55 3.4 Number of IMRI wave cycles in the Advanced LIGO band as a function of the central black. 132 5.2 The fractional slip of the inertial axes at the center of a model universe with a spherical axis.
Tidal interaction between compact objects and massive black holes
Tidal coupling between Earth and Moon
In this model, we give an outer edge to a spatially homogeneous, expanding, slowly rotating universe and examine the time evolution of the spin of a local inertial-driven gyroscope at the center. We discuss our results in terms of Mach's principle [21], which states that the inertial axes are related to the mean rotational motion of the matter that fills the distant universe.
Black hole tidal coupling
Noticeably different from the case of the Earth and the Moon, the tidal coupling in the case of a black hole has always been described in terms of the influence of the orbiting object's gravitational field on the hole's horizon - the perturbation of the horizon's 2-metric ( or the transformation of the tidal field to gravitational radiation at the horizon by gravitational blueshift and the energy and angular momentum carried inward by these waves [13]. We define our horizon phase shift to be the angle between the perturbing tidal field at the horizon of the horizon and displacement (i.e., the rate of change of shape) of the horizon, and we find, that the shift of the horizon leads to the disturbing.
IMRI waves for Advanced LIGO
Equation (1.4) is the hairless black hole theorem, which states that the entire multipole moment structure of a black hole's spacetime is determined solely by the black hole's mass and spin. The 'dive frequency' of gravitational waves can be used to indicate the existence and location of the horizon, allowing a massive black hole to be distinguished from a massive forest star [39].
Template waveforms for EMRI waves
The Press waveforms consistently outperform the quadrupole and quadrupole-octupole waveforms, but the improvement is not large. In the research presented in this chapter, I am primarily responsible for the development of the Press NK waveforms.
General relativity and Mach’s Principle
Motivations
Ryan's proof that the EMRI waves carry a map of the central body's metric completely ignored the influence of tidal coupling between the central body and the orbiting object. Because in Ryan's analysis the map is encoded redundantly in the inspiratory phase of the EMRI waves and in their modulations, it is reasonable to hope that the tidal coupling will break this redundancy in such a way as to allow extraction of both the map and details of the central body response on the tidal gravitational pull from the orbiting object [8].
Framework and results
Using this definition, we find that the shear of the central hole drives the perturbing tidal field on the horizon at an angle. 2.5 we study the phase shift between horizon deformation and applied tides and compare the result with the phase shift as defined by Hartle.
Problem setup
For the static disturbances generated by E20i j we impose a “regularity boundary condition”: the disturbances must be physically finite atr=2M. For the time-dependent perturbations generated by Eext,2±2i j we establish the “boundary condition for the incoming wave”: the perturbations exhibit the asymptotic behavior ∼e∓iωr∗ when approaching the horizon.
Time-dependent part of the perturbation
The perturbed metric
The Eexti j tidal field also sets the outer boundary condition for the problem: the O(r0) terms in the perturbed Eexti j field are known to go to Eexti j in the LARF [Eq. 26] but will be determined by the outer boundary condition in our problem; Fis the hypergeometric function with parameters [Eq.
Induced quadrupole moment in the LARF
Note that the induced quadrupole moment is proportional to the time derivative of the applied tidal field. The induced quadrupole moment and the applied tidal field are therefore completely out of phase with each other (π/4 phase shift in space, π/2 in time).
The static, axisymmetric part of the perturbation
Static induced quadrupole moment
Therefore, the remaining term P22 in Eq. 2.28) continues to grow quadratically as it becomes large, corresponding to a non-asymptotic flatness due to the presence of the moon. It is clear, therefore, that the shape of the horizon does acquire a small quadrupole component.
Ambiguity of the static induced quadrupole moment
Finally, we move from the LARF to the perturbed horizon and investigate the effect of the static perturbation there. This coupling, as we will show, contributes to the ambiguities in the definition of the SIQM.
