Gravitational Angular Momentum Research Articles

ALIGNMENT OF PROTOSTARS AND CIRCUMSTELLAR DISKS DURING THE EMBEDDED PHASE

Star formation proceeds via the collapse of a molecular cloud core over multiple dynamical timescales. Turbulence within cores results in a spatially non-uniform angular momentum of the cloud, causing a stochastic variation in orientation of the disk forming from the collapsing material. In the absence of star-disk angular momentum coupling, such disk-tilting would provide a natural mechanism for production of primordial spin-orbit misalignments in the resulting planetary systems. However, owing to high accretion rates in the embedded phase of star formation, the inner edge of the circumstellar disk extends down to the stellar surface, resulting in efficient gravitational and accretional angular momentum transfer between the star and the disk. Here, we demonstrate that the resulting gravitational coupling is sufficient to suppress any significant star-disk misalignment, with accretion playing a secondary role. The joint tilting of the star-disk system leads to a stochastic wandering of star-aligned bipolar outflows. Such wandering widens the effective opening angle of stellar outflows, allowing for more efficient clearing of the remainder of the protostar's gaseous envelope. Accordingly, the processes described in this work provide an additional mechanism responsible for sculpting the stellar Initial Mass Function (IMF).

A new gravitational wave generation algorithm for particle perturbations of the Kerr spacetime

We present a new approach to solve the 2+1 Teukolsky equation for gravitational perturbations of a Kerr black hole. Our approach relies on a new horizon penetrating, hyperboloidal foliation of Kerr spacetime and spatial compactification. In particular, we present a framework for waveform generation from point-particle perturbations. Extensive tests of a time domain implementation in the Teukode code are presented. The code can efficiently deliver waveforms at future null infinity. The accuracy and convergence of the waveforms’ phase and amplitude is demonstrated. As a first application of the method, we compute the gravitational waveforms from inspiraling and coalescing black-hole binaries in the large-mass-ratio limit. The smaller mass black hole is modeled as a point particle whose dynamics is driven by an effective-one-body-resummed analytical radiation reaction force. We compare the analytical, mechanical angular momentum loss (computed using two different prescriptions) to the gravitational wave angular momentum flux. We find that higher-order post-Newtonian corrections are needed to improve the consistency for rapidly spinning binaries. We characterize the multipolar waveform as a function of the black-hole spin. Close to merger, the subdominant multipolar amplitudes (notably the m = 0 ones) are enhanced for retrograde orbits with respect to prograde ones. We argue that this effect mirrors nonnegligible deviations from the circularity of the dynamics during the late-plunge and merger phase. For the first time, we compute the gravitational wave energy flux flowing into the black hole during the inspiral using a time-domain formalism proposed by Poisson. Finally, a self-consistent, iterative method to compute the gravitational wave fluxes at leading-order in the mass of the particle is developed. The method can be used alternatively to the analytical radiation reaction in cases where the analytical information is poor or not sufficient. Specifically, we apply it to compute dynamics and waveforms for a rapidly rotating black hole with dimensionless spin parameter . For this case, the simulation with the consistent flux differs from the one with the analytical flux by gravitational wave cycles over a total of about 250 cycles. In this simulation the horizon absorption accounts for about gravitational wave cycles.

The angular momentum of plane-fronted gravitational waves in the teleparallel equivalent of general relativity

We present a simplified expression for the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity (TEGR). The expression arises from the constraints equations of the Hamiltonian formulation of the TEGR. We apply this expression to the calculation of the angular momentum of plane-fronted gravitational waves in an arbitrary three-dimensional volume \(V\) of space and compare the results with those obtained for linearised gravitational waves.

Structure of the integrals of motion for a rotating body in the relativistic theory of gravitation

The energy-momentum tensor for the Kerr metric is calculated within the relativistic theory of gravitation. The external gravitational energy and angular momentum of a collapsing body are found. It is concluded from the analysis of their behavior near the gravitational radius that gravitational collapse is impossible.

Neutron Stars in Teleparallel Gravity

In this paper, we deal with neutron stars, which are described by a perfect fluid model, in the context of the teleparallel equivalent of general relativity. We use numerical computations (based on the RNS code) to find the relation between the angular momentum of the field and the angular momentum of the source. One such relation was established for each stable star resulting from the numerical computation with an equation of state, the central energy density, and the ratio between the polar and equatorial radii as input, the central energy density and the ratio between polar and equatorial radii. We also find a regime in which the relation between gravitational angular momentum and moment of inertia (as well as the angular velocity of the fluid) is linear. We give the spatial distribution of the gravitational energy and show that it depends linearly on the squared angular velocity of the source.

