When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free (STF) tensors: (i) the Weyl tensor's so-called "electric" part or tidal field E jk , which raises tides on the Earth's oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ξ k is ∆aj = −E jk ξ k ); and (ii) the Weyl tensor's so-called "magnetic" part or (as we call it) frame-drag field B jk , which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ξ k , as measured using a local inertial frame at the tail of ξ k , is ∆Ωj = B jk ξ k ). Being STF, E jk and B jk each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of E jk 's eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line's eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for B jk are frame-drag vortex lines or simply vortex lines, their vorticities, and vortexes.These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side-by-side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing current-quadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outward-and-backward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing mass-quadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become current-quadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary's tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary's mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate).In paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a ...

We explore prospects for detecting gravitational waves from stellar-mass compact objects spiraling into intermediate mass black holes (BHs) (M 50M to 350M ) with ground-based observatories. We estimate a rate for such intermediate-mass-ratio inspirals of &1-30 yr ÿ1 in Advanced LIGO. We show that if the central body is not a BH but its metric is stationary, axisymmetric, reflection symmetric and asymptotically flat, then the waves will likely be triperiodic, as for a BH. We suggest that the evolutions of the waves' three fundamental frequencies and of the complex amplitudes of their spectral components encode (in principle) details of the central body's metric, the energy and angular momentum exchange between the central body and the orbit, and the time-evolving orbital elements. We estimate that advanced ground-based detectors can constrain central body deviations from a BH with interesting accuracy. [3] and its international partners will increase the volume of the Universe searched a thousandfold or more. The most promising GW sources for this network are the inspiral and coalescence of black hole (BH) and/or neutron star (NS) binaries. Current inspiral searches target sources with total mass M & 40M : NS binaries with masses 1-3M , BH binaries with masses 3-40M , and NS-BH binaries with components in these mass ranges [4,5].Ultraluminous x-ray observations and simulations of globular cluster dynamics suggest the existence of intermediate-mass black holes (IMBHs) with masses M 10 2 -10 4 M [6]. The GWs from the inspiral of a NS or stellar-mass BH into an IMBH with mass M 50-350M will lie in the frequency band of AdvLIGO. These intermediate-mass-ratio inspirals (IMRIs) are analogous to the extreme-mass-ratio inspirals (EMRIs) of 10M objects spiraling into 10 6 M BHs, targeted by the planned LISA observatory [7]. We consider IMRIs containing NSs and BHs, as less compact objects (e.g., white dwarfs) are tidally disrupted at frequencies too low to be detectable in AdvLIGO.If we consider the possibility that the central body of an IMRI (or EMRI) is not a BH, but some other general relativistic object (e.g., a boson star or a naked singularity [8]), then we can quantify the accuracy with which it has the properties predicted for a BH that: (i) it obeys the BH no-hair theorem (its spacetime geometry is the Kerr metric, fully determined by its mass and spin), and (ii) its tidal coupling (tide-induced transfer of energy and angular momentum between orbit and body) agrees with BH predictions. Searching for non-BH objects may yield an unexpected discovery.We report on our initial explorations of the prospects for detecting GWs from IMRIs and probing the properties of IMRIs' central bodies. We report (i) IMRI event rate estimates in AdvLIGO, (ii) estimates of the efficacy of GW template families for IMRI searches, (iii) explorations of the character of the IMRI (EMRI) waves if the central body is not a BH, (iv) generalizations of Ryan's theorem concerning the information about the central body carried by IMRI and EMRI w...

I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a two degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about two degree of freedom systems. Evidence is given, in the form of orbit-crossing structure, that geodesics in SAV spacetimes might admit, a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).PACS numbers:

R-modes of a rotating neutron star are unstable because of the emission of gravitational radiation. We explore the saturation amplitudes of these modes determined by nonlinear mode-mode coupling. Modelling the star as incompressible allows the analytic computation of the coupling coefficients. All couplings up to n = 30 are obtained, and analytic values for the shear damping and mode normalization are presented. In a subsequent paper we perform numerical simulations of a large set of coupled modes.

Two mechanisms for nonlinear mode saturation of the r-mode in neutron stars have been suggested: the parametric instability mechanism involving a small number of modes and the formation of a nearly continuous Kolmogorov-type cascade. Using a network of oscillators constructed from the eigenmodes of a perfect fluid incompressible star, we investigate the transition between the two regimes numerically. Our network includes the 4995 inertial modes up to n ≤ 30 with 146,998 direct couplings to the r-mode and 1,306,999 couplings with detuning< 0.002 (out of a total of approximately 10 9 possible couplings). The lowest parametric instability thresholds for a range of temperatures are calculated and it is found that the r-mode becomes unstable to modes with 13 < n < 15. In the undriven, undamped, Hamiltonian version of the network the rate to achieve equipartition is found to be amplitude dependent, reminiscent of the Fermi-Pasta-Ulam problem. More realistic models driven unstable by gravitational radiation and damped by shear viscosity are explored next. A range of damping rates, corresponding to temperatures 10 6 K to 10 9 K, is considered. Exponential growth of the r-mode is found to cease at small amplitudes ≈ 10 −4 . For strongly damped, low temperature models, a few modes dominate the dynamics. The behavior of the r-mode is complicated, but its amplitude is still no larger than about 10 −4 on average. For high temperature, weakly damped models the r-mode feeds energy into a sea of oscillators that achieve approximate equipartition. In this case the r-mode amplitude settles to a value for which the rate to achieve equipartition is approximately the linear instability growth rate. In a previous paper [1] (henceforth Paper I), we presented details on the computation of the coupling coefficients and damping/driving rates for the generalized r-modes of a uniform density, uniformly rotating star in the limit of slow rotation. In this paper we explore the nonlinear dynamics of the oscillator network constructed in Paper I. We concentrate on the results that will have physical implications for the saturation of the r-mode instability in neutron stars. The astrophysical motivation for this problem was given in Paper I.Consider a sea of coupled oscillators obeying the amplitude equationṡwhere the coupling coefficients κ ABC , the driving/damping rates γ A and the frequencies w A have been derived from the fully nonlinear problem by means of second order perturbation theory. Only the most significant nonlinear and driving terms have been retained. Thus, we assume that modes never achieve large amplitudes c α , and explore the dynamics in the limit of weak nonlinearity. The validity of this assumption can be assessed in retrospect. The oscillator network for the r-mode problem has many distinct properties inherited from the original physical problem: The rescaled frequencies of the oscillators are bounded between -1 and 1, the connectedness of the coupling graph is determined by the selection rules and the general background...