The tidal phase shift
Phase of the tidal bulge on the horizon
Therefore, before we can consider the phases of quantities on the horizon, we must first define what we mean by time on the horizon. Also, as observed by Hartle, when the angular velocity Ω is not small compared to 1/M, the deformation of the horizon cannot be described in terms of a phase shift [14].
Phase shift between the tidal bulge and the moon
- Tidal phase shift between a rotating horizon and stationary moon
- Tidal phase shift between a non-rotating horizon and rotating moon
This should be compared to the location of the bump on the future horizon of the hole. Following Hartl, we compare the angular location of the tidal bulge, φbulge( ˜V ), to the angular location of the moon, φmoon( ˜V ), using zero angular momentum (ZAM) incoming zero rays to provide a relation between φ at the moon's orbit and φ at the horizon .
Concluding discussion
- Advanced LIGO and IMRIs
- IMRI and EMRI orbital dynamics in “SARSAF” spacetimes
- Overview of estimates on measurement accuracies for Advanced LIGO
- Outline of this chapter
Equation (2.77) is the value of the Regge-Wheeler function in the LARF; put it in Eq. This includes estimates of the detection range and number of wave cycles in the Advanced LIGO band.
Initial estimates
- Detection range
- Choice of IMRI waveform templates
- Number of wave cycles in the Advanced LIGO band
- Number of wave cycles contributed by tidal coupling
When calculating the integral Eq. 3.9a), we assume that the starting frequency is 10 Hz (the lower cutoff frequency of the advanced LIGO interferometers), and that the final frequency is the wave frequency (Fisco) corresponding to the inner stable circular orbit (isco) of the central black hole. Since we only consider the harmonicm=2 component of the gravitational wave, the number of wave cycles until the isco is reached.
Advanced LIGO’s accuracy for measuring tidal coupling
Review of parameter estimation
When the SNR is large,13 it can be shown that the measurement error distribution δα≡α˜ −αˆ follows a multivariate Gaussian distribution (Eq. The variance-covariance matrixΣ, defined as the inverse Fisher matrix, actually provides a lower bound (the so-called Cram´er- Rao's bound) to the least mean square error (see e.g. [59] and references therein).
Gravitational waveforms for circular equatorial orbits
This agrees with Gal'tsov's slow motion calculation [61]. 3.33b), we obtain the phase of the frequency domain waveform. This limitation does not concern us since most of the "information" is contained in the phase of the waveform [i.e. in Eq.
Approximate waveforms for circular inclined orbits
The pieceδΦis the precessional correction forΦc; it arises from the changing orientation of the orbital plane. Finally, the stretch is the polarization phase, which results from the changing projection of the orbital plane on the detector frame.
Results and comparison
The key quantity that determines δΦinϕ is the time-evolving ˆL, the direction of the orbital angular momentum. ACST introduced the post2-Newtonian precession equations for the orbital angular momentum and spin angular momentum, SandS0, of two bodies in a binary system.
Introduction
The principle of the kludges is to combine different prescriptions for the orbital evolution and the waveform emission (not necessarily in a self-consistent way). The first step of the NC is to produce an inspiring trajectory in "phase space" — i.e.
Waveform inventory
Teukolsky-based numerical waveforms
However, the improvement in the relatively strong field regime is only small - we did not find any regime where the Presse formula is a significant improvement of the quadrupole-octupole prescription. To date, the Teukolsky equation has been solved in the frequency domain for circularly inclined orbits [15] , eccentric-equatorial orbits [16] , and most recently for a handful of generic (eccentrically inclined) orbits [25] .
Analytic waveforms
Even for geodetic trajectories, AK and NK waveforms with the same physical parameters quickly go out of phase, since the frequency structure of the two waveform families is different. In the future, the effectiveness of the AK waveforms as detection templates will be investigated by using banks of AK templates to search for more accurate NK or TB waveforms embedded in noise.
