Hole Orbits Research Articles

Charge orbits of extremal black holes in five-dimensional supergravity

We derive the $U$-duality charge orbits, as well as the related moduli spaces, of ``large'' and ``small'' extremal black holes in nonmaximal ungauged Maxwell-Einstein supergravities with symmetric scalar manifolds in $d=5$ space-time dimensions. The stabilizer groups of the various classes of orbits are obtained by determining and solving suitable $U$-invariant sets of constraints, both in ``bare'' and ``dressed'' charge bases, with various methods. After a general treatment of attractors in real special geometry (also considering nonsymmetric cases), the $\mathcal{N}=2$ ``magic'' theories, as well as the $\mathcal{N}=2$ Jordan symmetric sequence, are analyzed in detail. Finally, the half-maximal ($\mathcal{N}=4$) matter-coupled supergravity is also studied in this context.

Computing waveforms for spinning compact binaries in quasi-eccentric orbits

Several scenarios have been proposed in which the orbits of binary black holes enter the band of a gravitational wave detector with significant eccentricity. To avoid missing these signals or biasing parameter estimation it is important that we consider waveform models that account for eccentricity. The ingredients needed to compute post-Newtonian (PN) waveforms produced by spinning black holes inspiralling on quasi-eccentric orbits have been available for almost two decades at 2 PN order, and this work has recently been extended to 2.5 PN order. However, the computational cost of directly implementing these waveforms is high, requiring many steps per orbit to evolve the system of coupled differential equations. Here we employ the standard techniques of a separation of time scales and a generalized Keplerian parameterization of the orbits to produce efficient waveforms describing spinning black hole binaries with arbitrary masses and spins on quasi-eccentric orbits to 1.5 PN order. We separate the fast orbital time scale from the slow spin-orbit precession time scale by solving for the orbital motion in a noninterial frame of reference that follows the orbital precession. We outline a scheme for extending our approach to higher post-Newtonian order.

Spacetime and orbits of bumpy black holes

Our universe contains a great number of extremely compact and massive objects which are generally accepted to be black holes. Precise observations of orbital motion near candidate black holes have the potential to determine if they have the spacetime structure that general relativity demands. As a means of formulating measurements to test the black hole nature of these objects, Collins and Hughes introduced "bumpy black holes": objects that are almost, but not quite, general relativity's black holes. The spacetimes of these objects have multipoles that deviate slightly from the black hole solution, reducing to black holes when the deviation is zero. In this paper, we extend this work in two ways. First, we show how to introduce bumps which are smoother and lead to better behaved orbits than those in the original presentation. Second, we show how to make bumpy Kerr black holes -- objects which reduce to the Kerr solution when the deviation goes to zero. This greatly extends the astrophysical applicability of bumpy black holes. Using Hamilton-Jacobi techniques, we show how a spacetime's bumps are imprinted on orbital frequencies, and thus can be determined by measurements which coherently track a small orbiting body's orbital phase. We find that weak-field orbits of bumpy black holes are modified exactly as expected from a Newtonian analysis of a body with a prescribed multipolar structure, reproducing well-known results from the celestial mechanics literature. The impact of bumps on strong-field orbits is especially strong, suggesting that this framework will allow observations to set robust limits on the extent to which a spacetime's multipoles deviate from the black hole expectation.

Homoclinic orbits around spinning black holes. II. The phase space portrait

In paper I in this series, we found exact expressions for the equatorial homoclinic orbits: the separatrix between bound and plunging, whirling and not whirling motion. As a companion to that physical space study, in this paper we paint a phase space portrait of the homoclinic orbits that includes exact expressions for the actions and fundamental frequencies. Additionally, we develop a reduced Hamiltonian description of Kerr motion that allows us to track groups of trajectories with a single global clock. This facilitates a variational analysis, whose stability exponents and eigenvectors could potentially be useful for future studies of families of black hole orbits and their associated gravitational waveforms.

