Geodesic Orbits Research Articles

Worldtube excision method for intermediate-mass-ratio inspirals: Self-consistent evolution in a scalar-charge model

This is a third installment in a program to develop a method for alleviating the scale disparity in binary black hole simulations with mass ratios in the intermediate astrophysical range, where simulation cost is prohibitive while purely perturbative methods may not be adequate. The method is based on excising a “worldtube” around the smaller object, much larger than the object itself, replacing it with an analytical model that approximates a tidally deformed black hole. Previously [N. A. Wittek , Worldtube excision method for intermediate-mass-ratio inspirals: Scalar-field model in 3+1 dimensions, ] we have tested the idea in a toy model of a scalar charge in a fixed circular geodesic orbit around a Schwarzschild black hole, solving for the massless Klein-Gordon field in 3+1 dimensions on the p platform. Here we take the significant further step of allowing the orbit to evolve radiatively, in a self-consistent manner, under the effect of back-reaction from the scalar field. We compute the inspiral orbit and the emitted scalar-field waveform, showing a good agreement with perturbative calculations in the adiabatic approximation. We also demonstrate how our simulations accurately resolve postadiabatic effects (for which we do not have perturbative results). In this work we focus on quasicircular inspirals. Our implementation is publicly accessible in the p numerical relativity code. Published by the American Physical Society 2024

Gravitational waves for eccentric extreme mass ratio inspirals of self-dual spacetime

In this paper, we calculate the frequencies of geodesic orbits in self-dual spacetime on the equatorial plane and obtain the leading-order effects of loop quantum parameters P on the energy flux and angular momentum flux in eccentric extreme mass ratio inspirals. The gravitational waveform under different eccentricity is carried out by improved “analytic-kludge” method. We calculate the waveform mismatches for the LISA detector and the measurement error on loop quantum parameters. The constraint capability on P will be improved by a factor of 3 to 10, compared to the weak field experiments in the solar system.

Fate of initially bound timelike geodesics in spherical boson stars

Boson stars are horizonless compact objects and they could possess novel geodesic orbits under the equilibrium assumption, which differ from those in black hole backgrounds. However, unstable boson stars may collapse into black holes or migrate to stable states, resulting in an inability to maintain the initially bound geodesic orbits within the backgrounds of unstable boson stars. To elucidate the fate of initially bound geodesic orbits in boson stars, we present a study of geodesics within the spherical space times of stable, collapsing, and migrating boson stars. We focus on timelike geodesics that are initially circular or reciprocating. We verify that orbits initially bound within a stable boson star persist in their bound states. For a collapsing boson star, we show that orbits initially bound and reciprocating finally either become unbound or plunge into the newly formed black hole, depending on their initial maximal radii. For initially circular geodesics, we have discovered the existence of a critical radius. Orbits with radii below this critical value are found to plunge into the newly formed black hole, whereas those with radii larger than the critical radius continue to orbit around the vicinity of the newly formed black hole, exhibiting nonzero eccentricities. For the migrating case, a black hole does not form. In this case, the reciprocating orbits span a wider radial range. For initially circular geodesics, orbits with small radii become unbound, and orbits with large radii remain bound with nonvanishing eccentricities. This geodesic study provides an approach to investigating the gravitational collapse and migration of boson stars. Published by the American Physical Society 2024

Photon emissions from Kerr equatorial geodesic orbits

We consider the light emitters moving freely along the geodesics on the equatorial plane near a Kerr black hole and study the observability of these emitters. To do so, we assume these emitters emit photons isotropically and monochromatically, and we compute the photon escaping probability (PEP) and the maximum observable blueshift (MOB) of the photons that reach infinity. We obtain numerical results of PEP and MOB for the emitters along various geodesic orbits, which exhibit distinct features for the trajectories of different classes. In particular, we find that the plunging emitters could have considerable observability even in the near-horizon region. This interesting observational feature becomes more significant for the high-energy emitters near a high-spin black hole. As the radiatively-inefficient accretion flow may consist of plunging emitters, the present work could be of great relevance to the astrophysical observations.

Quantum geodesics reflecting the internal structure of stars composed of shells

In general relativity, an external observer cannot distinguish distinct internal structures between two spherically symmetric stars that have the same total mass M. However, when quantum corrections are taken into account, the external metrics of the stars will receive quantum corrections depending on their internal structures. In this paper, we obtain the quantum-corrected metrics at linear order in curvature for two spherically symmetric shells characterized by different internal structures: one with an empty interior and the other with N internal shells. The dependence on the internal structures in the corrected metrics tells us that geodesics on these backgrounds would be deformed according to the internal structures. We conduct numerical computations to find out the angle of geodesic precession and show that the presence of internal structures amplifies the precession angle reflecting the discrepancy between the radial and orbital periods within the geodesic orbit. The amount of the precession angle increases monotonically as the number of internal shells increases and it eventually converges to a certain value for N ⟶ ∞.