Modification of the exact pole placement method and its application for the control of spacecraft motion

For linear stationary models of the spacecraft motion, a modification of the exact pole placement method is developed that makes it possible to design a modal control for any class of linear stationary systems. In the case of circular orbits, the modified method is used to obtain an analytical solution of the problem of gravitational angular momentum unload problem for inertial actuators of a spacecraft in the gyroscopically coupled roll-yaw channels.

Binary black hole coalescence in the large-mass-ratio limit: The hyperboloidal layer method and waveforms at null infinity

We compute and analyze the gravitational waveform emitted to future null infinity by a system of two black holes in the large mass ratio limit. We consider the transition from the quasi-adiabatic inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To compute the waveforms we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foliation which coincides with the standard Schwarzschild foliation in the region including the motion of the small black hole, and is globally hyperboloidal, allowing us to include future null infinity in the computational domain by compactification. This method is called the hyperboloidal layer method, and is discussed here for the first time in a study of the gravitational radiation emitted by black hole binaries. We consider binaries characterized by five mass ratios, $\nu=10^{-2,-3,-4,-5,-6}$, that are primary targets of space-based or third-generation gravitational wave detectors. We show significative phase differences between finite-radius and null-infinity waveforms. We test, in our context, the reliability of the extrapolation procedure routinely applied to numerical relativity waveforms. We present an updated calculation of the gravitational recoil imparted to the merger remnant by the gravitational wave emission. As a self consistency test of the method, we show an excellent fractional agreement (even during the plunge) between the 5PN EOB-resummed mechanical angular momentum loss and the gravitational wave angular momentum flux computed at null infinity. New results concerning the radiation emitted from unstable circular orbits are also presented.

Gravitational energy, momentum and angular momentum of Schwarzschild Anti-de Sitter space-times in the teleparallel equivalent of general relativity

We give a class of spherically symmetric-Anti de Sitter (Ads), exact solution in the teleparallel equivalent of general relativity (TEGR). The solution depends on an arbitrary function \(\mathcal{F}(R)\) and reproduce the metric of Schwarzschild Ads space-time. In the context of the Hamiltonian formulation of the TEGR we compute the gravitational energy of this class. The calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. We show that the form of the energy depends on the arbitrary function. We make a constrain on this arbitrary function to give the correct form of energy.

Transition from adiabatic inspiral to plunge into a spinning black hole

A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M >> mu will slowly inspiral as gravitational radiation extracts energy and angular momentum from its orbit. This inspiral can be considered adiabatic when the orbital period is much shorter than the timescale on which energy is radiated, and quasi-circular when the radial velocity is much less than the azimuthal velocity. Although the inspiral always remains adiabatic provided mu << M, the quasi-circular approximation breaks down as the particle approaches the innermost stable circular orbit (ISCO). In this paper, we relax the quasi-circular approximation and solve the radial equation of motion explicitly near the ISCO. We use the requirement that the test particle's 4-velocity remain properly normalized to calculate a new contribution to the difference between its energy and angular momentum. This difference determines how a black hole's spin changes following a test-particle merger, and can be extrapolated to help predict the mass and spin of the final black hole produced in finite-mass-ratio black-hole mergers. Our new contribution is particularly important for nearly maximally spinning black holes, as it can affect whether a merger produces a naked singularity.

Gravitational radiation and angular momentum flux from a slowly rotating dynamical black hole

A four-dimensional asymptotic expansion scheme is used to study the next-order effects of the nonlinearity near a spinning dynamical black hole. The angular-momentum flux and energy flux formula are then obtained by constructing the reference frame in terms of the compatible constant spinors and the compatibility of the coupling leading-order Newman-Penrose equations. By using the slow rotation and small-tide approximation for a spinning black hole, the horizon cross-section we chose is spherical symmetric. It turns out the flux formula is rather simple and can be compared with the known results. Directly from the energy flux formula of the slow-rotating dynamical horizon, we find that the physically reasonable condition on requiring the positivity of the gravitational energy flux yields that the shear will monotonically decrease with time. Thus a slow-rotating dynamical horizon will asymptotically approach an isolated horizon during late time.