Verifying black hole orbits with gravitational spectroscopy

Gravitational waves from test masses bound to geodesic orbits of rotating black holes are simulated, using Teukolsky's black hole perturbation formalism, for about ten thousand generic orbital configurations. Each binary radiates power exclusively in modes with frequencies that are integer-linear combinations of the orbit's three fundamental frequencies. General spectral properties are found with a survey of orbits about a black hole taken to be rotating at 80% of the maximal spin. The orbital eccentricity is varied from 0.1 to 0.9. Inclination ranges from 20\ifmmode^\circ\else\textdegree\fi{} to 160\ifmmode^\circ\else\textdegree\fi{} and comes to within 20\ifmmode^\circ\else\textdegree\fi{} of polar. Semilatus rectum is varied from 1.2 to 3 times the value at the innermost stable circular orbits. The following general spectral properties are found: (i) 99% of the radiated power is typically carried by a few hundred modes, and at most by about a thousand modes, (ii) the dominant frequencies can be grouped into a small number of families defined by fixing two of the three integer frequency multipliers, and (iii) the specifics of these trends can be qualitatively inferred from the geometry of the orbit under consideration. Detections using triperiodic analytic templates modeled on these general properties would constitute a verification of radiation from an adiabatic sequence of black hole orbits and would recover the evolution of the fundamental orbital frequencies. In an analogy with ordinary spectroscopy, this would compare to observing the Bohr model's atomic hydrogen spectrum without being able to rule out alternative atomic theories or nuclei. The suitability of such a detection technique is demonstrated using snapshots computed at 12-hour intervals throughout the last three years before merger of a kludged inspiral. The system chosen is typical of those thought to occur in galactic nuclei and to be observable with space-based gravitational wave detectors like LISA. Because of circularization, the number of excited modes decreases as the binary evolves. A hypothetical detection algorithm that tracks mode families dominating the first 12 hours of the inspiral would capture 98% of the total power over the remaining three years.

Erratum: Evolution of circular, nonequatorial orbits of Kerr black holes due to gravitational-wave emission [Phys. Rev. D61, 084004 (2000)

Received 1 September 2008DOI:https://doi.org/10.1103/PhysRevD.78.109902©2008 American Physical Society

Computational efficiency of frequency- and time-domain calculations of extreme mass-ratio binaries: Equatorial orbits

Gravitational waveforms and fluxes from extreme mass-ratio inspirals can be computed using time-domain methods with accuracy that is fast approaching that of frequency-domain methods. We study in detail the computational efficiency of these methods for equatorial orbits of fast spinning Kerr black holes, and find the number of modes needed in either method---as functions of the orbital parameters---in order to achieve a desired accuracy level. We then estimate the total computation time and argue that for high-eccentricity orbits the time-domain approach may be more efficient computationally. We suggest that in practice low-$m$ modes are computed using the frequency-domain approach, and high-$m$ modes are computed using the time-domain approach, where $m$ is the azimuthal mode number.

Band warping effect appeared in commensurability oscillations in antidot lattices of a two‐dimensional hole gas

AbstractWe have confirmed band warping effect of twodimensional holes confined at GaAs/AlGaAs singleheterojunction through classical commensurability resistance oscillations in anti‐dot lattices (ADLs). The oscillations in the triangular ADLs are decomposed into three peaks, two of which are assigned to localized orbits of holes with a lighter effective math (l‐hole), and the other is assigned to that of holes with heavier average effective math (h‐hole). The latter assignment is possible only when the h‐hole band is heavily warped, thus evidencing the warping. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Anisotropic behaviours of massless Dirac fermions in graphene under periodic potentials

The propagation of charge carriers in graphene under an imposed periodic potential can become strongly anisotropic, suggesting a way of making electronic circuits with appropriately patterned surface electrodes without the need for cutting nanoscale structure into graphene. Graphene’s conical valence and conduction bands give rise to charge carriers that have neutrino-like linear energy dispersion and exhibit chiral behaviour near the Dirac points where these bands meet1,2,3,4,5,6. Such characteristics offer exciting opportunities for the occurrence of new phenomena and the development of high performance electronic devices. Making high quality devices from graphene, which typically involves etching it into nanoscale structures7,8,9,10, however, has proven challenging. Here we show that a periodic potential applied by suitably patterned modifications or contacts on graphene’s surface leads to further unexpected and potentially useful charge carrier behaviour. Owing to their chiral nature, the propagation of charge carriers through such a graphene superlattice is highly anisotropic, and in extreme cases results in group velocities that are reduced to zero in one direction but are unchanged in another. Moreover, we show that the density and type of carrier species (electron, hole or open orbit) in a graphene superlattice are extremely sensitive to the potential applied, and they may further be tuned by varying the Fermi level. As well as addressing fundamental questions about how the chiral massless Dirac fermions of graphene propagate in a periodic potential, our results suggest the possibility of building graphene electronic circuits from appropriately engineered periodic surface patterns, without the need for cutting or etching.

