Linkages in general relativity

In this paper, we obtain a formula for time in general relativity from solutions of the Einstein-Hamilton-Jacobi equation along paths of steepest descent in the configuration space of general relativity. The formula for time emerges from the geometry of configuration space via the length of the steepest descent paths which is measured with respect to the de Witt metric on configuration space. The consequences of the indefinite signature of the de Witt metric are discussed in detail. We illustrate the general formula for time in various examples including de Sitter spacetime and asymptotically anti–de Sitter spacetimes. Published by the American Physical Society 2025

A limiting diagram for the Segre classification in 5-dimensional space–times is obtained, extending a recent work on limits of the energy–momentum tensor in general relativity. Some of Geroch’s results on limits of space–times in general relativity are also extended to the context of five-dimensional Kaluza–Klein space–times.

Recent attempts at explaining free-fall motion (i.e. motion only under the influence of gravity) in the context of the General Theory of Relativity have invoked a concept of time passing/progressing. Although students who have studied the Special Theory of Relativity will likely know that this concept does not feature in Special Relativity, they may incorrectly accept the concept of time passing/progressing as an aspect of General Relativity if they have encountered these accounts of free-fall motion. It is pointed out that time passing/progressing is not part of General Relativity theory and is briefly shown that free-fall motion in General Relativity is quantitatively described without such a concept.

It is shown - in Ashtekar's canonical framework of General Relativity - that the constraints of spherically symmetric (Schwarzschild) gravity in 4 dimensional space-time can be solved completely yielding two canonically conjugate observables for asymptotically flat spacetimes, namely mass and - surprisingly - time. The emergence of the time observable is a consequence of the Hamiltonian formulation and its subtleties concerning the slicing of space and time and is not in contradiction to Birkhoff's theorem. Our results can be expressed within the ADM formalism, too, and their relation to the equivalent ones Kuchař obtained recently are briefly discussed. Quantization of the system and the associated Schrödinger equation depend on the allowed spectrum of the masses.

Gravitational wave (GW) observations of binary black hole (BBH) coalescences provide a unique opportunity to test general relativity (GR) in the strong-field regime. To ensure the reliability of these tests, it is essential to identify and address potential sources of error, particularly those arising from missing physics in the waveform models used in GW data analysis. This paper investigates potential biases in these tests arising from strong gravitational lensing, an effect not currently incorporated into the standard framework for GR tests. In the geometric optics approximation, strong lensing produces three types of images: Type I, Type II, and Type III. While Type I and Type III images do not distort the signal, Type II images introduce a characteristic phase shift that can mimic GR deviations for signals with higher-order modes, precession, or eccentricity. We assess the response of four standard GR tests on simulated Type II lensed BBH signals, including the two parameterized tests (TIGER and FTI), the modified dispersion relation test and the inspiral-merger-ringdown consistency test. We focus on precessing waveforms for binaries with total masses of $20M_{\odot}$ and $80M_{\odot}$, and dimensionless spins of 0.5 and 0.95, considering a fixed signal-to-noise ratio of 25 using the design A+ sensitivity of the LIGO-Virgo network. Our findings indicate that more mass-asymmetric and higher-spin binaries show larger false deviations from GR in the TIGER and modified dispersion relation tests when applying GR tests to Type II lensed signals. These results highlight the risk of false GR violations as detector sensitivity improves in future observational runs. Therefore, it is crucial to consider the possibility of strong lensing before drawing conclusions about deviations from GR in GW signals.

