Gravitational waves and Higgs-like potential from Alena Tensor

<p indent="0mm">The direct detection of gravitational waves from stellar-mass compact binary merger by ground-based laser interferometer gravitational wave detector LIGO/Virgo has verified the prediction of general relativity and opened a new chapter in gravitational wave astronomy. Up to now, a total of 50 gravitational wave events have been detected and published in GWTC-1 and GWTC-2 catalogue. In the near future, the third-generation ground based gravitational wave detector, such as the Einstein Telescope (ET), will be constructed with sensitivity improved by at least a factor of 10. Tens of thousands of gravitational wave signals are expected to be detected per year in the third-generation detector era. These gravitational wave signals will inevitably overlap with foreground massive celestial bodies (such as black hole, galaxy and galaxy cluster), thus leading to lensed gravitational wave signals which will undoubtedly be another important test of general relativity once detected. Furthermore, strongly lensed gravitational wave signals by galaxy from massive binary black hole could possibly be detected by future space detector, e.g., LISA and DECIGO. Since the wavelengths of gravitational waves are comparable with the size of some lens, the lensed gravitational waves play a unique role in studying the phenomena of wave nature, e.g., interference and diffraction. Lensed gravitational wave-electromagnetic wave system will have a wide range of applications in fundamental physics, cosmology and astrophysics when a series of lensed gravitational wave events and their corresponding electromagnetic counterparts have been detected. The most obvious advantage of lensed gravitational wave-electromagnetic wave system lies in that gravitational wave could provide time delay information with high accuracy, and electromagnetic wave could provide Fermat potential difference with high precision because a relatively complete arc of light could be obtained by electromagnetic wave observations and this is the most important step in measuring the Fermat potential. Thus, by combining the information from both approaches, lensed gravitational wave-electromagnetic wave system could be applied to study the speed of gravitational waves, constrain cosmological parameters, explore the substructure of the dark matter halo and investigate the lens model and so on. In this paper, we will review in detail how to use geodesic equation, lens equation, as well as wave equation to tackle the stationary scattering problem of lensed gravitational waves, and introduce how lensed gravitational wave-electromagnetic wave system could be applied to study the tensor properties, interference and diffraction effects of gravitational wave, as well as its applications in gravitational wave velocity, Hubble constant, cosmic curvature, lens mass, substructure and so on.

Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition, we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

The problem of obtaining an explicit representation for the fourth invariant of geodesic motion (generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated from an Ernst Potential is considered. The coupling between the non-local curvature content of the spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a constructive, algebraic test for a fourth order Killing tensor suggested. The approach used exploits the variables defined for the B\"{a}ckland transformations to clarify the relationship between Weyl curvature, constants of geodesic motion, expressed as Killing tensors, and the solution generation techniques. A new symmetric non-covariant formulation of the Killing equations is given. This formulation transforms the problem of looking for fourth-order Killing tensors in 4D into one of looking for four interlocking two-manifolds admitting fourth-order Killing tensors in 2D.

We introduce the Higgs mechanism for the self-dual spin connection (also known as the Ashtekar connection), using the Plebański formulation of gravity. We develop our formalism within the framework of the chiral action and derive the equations of motion of the theory. One particular test model is explored: since anisotropy is an intrinsic property of the theory, a modified version of the spatially flat Bianchi I model with two different scale factors is considered. We apply our formalism and derive the Friedmann equations which regulate the scale factors and the Higgs field. We also present a Proca-like term for the connection, which when reduced to minisuperspace with a positive $$\Lambda $$ Λ yields a De Sitter universe with an effective cosmological constant that depends on the mass of the gauge fields. We finally investigate the effect of these mass terms on gravitational waves and find that the wave equation remains unchanged relatively to GR; however the Weyl tensor is scaled by a constant which depends on the mass of the connection components.

The second order Killing and conformal tensors are analyzed in terms of their spectral decomposition, and some properties of the eigenvalues and the eigenspaces are shown. When the tensor is of type I with only two different eigenvalues, the condition to be a Killing or a conformal tensor is characterized in terms of its underlying almost-product structure. A canonical expression for the metrics admitting these kinds of symmetries is also presented. The space–time cases 1+3 and 2+2 are analyzed in more detail. Starting from this approach to Killing and conformal tensors a geometric interpretation of some results on quadratic first integrals of the geodesic equation in vacuum Petrov-Bel type D solutions is offered. A generalization of these results to a wider family of type D space–times is also obtained.