Binary black hole merger in the extreme-mass-ratio limit: A multipolar analysis

Building up on previous work, we present a new calculation of the gravitational wave (GW) emission generated during the transition from quasi-circular inspiral to plunge, merger and ringdown by a binary system of nonspinning black holes, of masses $m_1$ and $m_2$, in the extreme mass ratio limit, $m_1 m_2\ll(m_1+m_2)^2$. The relative dynamics of the system is computed {\it without making any adiabatic approximation} by using an effective one body (EOB) description, namely by representing the binary by an effective particle of mass $\mu=m_1 m_2/(m_1+m_2)$ moving in a (quasi-)Schwarzschild background of mass $M=m_1+m_2$ and submitted to an $\O(\nu)$ 5PN-resummed analytical radiation reaction force, with $\nu=\mu/M$. The gravitational wave emission is calculated via a multipolar Regge-Wheeler-Zerilli type perturbative approach (valid in the limit $\nu\ll 1$). We consider three mass ratios, $\nu={10^{-2},10^{-3},10^{-4}}$,and we compute the multipolar waveform up to $\ell=8$. We estimate energy and angular momentum losses during the quasi-universal and quasi-geodesic part of the plunge phase and we analyze the structure of the ringdown. We calculate the gravitational recoil, or "kick", imparted to the merger remnant by the gravitational wave emission and we emphasize the importance of higher multipoles to get a final value of the recoil $v/(c\nu^2)=0.0446$. We finally show that there is an {\it excellent fractional agreement} ($\sim 10^{-3}$) (even during the plunge) between the 5PN EOB analytically-resummed radiation reaction flux and the numerically computed gravitational wave angular momentum flux. This is a further confirmation of the aptitude of the EOB formalism to accurately model extreme-mass-ratio inspirals, as needed for the future space-based LISA gravitational wave detector.

Energy and angular momentum densities in a Gödel-type universe in teleparallel geometry

The main scope of this research consists in evaluating the energy-momentum (gravitational field plus matter) and gravitational angular momentum densities in the universe with global rotation, considering the Gödel—Obukhov metric. To do that, we use the Hamiltonian formalism of the Teleparallel equivalent of general relativity (TEGR), which is justified for presenting covariant expressions for the quantities under consideration. We have found that the total energy density calculated by the TEGR method is in agreement with the results reported by other authors in the literature using pseudotensors. The result found for the angular momentum density depends on the rotational parameter, as expected. We also show explicitly the equivalence among the field equations of the TEGR and the Einstein equations (GR), considering a perfect fluid and the Gödel—Obukhov metric.

SELF-GRAVITY AND ANGULAR MOMENTUM TRANSPORT IN EXTENDED GALACTIC DISKS

We demonstrate a significant difference in the angular momentum transport properties of galactic disks between regions in which the interstellar medium is single phase or two phase. Our study is motivated by observations of HI in extended galactic disks which indicate velocity dispersions of nonthermal origin, suggesting that turbulence in the gas may be contributing significantly to the observed dispersion. To address this, we have implemented a shearing-box framework within the FLASH code. The new code was used to perform local simulations of galactic disks that incorporate differential rotation, self-gravity, vertical stratification, hydrodynamics and cooling. These simulations explore plausible mechanisms for driving turbulent motions via the thermal and self-gravitational instabilities coupling to differential rotation. Where a two-phase medium develops, gravitational angular momentum transporting stresses are much greater, creating a possible mechanism for transferring energy from galactic rotation to turbulence. In simulations where the disk conditions do not trigger the formation of a two-phase medium, it is found that perturbations to the flow damp without leading to a sustained mechanism for driving turbulence. The differing angular momentum transport properties of the single- and two-phase regimes of the disk suggest that a significant, dynamically motivated division can be drawn between the two, even when this division occurs far outside the star formation cutoff in a galactic disk.

On the gravitational angular momentum of rotating sources

The gravitational energy-momentum and angular momentum satisfy the algebra of the Poincare group in the full phase space of the teleparallel equivalent of general relativity. The expression for the gravitational energy-momentum may be written as a surface integral in the three-dimensional spacelike hypersurface, whereas the definition for the angular momentum is given by a volume integral. It turns out that in practical calculations of the angular momentum of the gravitational field generated by localized sources like rotating neutron stars, the volume integral reduces to a surface integral, and the calculations can be easily carried out. Similar to previous investigations in the literature, we show that the total angular momentum is finite provided a certain asymptotic behaviour is verified. We discuss the dependence of the gravitational angular momentum on the frame, and argue that it is a measure of the dragging of inertial frames.

Inertial frame rotation induced by rotating gravitational waves

We calculate the rotation of the inertial frame within an almost flat cylindrical region surrounded by a pulse of non-axially-symmetric gravitational waves that rotate about the axis of our cylindrical polar coordinates. Our spacetime has only one Killing vector. It is along the z-axis and hypersurface orthogonal. We solve the Einstein equations to first order in the wave amplitude and superpose such linearized solutions to form a wave pulse. We then solve the relevant Einstein equation to second order in the amplitude to find the rotation of inertial frames produced by the pulse. The rotation is without time delay. The influence of gravitational wave angular momentum on the inertial frame demonstrates that Mach's principle cannot be expressed in terms of the influence of the stress–energy–momentum tensor alone but must also involve influences of gravitational wave energy and angular momentum.