Detections of gravitational waves emitted by binary black holes allow for tests of General Relativity in the strong-field regime. In particular, deviations from General Relativity can be observed by comparing incoming signals to waveform templates that include parametrized deviations from General Relativity. However, it is essential that the General Relativity sector of these templates accounts for all predictable physics. Otherwise, missing physics might be mimicked by the "beyond General Relativity" sector of the templates, leading the analysis to report apparent deviations from General Relativity. Current parametrized tests implement templates that omit physical phenomena such as orbital eccentricity and higher-order modes. In this paper, we show how the omission of higher modes can lead to false deviations from General Relativity when these effects are strong enough. We study the extent of these deviations as a function of the mass ratio and the orbital orientation. We find that significant false deviations can arise when current tests are performed on signals emitted by asymmetric binaries whose orbital angular momentum is orthogonal to the line-of-sight. We estimate that the Advanced LIGO-Virgo network operating at its design sensitivity can observe false violations with a significance above $5 \sigma$ as often as once per year. Similar results are expected for other tests of General Relativity that use modified waveform models. Hence, we stress the necessity of accurate waveform models that include physical effects such as higher-order modes to trust future tests of General Relativity.

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc. In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz. In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time. In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements. For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.

In general relativity there is a long-standing problem of defining momentum and angular momentum in a general curved space-time. In this chapter, we describe an application of twistor theory which aims to provide a solution to this problem. As we shall see, this aim is not entirely achieved and the programme has both successes and failures. We recall that in general relativity, all the local matter content is described by the stress–energy tensor T ab . Gravitational energy, whether in gravitational waves or in the form of gravitational potential energy, is notoriously non-local and one cannot expect to characterise it by a local density. Instead, Penrose (1982) has suggested that one should seek a non-local invariant associated to any two-surface S and representing the total momentum–angular momentum flux through that surface. This non-local invariant will be constructed by twistorial techniques tailored to give the right answer for linearised general relativity where there is a clear right answer. We begin by reviewing the definition of momentum and angular momentum in special relativity and in linearised general relativity. A material system in special relativity is defined by its stress–energy tensor T ab which we may suppose for definiteness to have support within a world tube W in Minkowski space, M . (The case of say electromagnetic fields spread throughout M is a simple generalisation.)

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

Gravitational waves can act like gravitational lenses, affecting the observed positions, brightnesses, and redshifts of distant objects. Exact expressions for such effects are derived here in general relativity, allowing for arbitrarily-moving sources and observers in the presence of plane-symmetric gravitational waves. At least for freely falling sources and observers, it is shown that the commonly-used predictions of linear perturbation theory can be generically overshadowed by nonlinear effects; even for very weak gravitational waves, higher-order perturbative corrections involve secularly-growing terms which cannot necessarily be neglected when considering observations of sufficiently distant sources. Even on more moderate scales where linear effects remain at least marginally dominant, nonlinear corrections are qualitatively different from their linear counterparts. There is a sense in which they can, for example, mimic the existence of a third type of gravitational wave polarization.

Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dimensions. This completion is used to analyze the asymptotic symmetries, introduce the analog of the 4-dimensional Bondi energy-momentum and write down a flux formula. The analysis is also of interest from a purely 3-dimensional perspective because it pertains to a diffeomorphism invariant 3-dimensional field theory with {\it local} degrees of freedom, i.e., to a midi-superspace. Furthermore, due to certain peculiarities of 3 dimensions, the description of null infinity does have a number of features that are quite surprising because they do not arise in the Bondi-Penrose description in 4 dimensions.

Scalar-tensor theories leaving significant modifications of gravity at cosmological scales rely on screening mechanisms to recover general relativity (GR) in high-density regions and pass stringent tests with astrophysical objects. Much focus has been placed on the signatures of such modifications of gravity on the propagation of gravitational waves (GWs) through cosmological distances while typically assuming their emission from fully screened regions with the wave generation strictly abiding by GR. Here, we closely analyze the impact of screening mechanisms on the inspiral GW waveforms from compact sources by employing a scaling method that enables a post-Newtonian (PN) expansion in screened regimes. Particularly, we derive the leading-order corrections to a fully screened emission to first PN order in the near zone, and we also compute the modifications in the unscreened radiation zone to second PN order. For a concrete example, we apply our results to a cubic Galileon model. The resulting GW amplitude from a binary black hole inspiral deviates from its GR counterpart at most by one part in ${10}^{2}$ for the modifications in the radiation zone and at most one part in ${10}^{11}$ due to next-order corrections to the fully screened near zone. We expect such modifications to be undetectable by the current generation of GW detectors, but the deviation is not so small as to remain undetectable in future experiments.

