Disc around a Schwarzschild Black Hole

A stationary and axisymmetric (in fact circular) metric is reviewed which describes the first-order perturbation of a Schwarzschild black-hole space-time due to a rotating finite thin disc encircling the hole symmetrically. The key Green functions of the problem (corresponding to an infinitesimally thin ring)—the one for the gravitational potential and the one for the dragging angular velocity—were already derived, in terms of infinite series, by Will in 1974, but we have now put them into closed forms using elliptic integrals. Such forms are more practical for numerical evaluation and for integration in problems involving extended sources. This last point mostly remains difficult, but we illustrate that it may be workable by using the simple case of a finite thin disc with constant Newtonian surface density.

Ellipticity of Bartnik Boundary Data for Stationary Vacuum Spacetimes

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $\Pi$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $\Pi$ is Fredholm by showing that the stationary vacuum equations (combined with proper gauge terms) and the Bartnik boundary conditions form an elliptic boundary value problem. As an application, we show that the Bartnik boundary data near the standard flat boundary data admits a unique (up to diffeomorphism) stationary vacuum extension locally.

A radiating Kerr black hole and Hawking radiation

This study proposes an axisymmetric generalization of the Vaidya metric, namely the Vaidya–Kerr metric, to describe a radiating rotating black hole, and presents its Hawking radiation temperature. This study is an improved version of our previous research via ellipsoid coordinate transformation, and the Einstein field equations are solved concisely and intuitively by an orthogonal ansatz. The results demonstrate that the energy–momentum tensor of the derived radiating Kerr metric satisfies the energy-conservation law and is classified as a Petrov type II fluid, whereas the stationary Kerr metric is a Petrov type IV vacuum. The inner and outer event-horizon radii, the ergosphere radii, as well as the angular velocity at the event horizon are solved, and then, surface gravity, entropy, and Hawking radiation are derived. We estimate the Hawking-radiation temperature of the black holes with the angular momentum and the same mass of Pluto and the sun, as well as the supermassive black hole in the core of the M87 galaxy to be 9.42K, 6.08×10−8K, and 8.78×10−18K, respectively. Only the value of the rotating Pluto-mass black hole is slightly greater than the 3K cosmic microwave background radiation and may be detected by high-resolution tools in the future.