Circular and noncircular nearly horizon-skimming orbits in Kerr spacetimes

We have performed a detailed analysis of orbital motion in the vicinity of a nearly extremal Kerr black hole. For very rapidly rotating black holes---spin parameter $a\ensuremath{\equiv}J/M>0.9524M$---we have found a class of very strong-field eccentric orbits whose orbital angular momentum ${L}_{z}$ increases with the orbit's inclination with respect to the equatorial plane, while keeping latus rectum and eccentricity fixed. This behavior is in contrast with Newtonian intuition, and is in fact opposite to the normal behavior of black hole orbits. Such behavior was noted previously for circular orbits; since it only applies to orbits very close to the black hole, they were named ``nearly horizon-skimming orbits.'' Our current analysis generalizes this result, mapping out the full generic (inclined and eccentric) family of nearly horizon-skimming orbits. The earlier work on circular orbits reported that, under gravitational radiation emission, nearly horizon-skimming orbits exhibit unusual inspiral, tending to evolve to smaller orbit inclination, toward prograde equatorial configuration. Normal orbits, by contrast, always demonstrate slowly growing orbit inclination---orbits evolve toward the retrograde equatorial configuration. Using up-to-date Teukolsky-based fluxes, we have concluded that the earlier result was incorrect---all circular orbits, including nearly horizon-skimming ones, exhibit growing orbit inclination under radiative backreaction. Using kludge fluxes based on a Post-Newtonian expansion corrected with fits to circular and to equatorial Teukolsky-based fluxes, we argue that the inclination grows also for eccentric nearly horizon-skimming orbits. We also find that the inclination change is, in any case, very small. As such, we conclude that these orbits are not likely to have a clear and peculiar imprint on the gravitational waveforms expected to be measured by the space-based detector LISA.

Crystal structure, Fermi surface calculations and Shubnikov-de Haas oscillation spectrum of the organic metal θ-(BETS) 4HgBr 4(C 6H 5Cl) at low temperature

The organic metal θ-(BETS) 4HgBr 4(C 6H 5Cl) is known to undergo a phase transition as the temperature is lowered down to about 240 K. X-ray data obtained at 200 K indicate a corresponding modification of the crystal structure, the symmetry of which is lowered from quadratic to monoclinic. In addition, two different types of cation layers are observed in the unit cell. The Fermi surface (FS), which can be regarded as a network of compensated electron and hole orbits according to band structure calculations at room temperature, turns to a set of two alternating linear chains of orbits at low temperature. The field and temperature dependence of the Shubnikov-de Haas oscillation spectrum have been studied up to 54 T. Eight frequencies are observed which, in any case, point to an FS much more complex than predicted by band structure calculations at room temperature, even though some of the observed Fourier components might be ascribed to magnetic breakdown or frequency mixing. The obtained spectrum could result from either an interaction between the FSs linked to each of the two cation layers or to an eventual additional phase transition in the temperature range below 200 K.

Magnetic quantum oscillations of electrons on a two-dimensional lattice: Numerical simulations and the magnetic breakdown approach

We report on analytical and numerical studies of the magnetic quantum oscillations---the de Haas--van Alphen (dHvA) effect---in a model of spinless electrons on a two-dimensional (2D) square lattice in a magnetic field $B$ known as the Azbel-Hofstadter (AH) problem. The numerical simulations of the dHvA oscillations are based on the magnetic band structure, which is obtained by numerical diagonalization of the one-electron model Hamiltonian on a square lattice in the magnetic field. At zero magnetic field and for a filling of 0.55 electrons per unit cell, the shape of the Fermi surface (FS) is close to a chessboard in which white and black cells are related to holes and electrons, respectively. The numerical simulations of the dHvA oscillations for this FS clearly display the magnetic breakdown (MB) features in the Fourier spectrum: the combinational and forbidden frequencies, which cannot be explained within the standard Lifshitz-Kosevich theory. Because of the idealized conditions of our numerical simulation (zero temperature, ideal crystal lattice) the only reason for deviation of the dHvA oscillations from the sawtooth magnetization of a 2D electron gas is the fractal magnetic band structure of the AH problem. We have also developed a semiclassical analytical approach to the dHvA oscillations taking account of the MB at the corners of the hole and electron orbits of the FS, the probability of which $0\ensuremath{\le}W(B)=\mathrm{exp}(\ensuremath{-}{B}_{\mathrm{MB}}∕B)\ensuremath{\le}1$ depends on $B$ and the MB field ${B}_{\mathrm{MB}}$. We have shown that the dHvA oscillations in this case are completely determined by the quantization of the closed hole and electron orbits and by special MB factors for these orbits ${I}^{e(h)}(p)$ ($p=1,2,3,\dots{}$, is the harmonic index). The factors ${I}^{e(h)}(p)$ not only strongly modulate the amplitudes of harmonics in the magnetization but act also through the changes in the chemical potential (CP) oscillations. The latter are very important: they control the shape of the magnetization (which can be close either to a direct or an inverted sawtoothlike profile) and add forbidden peaks in the Fourier spectrum. In the case of a fixed CP the forbidden peaks vanish and the magnetization profile is close to the direct sawtooth. As a function of the MB probability $W(B)$, the MB factors ${I}^{e(h)}(p)$ are polynomials of degree $p$ with the following fixed values at the ends: 1 at $W(B)=0$ and 0 at $W(B)=1$. Our analytic expression for the magnetization as a function of magnetic field $B$ agrees pretty well with our numerical simulations using only one fitting parameter---the MB field ${B}_{\mathrm{MB}}$. Small deviations are due to the fractal fine structure of the magnetic bands.