Plane walls (including plane domain walls) without reflection symmetry are studied in the framework of Einstein's general relativity. Using the distribution theory, all the Einstein field equations and Bianchi identities are split into two groups: one holding in the regions outside of the wall and the other holding at the wall. The Einstein field equations at the wall are found to take a very simple form, and given explicitly in terms of the discontinuities of the metric coefficients and their derivatives. The Bianchi identities at the wall are also given explicitly. Using the latter, the interaction of a plane wall with gravitational waves and some specific matter fields is studied. In particular, it is found that, when a gravitational plane wave passes through a wall, if the wall has no reflection symmetry, the phenomena, such as reflection, stimulation, or absorption, in general, occur. It is also found that, unlike for gravitational waves, a massless scalar wave or an electromagnetic wave continuously passes through a wall without any reflection. The repulsion and attraction of a plane wall are also studied. It is found that the acceleration of an observer who is at rest relative to the wall usually consists of threemore » parts: one is due to the force produced by the wall, the second is due to the force produced by the space-time curvature, which is zero if the wall has reflection symmetry, and the last is due to the accelerated motion of the wall. As a result, a repulsive (attractive) plane wall may not be repulsive (attractive) at all. Finally, the collision and interaction among the walls are studied.« less

In general relativity (GR), linearized gravitational waves propagating in empty Minkowski spacetime along a fixed spatial direction have the property that the wave front is the Euclidean plane. Beyond the linear regime, exact plane waves in GR have been studied theoretically for a long time and many exact vacuum solutions of the gravitational field equations are known that represent plane gravitational waves. These have parallel rays and uniform wave fronts. It turns out, however, that GR also admits exact solutions representing gravitational waves propagating along a fixed direction that are nonplanar. The wave front is then nonuniform and the bundle of rays is twisted. We find a class of solutions representing nonplanar unidirectional gravitational waves and study some of the properties of these twisted waves.

We perform tests of general relativity (GR) with gravitational waves (GWs) from the inspiral stage of compact binaries using a theory-independent framework, which adds generic phase corrections to each multipole of a GR waveform model in frequency domain. This method has been demonstrated on Laser Interferometer Gravitational Wave Observatory-Virgo observations to provide stringent constraints on post-Newtonian predictions of the inspiral and to assess systematic biases that may arise in such parametrized tests. Here, we detail the anatomy of our framework for aligned-spin waveform models. We explore the effects of higher modes in the underlying signal on tests of GR through analyses of two unequal-mass simulated binary signals similar to GW190412 and GW190814. We show that the inclusion of higher modes improves both the precision and the accuracy of the measurement of the deviation parameters. Our testing framework also allows us to vary the underlying baseline GR waveform model and the frequency at which the non-GR inspiral corrections are tapered off. We find that to optimize the GR test of high-mass binaries, comprehensive studies would need to be done to determine the best choice of the tapering frequency as a function of the binary's properties. We also carry out an analysis on the binary neutron-star event GW170817 to set bounds on the coupling constant ${\ensuremath{\alpha}}_{0}$ of Jordan-Fierz-Brans-Dicke gravity. We take two plausible approaches; the first approach involves translating directly the ``theory-agnostic'' bound on dipole-radiation into a bound on ${\ensuremath{\alpha}}_{0}$ for different neutron-star equations of state (EOS). The second ``theory-specific'' approach involves reparametrizing the test such that the deviation parameter is ${\ensuremath{\alpha}}_{0}$ itself. The two approaches provide slightly different bounds, namely, ${\ensuremath{\alpha}}_{0}\ensuremath{\lesssim}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}$ and ${\ensuremath{\alpha}}_{0}\ensuremath{\lesssim}4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}$, respectively, at 68% credible level. These differences arise mainly since in the theory-specific approach the tidal and scalar-charge parameters are fixed coherently for each neutron-star EOS and mass.