On geodesic orbit nilmanifolds

The paper is devoted to the study of geodesic orbit Riemannian metrics on nilpotent Lie groups. The main result is the construction of continuous families of pairwise non-isomorphic connected and simply connected nilpotent Lie groups, every of which admits geodesic orbit metrics. The minimum dimension of groups in the constructed families is 10.

Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds

Abstract In this article, we study the geodesic orbit Randers spaces of the form ( G / H , F ) {(G/H,F)} , such that G is one of the compact classical Lie groups SO ⁢ ( n ) {{\mathrm{S}}{\mathrm{O}}(n)} , SU ⁢ ( n ) {{\mathrm{S}}{\mathrm{U}}(n)} , Sp ⁢ ( n ) {{\mathrm{S}}{\mathrm{p}}(n)} , and H is a diagonally embedded product H 1 × ⋯ × H s {H_{1}\times\cdots\times H_{s}} , where H i {H_{i}} is of the same type as G. Such spaces include spheres, Stiefel manifolds, Grassmann manifolds, and flag manifolds. The present work is a contribution to the study of geodesic orbit Randers spaces ( G / H , F ) {(G/H,F)} with H semisimple. We construct new examples of non-Riemannian Randers g.o. metrics in homogeneous bundles over generalized Stiefel manifolds which are not naturally reductive. Also, we obtain the specific expressions of these Randers g.o. metrics.

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The geodesic orbit property is useful and interesting in itself, and it plays a key role in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we extend Riemannian and Lorentz results to trans-Lorentz nilmanifolds. Those are the geodesic orbit pseudo Riemannian manifolds M=G/H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M = G/H$$\\end{document} of signature (n-2,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2,2)$$\\end{document} such that a nilpotent analytic subgroup of G is transitive on M. For that we suppose that there is a reductive decomposition g=h⊕n(vector space direct sum) with[h,n]⊂n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {g}}= {\\mathfrak {h}}\\oplus {\\mathfrak {n}}\ ext { (vector space direct sum) with } [{\\mathfrak {h}},{\\mathfrak {n}}] \\subset {\\mathfrak {n}}$$\\end{document} and n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} nilpotent. When the metric is nondegenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is abelian or 2-step nilpotent. That is the same result as for geodesic orbit Riemannian and Lorentz nilmanifolds. When the metric is degenerate on [n,n]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[{\\mathfrak {n}},{\\mathfrak {n}}]$$\\end{document} we show that n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {n}}$$\\end{document} is a double extension of a geodesic orbit nilmanifold of either Riemannian or Lorentz signature.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

We study the relation between two special classes of Riemannian Lie groups [Formula: see text] with a left-invariant metric [Formula: see text]: The Einstein Lie groups, defined by the condition [Formula: see text], and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups [Formula: see text] are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the [Formula: see text]-invariant geodesic orbit metrics on Lie groups [Formula: see text] for a wide class of subgroups [Formula: see text] that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.

ГЕОДЕЗИЧЕСКИЕ ОРБИТЫ И ЭКСПОНЕНТЫ ЛЯПУНОВА ЧЕРНОЙ ДЫРЫ ФРОЛОВА

Binary black holes maintain unstable orbits at very close distances. In the simplest case of geodesics around a Schwarzschild black hole, the orbits, although unstable, are regular and depend only on the mass. In more complex cases, geodesics may depend on charge, rotation, and other parameters. When perturbed, unstable orbits can become a source of chaos. All unstable orbits, whether regular or chaotic, can be quantified by their Lyapunov exponents. Exponents are important for observations because the phase of gravitational waves can decohere in Lyapunov time. If the time scale of dissipation due to gravitational waves is shorter than the Lyapunov time, the chaos will be damped and practically unobservable. These two time scales can be compared. Lyapunov exponents should be used with caution for several reasons: they are relative and dependent on the coordinate system used, they vary from orbit to orbit, and finally, they can be deceptively diluted by transitional behavior for orbits that pass in and out of unstable regions. The stability of circular geodesic orbits of Frolov's black hole space-time is studied in this work. The influence of the black hole charge and the scale parameter on the stability of geodesic orbits and the Lyapunov exponent is analyzed. It is shown that the region of stable circular orbits increases with the black hole charge Q and the scale parameter ℓ . The largest region of stable circular orbits of Frolov's black hole is reached at Q = M and ℓ = 0.75M.