Chaos and order in models of black hole pairs

Chaos in the orbits of black hole pairs has by now been confirmed by several independent groups. While the chaotic behavior of binary black hole orbits is no longer argued, it remains difficult to quantify the importance of chaos to the evolutionary dynamics of a pair of comparable mass black holes. None of our existing approximations are robust enough to offer convincing quantitative conclusions in the most highly nonlinear regime. It is intriguing to note that, in three different approximations to a black hole pair built of a spinning black hole and a nonspinning companion, two approximations exhibit chaos and one approximation does not. The fully relativistic scenario of a spinning test mass around a Schwarzschild black hole shows chaos, as does the post-Newtonian Lagrangian approximation. However, the approximately equivalent post-Newtonian Hamiltonian approximation does not show chaos when only one body spins. It is well known in dynamical systems theory that one system can be regular while an approximately related system is chaotic, so there is no formal conflict. However, the physical question remains: Is there chaos for comparable mass binaries when only one object spins? We are unable to answer this question given the poor convergence of the post-Newtonian approximation to the fully relativistic system. A resolution awaits better approximations that can be trusted in the highly nonlinear regime.

Coulomb-interaction effects on the electronic structure of radially polarized excitons in nanorings

The electronic structure of radially polarized excitons in structured nanorings is analyzed, with emphasis in the ground-state properties and their dependence under applied magnetic fields perpendicular to the ring plane. The electron-hole Coulomb attraction has been treated rigorously, through numerical diagonalization of the full exciton Hamiltonian in the non-interacting electron-hole pairs basis. Depending on the relative weight of the kinetic energy and Coulomb contributions, the ground-state of polarized excitons has "extended" or "localized" features. In the first case, corresponding to small rings dominated by the kinetic energy, the ground-state shows Aharonov-Bohm (AB) oscillations due to the individual orbits of the building particles of the exciton. In the localized regime, corresponding to large rings dominated by the Coulomb interaction, the only remaining AB oscillations are due to the magnetic flux trapped between the electron and hole orbits. This dependence of the exciton, a neutral excitation, on the flux difference confirms this feature as a signature of Coulomb dominated polarized excitons. Analytical approximations are provided in both regimens, which accurate reproduce the numerical results.

Quasicircular orbits of conformal thin-sandwich puncture binary black holes

I construct initial data for equal-mass irrotational binary black holes using the conformal thin-sandwich puncture approach. I locate quasicircular orbits using the effective-potential method, and estimate the location of the innermost stable circular orbit (ISCO). The ISCO prediction is consistent with results for conformal thin-sandwich data produced using excision techniques. These results also show that the ISCOs predicted by the effective-potential and ADM-Komar mass-comparison methods agree for conformal thin-sandwich data, just as they did for Bowen-York data.

Electronic structure and impurity states in GaN quantum dots

We study the electronic structure of spherical GaN quantum dots (QD's) with a substitutional acceptor impurity at the center. The size-dependent energy spectra are calculated within the sp 3 s* tight-binding model, which yields a good agreement with the confinement-induced blue shifts observed in undoped QD's. The acceptor binding energy is strongly enhanced in a QD and decreases with increasing size following a scaling law that extrapolates to the bulk experimental value. The size-dependent average radius of the hole orbit is also calculated. The results are in agreement with the available experimental data for Mg impurity in bulk GaN.