Exoplanet properties from Lick, Keck and AAT

Doppler-shift measurements with a remarkable precision of Δλ/λ=3×10−9, corresponding to velocities of 1 m s−1, have been made repeatedly of 2500 stars located within 300 light years. The observed gravitational perturbations of the stars have revealed 250 orbiting planets, with 27 that cross in front of the host star, blocking a fraction of the starlight to allow measurement of the planet's mass, radius and density. Two new discoveries are the first good analog of Jupiter (HD 154345b) and the first system of five planets (55 Cancri). The predominantly eccentric orbits of exoplanets probably result from planet–planet gravitational interactions or angular momentum exchange by mean-motion resonances. The planet mass distribution ranges from ∼15 M JUP to as low as ∼5 M Earth and rises toward lower masses as dN/dM∝M −1.1. The distribution with orbital distance, a, rises (in logarithmic intervals) as dN/d log a∝a +0.4. Extrapolation and integration suggests that 19% of all Sun-like stars harbor a gas-giant planet within 20 AU, but there remains considerable incompleteness for large orbits. Beyond 20 AU, the occurrence of gas-giant planets may be less than a few per cent as protoplanetary disk material there has lower densities and is vulnerable to destruction. Jupiter-mass planets occur more commonly around more massive stars than low mass stars. The transit of the Neptune-mass planet, Gliese 436b, yields a density of 1.55 g cm−3 suggesting that its interior has an iron–silicate core surrounded by an envelope of water–ice and an outer H–He shell. Planets with masses as low as five Earth-masses may be commonly composed of iron–nickel, rock and water along with significant amounts of H and He, making the term ‘super-Earth’ misleading. The transiting planet HD147506b has high orbital eccentricity but no significant orbital inclination to the line of sight, presenting a puzzle about its history. Its orbit together with the mean motion resonances of 4 of the 22 multi-planet systems provides further evidence for the role of planet–planet interactions in shaping planetary architectures.

The angular momentum of the gravitational field and the Poincaré group

We redefine the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity. Similar to the gravitational energy–momentum, the new definition for the gravitational angular momentum is coordinate independent. By considering the Poisson brackets in the phase space of the theory, we find that the gravitational energy–momentum and angular momentum correspond to a representation of the Poincaré group. This result allows us to define Casimir type invariants for the gravitational field.

Accurate Evolutions of Orbiting Black-Hole Binaries without Excision

We present a new algorithm for evolving orbiting black-hole binaries that does not require excision or a corotating shift. Our algorithm is based on a novel technique to handle the singular puncture conformal factor. This system, based on the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein's equations, when used with a "precollapsed" initial lapse, is nonsingular at the start of the evolution and remains nonsingular and stable provided that a good choice is made for the gauge. As a test case, we use this technique to fully evolve orbiting black-hole binaries from near the innermost stable circular orbit regime. We show fourth-order convergence of waveforms and compute the radiated gravitational energy and angular momentum from the plunge. These results are in good agreement with those predicted by the Lazarus approach.

Angular Momentum of the BTZ Black Hole in the Teleparallel Geometry

We carry out the Hamiltonian formulation of the three- dimensional gravitational teleparallelism without imposing the time gauge condition, by rigorously performing the Legendre transform. Definition of the gravitational angular momentum arises by suitably interpreting the integral form of the constraint equation Gama^ik=0 as an angular momentum equation. The gravitational angular momentum is evaluated for the gravitational field of a rotating BTZ black hole.

Rotating neutron stars: an invariant comparison of approximate and numerical space-time models

We compare three different models of rotating neutron star spacetimes: i) the Hartle-Thorne (1968) slow-rotation approximation, keeping terms up to second order in the stellar angular velocity; ii) the exact analytic vacuum solution of Manko et al. (2000a, 2000b); and iii) a numerical solution of the full Einstein equations. In the first part of the paper we estimate the limits of validity of the slow-rotation expansion by computing relative errors in the spacetime’s quadrupole moment Q and in the corotating and counterrotating radii of Innermost Stable Circular Orbits (ISCOs) R±. We integrate the Hartle-Thorne structure equations for five representative equations of state. Then we match these models to numerical solutions of the Einstein equations, imposing the condition that the gravitational mass and angular momentum of the models be the same. We find that the Hartle-Thorne approximation gives very good predictions for the ISCO radii, with R± accurate to better than 1% even for the fastest millisecond pulsars. At these rotational rates the accuracy on Q is � 20%, and better for longer periods. In the second part of the paper we focus on the exterior vacuum spacetimes, comparing the Hartle-Thorne approximation and the Manko analytic solution to the numerical models using Newman-Penrose (NP) coordinateindependent quantities. For all three spacetimes we introduce a physically motivated ‘quasi-Kinnersley’ NP frame. In this frame we evaluate a quantity, the speciality index S, measuring the deviation of each stellar model from Petrov Type D. Deviations from speciality on the equatorial plane are smaller than 5% at star radii for the faster rotating models, and rapidly decrease for slower rotation rates and with distance. We find that, at leading order, the deviation from Type D is proportional to (Q QKerr). Our main conclusion is that the Hartle-Thorne approximation is very reliable for most astrophysical applications.