Astrophysical observables for regular black holes with sub-Planckian curvature* *Supported in part by the Natural Science Foundation of China (11875053, 12035016). It is also supported by the Beijing Natural Science Foundation (1222031), and the Sichuan Youth Science and Technology Innovation Research Team with (21CXTD0038), Central Government Funds of Guiding Local Scientific and Technological Development for Sichuan Province with (2021ZYD0032)

We investigate the photon sphere and marginally stable circular orbit of massive particles over the recently proposed regular black holes with sub-Planckian curvature and a Minkowskian core. We derive the effective potential for geodesic orbits and determine the radius of circular photon orbits, with an analysis of the stability of these orbits. We extend our analysis to the background of a compact massive object (CMO) without a horizon, whose mass is below the lowest bound for the formation of a black hole. For massive particles, marginally stable circular orbits become double-valued in the CMO phase. Through a comparison with Bardeen and Hayward black holes, we also find that the locations of the photon sphere and marginally stable circular orbit in the CMO phase with a Minkowskian core are evidently different from those in the CMO phase with a dS core, which potentially provides a way to distinguish between these two types of black holes by astronomical observation. Finally, we present the observational constraint on the deviation parameter for such regular black holes using observed data from the black hole M87*.

Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Axisymmetric, stationary collisionless gas configurations surrounding Schwarzschild black holes

The properties of a stationary gas cloud surrounding a black hole are discussed, assuming that the gas consists of collisionless, identical massive particles that follow spatially bound geodesic orbits in the Schwarzschild spacetime. Several models for the one-particle distribution function are considered, and the essential formulae that describe the relevant macroscopic observables, like the current density four-vector and the stress–energy–momentum tensor are derived. This is achieved by rewriting these observables as integrals over the constants of motion and by a careful analysis of the range of integration. In particular, we provide configurations with finite total mass and angular momentum. Differences between these configurations and their nonrelativistic counterparts in a Newtonian potential are analyzed. Finally, our configurations are compared to their hydrodynamic analogues, the ‘polish doughnuts’.

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $$M = G/H$$ such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition $${\mathfrak {g}}= {\mathfrak {h}}\oplus {\mathfrak {n}}$$ (vector space direct sum) with $${\mathfrak {n}}$$ nilpotent. When the metric is nondegenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded.

Homogeneous geodesics in sub-Riemannian geometry

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

The infrared behavior of tame two-field cosmological models

We study the first order infared behavior of tame hyperbolizable two-field cosmological models, defined as those classical two-field models whose scalar manifold is a connected, oriented and topologically finite hyperbolizable Riemann surface (Σ,G) and whose scalar potential Φ admits a positive and Morse extension to the end compactification of Σ. We achieve this by determining the universal forms of the asymptotic gradient flow of the classical effective potential V with respect to the uniformizing metric G near all interior critical points and ends of Σ, finding that some of the latter act like fictitious but exotic stationary points of the gradient flow. We also compare these results with numerical studies of cosmological orbits. For critical cusp ends, we find that cosmological curves have transient quasiperiodic behavior but are eventually attracted or repelled by the cusp along principal geodesic orbits determined by the extended effective potential. This behavior is approximated in the infrared by that of gradient flow curves near the cusp.

Electromagnetic self-force on a charged particle on Kerr spacetime: Equatorial circular orbits

We calculate the self-force acting on a charged particle on a circular geodesic orbit in the equatorial plane of a rotating black hole. We show by direct calculation that the dissipative self-force balances with the sum of the flux radiated to infinity and through the black hole horizon. Prograde orbits are found to stimulate black hole superradiance, but we confirm that the condition for floating orbits cannot be met. We calculate the conservative component of the self-force by application of the mode sum regularization method, and we present a selection of numerical results. By numerical fitting, we extract the leading-order coefficients in post-Newtonian expansions. The self-force on the innermost stable circular orbits of the Kerr spacetime is calculated, and comparisons are drawn between the electromagnetic and gravitational self forces.

Standard homogeneous (α1,α2)${\big(\alpha _1,\alpha _2\big)}$‐metrics and geodesic orbit property

AbstractIn this paper, we introduce the notion of standard homogeneous ‐metric, as a generalization for the normal homogeneity in Finsler geometry with relatively good computability. We explore the geodesic orbit (g.o. in short) property of this metric. Especially, when it is associated with a triple of compact Lie groups, we find new examples of g.o. Finsler spaces from H. Tamaru's classification list. Meanwhile, we prove that all standard g.o. ‐metrics on the Wallach spaces, , and , must be the normal homogeneous Riemannian metrics.

Geodesic orbit spaces of compact Lie groups of rank two

Geodesic orbit spaces are those Riemannian homogeneous spaces (G/H, g) whose geodesics are orbits of one-parameter subgroups of G. We classify the simply connected geodesic orbit spaces where G is a compact Lie group of rank two. We prove that the only such spaces for which the metric g is not induced from a bi-invariant metric on G are certain spheres and projective spaces, endowed with metrics induced from Hopf fibrations.

On the Geodesic Orbit Property for Lorentz Manifolds

On the Geodesic Orbit Property for Lorentz Manifolds