Intermediate‐Mass Black Hole(s) and Stellar Orbits in the Galactic Center

Many young stars reside within the central half-parsec from Sgr A*, the supermassive black hole in the Galactic center. The origin of these stars remains a puzzle. Recently, Hansen & Milosavljevic (HM) have argued that an intermediate-mass black hole (IMBH) could have delivered the young stars to the immediate vicinity of Sgr A*. Here we focus on the final stages of the HM scenario. Namely, we integrate numerically the orbits of stars that are initially bound to the IMBH but are stripped from it by the tidal field of Sgr A*. Our numerical algorithm is a symplectic integrator designed specifically for the problem at hand; however, we have checked our results with SYMBA, a version of the widely available SWIFT code. We find that the distribution of the postinspiral orbital parameters is sensitive to the eccentricity of the inspiraling IMBH. If the IMBH is on a circular orbit, then the inclinations of numerically computed orbits relative to the inspiral plane are almost always smaller than 10°, and therefore (1) the simulations are in good agreement with the observed motions of stars in a clockwise-moving stellar disk; and (2) the simulations never reproduce the orbits of stars outside this disk, which include those in the second thick ring of stars and the randomly oriented unrelaxed orbits of some of the S stars. If the IMBH's orbital eccentricity is e = 0.6, then approximately half of the stars end up with orbital inclinations below 10°, and another half have inclinations anywhere between 0° and 180°; this is somewhat closer to what is observed. We also show that if the IRS 13 cluster is bound by an IMBH, as has been argued by Maillard et al., then the same IMBH could not have delivered all of the young stars to their present location.

Towards a formalism for mapping the spacetimes of massive compact objects: Bumpy black holes and their orbits

Observations have established that extremely compact, massive objects are common in the universe. It is generally accepted that these objects are black holes. As observations improve, it becomes possible to test this hypothesis in ever greater detail. In particular, it is or will be possible to measure the properties of orbits deep in the strong field of a black hole candidate (using x-ray timing or with gravitational-waves) and to test whether they have the characteristics of black hole orbits in general relativity. Such measurements can be used to map the spacetime of a massive compact object, testing whether the object's multipoles satisfy the strict constraints of the black hole hypothesis. Such a test requires that we compare against objects with the ``wrong'' multipole structure. In this paper, we present tools for constructing bumpy black holes: objects that are almost black holes, but that have some multipoles with the wrong value. The spacetimes which we present are good deep into the strong field of the object -- we do not use a large r expansion, except to make contact with weak field intuition. Also, our spacetimes reduce to the black hole spacetimes of general relativity when the ``bumpiness'' is set to zero. We propose bumpy black holes as the foundation for a null experiment: if black hole candidates are the black holes of general relativity, their bumpiness should be zero. By comparing orbits in a bumpy spacetime with those of an astrophysical source, observations should be able to test this hypothesis, stringently testing whether they are the black holes of general relativity. (Abridged)

Combination frequencies of magnetic oscillations inβ″−(BEDT−TTF)4(NH4)[Fe(C2O4)3]⋅DMF

Interlayer magnetoresistance of the quasi-two-dimensional organic superconductor ${\ensuremath{\beta}}^{\ensuremath{''}}\ensuremath{-}(\mathrm{BEDT}\ensuremath{-}\mathrm{TTF}{)}_{4}({\mathrm{NH}}_{4})[\mathrm{Fe}({\mathrm{C}}_{2}{\mathrm{O}}_{4}{)}_{3}]\ensuremath{\cdot}\mathrm{DMF}$ has been investigated in pulsed magnetic fields of up to 55 T in the temperature range from 1.6 K to 4.2 K. According to band-structure calculations, the Fermi surface of this compound originates from hybridized intersecting hole tubes with a cross sectional area equal to the first Brillouin-zone (FBZ) area, leading to two compensated electron and hole orbits. Actually, the Fourier spectrum of the oscillatory magnetoresistance exhibits various frequencies which can be regarded as linear combinations of the two frequencies with the highest amplitude $[{F}_{a}=(48\ifmmode\pm\else\textpm\fi{}2)T$ and ${F}_{b}=(241\ifmmode\pm\else\textpm\fi{}5)T]$ corresponding to 1.2% and 6.0% of the FBZ area, respectively. The oscillatory spectrum can be accounted for by three compensated electron and hole orbits and combination frequencies typical of networks of orbits coupled by magnetic breakthrough.

Ring-shaped Andreev billiards in quantizing magnetic fields

We present a detailed semiclassical study of a clean disk-shaped insulator--normal-metal--superconductor hybrid system in a magnetic field. It is based on an exact secular equation that we derived within the microscopic Bogoliubov--de Gennes (BdG) formalism. Results obtained from a classification of electron and hole orbits are in excellent agreement with those from an exact numerical diagonalization of the BdG equation. Our analysis opens up different possibilities for determining thermodynamic properties of mesoscopic hybrid systems.