Semi-relativistic numerical kludge waveforms
This arises because AK uses a Keplerian orbital parameterization, compared to the geodesic parameterization used in NK. For an equatorial orbit with semi-latus rectump =30Mand eccentricitye=0.3 around a 106Mb black hole spina=0.8, the azimuthal frequency of the NK waveform is 0.196mHz compared to 0.216mHz for AK.
Generation of numerical kludge waveforms
Construct orbital trajectory
For a given E,LzandQ, the roots of Vr determine the turning points of the radial motion - the periastron rp and apastronra. This leads to obvious inconsistencies in approach - e.g. non-conservation of flat space energy momentum.
Compute gravitational waveforms from particle trajectory
The waveform in the standard transverse-trackless (TT) meter is given by the TT projection of the above expressions. It is important to underline the physical assumptions made in the derivation of the quadrupole formula (4.16), the quadrupole-octupole formula (4.17) and the Press formula (4.19) to understand their generic limitations.
Kludge waveforms: Results and comparison
The overlap between waveforms
The PSD in the denominator of Eq. 4.35) serves to suppress those frequency components of the signal where the detector noise is large. We conclude our discussion of the overlap with a description of our approach to the LISA noiseSh(f).
Time-domain comparison
The waveforms are scaled in units of D/µ, where Dis is the radial distance of the observation point from the source and µ is the mass of the test body. The waveforms are scaled in units of D/µ, where D is the radial distance of the observation point from the source and µ is the mass of the test body.
Frequency-domain comparison
As previously mentioned, the flat-space emission formulas used in the production of NK waveforms do not account for all scattering effects due to background curvature. In addition to the harmonics coming from the azimuthal motion, there are many additional components coming from the flickering between the harmonics of the fundamental frequencies of the motion φ-, θ-, etc.: Ωφ, Ωθ, Ωr (see [67] for the Fourier decomposition of the orbital motion).
Concluding discussion
As the universe expands, this frame drag weakens, and at late times near-Earth inertia breaks free from the grip of the universe's rotating matter. Apparently, the weakness of the Earth's frame drag is due to the Earth's large size compared to the equivalent black hole: R ~104 km2GM/c2 ~ 1 cm.
Frame dragging: The space-drag and gravitomagnetic descriptions
Fluid like space-drag description
The simplest and clearest version of Mach's principle states that, when there are no massive spinning bodies nearby (such as Earth, a neutron star, or a black hole), axes of inertia are associated with the mean rotational motion of the matter that fills the distant universe. This is similar to the speed of the air in a tornado or the water in a vortex: greater near the center than far away.
Gravitomagnetic description
Inserting the expression (5.6) for~vspace yields the exact angular velocity of the drag frame predicted by general relativity, Eq. the force between like particles is attractive as opposed to repulsive electromagnetic force; factor 4 can be traced to the "spin-2" nature of weak gravity, which is considered canonical field theory, as opposed to the spin-1 nature of electromagnetism. If we translate the magnetic precession equation to the gravitomagnetic case2, we get for the gyroscopic precession angular velocity.
Relation of space-drag and gravitomagnetic descriptions of frame dragging
Like a spinning sphere, the charge generates a dipolar magnetic field via Eq. 5.9), so the rotating Earth generates a dipolar magnetic field via Eq. Since the gravitomagnetic force on a massive element inside a gyroscope has the standard form of the Lorenz force (~v/c)×H, the rotation of a gyroscope progresses around a gravitomagnetic field in the same way as a rotating particle with a magnetic dipole. spin moment about an ordinary magnetic field [22].
Mach’s Principle
- Overview
- Spatially homogeneous model universe that rotates slowly and rigidly
- Universe with an outer edge
- No rotation
- Exterior of the rotating universe
- Matching interior and exterior of rotating universe
- Universe with dark energy and an outer edge
- Non-rotating universe and exterior
- Slowly rotating universe and its exterior
The matching of the 3-metric and the outer curvature at the boundary of the universe can be done in the same way as in Sec. Let 4 be the velocity of the universe surface in the outer SdS geometry and let be the unit normal vector.