This paper considers the statistical theory of cosmogonical body formation in order to derive field equations and investigate gravitational waves in the gravitational field of a condensing cosmogonical body. The models and evolution equations of the statistical mechanics have been proposed in this theory, while well-known problems of gravitational condensation of infinitely spread cosmic media have been solved on the basis of the proposed statistical model of a spheroidal body. Within the framework of the statistical theory, the presence of an ethereal medium (or a dark matter) around gravitationally interacting masses serves as the basis for their gravitational interactions and gravitational wave propagation. First, equation of motion of a “liquid” particle in the “quasi-solid” approximation inside a spheroidal body in gravitational and inertial fields is considered. Using the vector gravimagnetic potential introduced the equation of moving solid particle with an ethereal surrounding in gravitational and gravimagnetic fields is obtained. Using both Lagrange function of a moving “liquid” gaseous–ethereal particle (with a “solid” core) in the gravitational/gravimagnetic field and Lagrange function of the same field, the expression of a total action for the gravitational/gravimagnetic field and matter is obtained. Applying the principle of least action and varying only the gravitational potentials together with the anti-diffusion velocity (but not the coordinates) the system of equations of the gravitational/gravimagnetic field of a condensing spheroidal body is derived. The equations of plane gravitational waves are obtained. This paper shows that Newtonian approximation (following from Einstein’s GR) does not fully describe the gravitational field of gravitating masses while in the framework of the proposed statistical theory the equation of particle motion in an arbitrary non-inertial frame of reference is derived.

We find and study the properties of black hole solutions for a subclass of Horndeski theory including the cubic Galileon term. The theory under study has shift symmetry but not reflection symmetry for the scalar field. The Galileon is assumed to have linear time dependence characterized by a velocity parameter. We give analytic 3-dimensional solutions that are akin to the BTZ solutions but with a non-trivial scalar field that modifies the effective cosmological constant. We then study the 4-dimensional asymptotically flat and de Sitter solutions. The latter present three different branches according to their effective cosmological constant. For two of these branches, we find families of black hole solutions, parametrized by the velocity of the scalar field. These spherically symmetric solutions, obtained numerically, are different from GR solutions close to the black hole event horizon, while they have the same de-Sitter asymptotic behavior. The velocity parameter represents black hole primary hair.

Forms are obtained for all of the spacetime metrics which admit or are conformal to those which admit Killing tensors whose Segre characteristics are [(11) (11)] and whose eigenvalues λ(1) and λ(2) are not constants. No a priori assumptions are made concerning separability, isometries, invertibility, Petrov type, or the matter tensor. Some of our metrical forms, viz., those for which the Hamilton–Jacobi equation is separable or partially separable after multiplication by an integrating factor, are already known; in particular, they include all of Carter’s Hamilton–Jacobi separable spacetimes. Also, some of our metrical forms are new and may be applicable to finding interesting nonvacuum metrics which admit [(11) (11)] Killing tensors, but which do not necessarily have any Killing vectors. Though our results include a class of metrics for which λ(1) and/or λ(2) are constant, they do not include all such metrics; in an appendix, we prove that the set of all metrics which admit [(11) (11)] Killing tensors with constant λ(1) and λ(2) is identical except for conformal factors with the set of all metrics which admit [(11) (11)] conformal Killing tensors.

The effect of noise induced by gravitons has been investigated using a Bose-Einstein condensate. The general complex scalar field theory with a quadratic self-interaction term has been considered in the presence of a gravitational wave. The gravitational wave perturbation is then considered as a sum of discrete Fourier modes in the momentum space. Varying the action and making use of the principle of least action, one obtains two equations of motion corresponding to the gravitational perturbation and the time-dependent part of the pseudo-Goldstone boson. Coming to an operatorial representation and quantizing the phase space variables via appropriately introduced canonical commutation relations between the canonically conjugate variables corresponding to the graviton and bosonic part of the total system, one obtains a proper quantum gravity setup. Then we obtain the Bogoliubov coefficients from the solution of the time-dependent part of the pseudo-Goldstone boson and construct the covariance metric for the bosons initially being in a squeezed state. The entries of the covariance matrix now involves a stochastic contribution which results in an operatorial stochastic structure of the quantum Fisher information. Using the stochastic average of the Fisher information, we obtain a lower bound on the amplitude parameter of the gravitational wave. As the entire calculation is done at zero temperature, the bosonic system, by construction, will behave as a Bose-Einstein condensate. For a Bose-Einstein condensate with a single mode, we observe that the lower bound of the expectation value of the square of the uncertainty in the amplitude measurement does not become infinite when the total observational term approaches zero. It always has a finite value if the gravitons are initially in a squeezed state with high enough squeezing. In order to sum over all possible momentum modes, we next consider a noise term with a suitable Gaussian weight factor which decays over time. We then obtain the lower bound on the final expectation value of the square of the variance in the amplitude parameter. Because of the noise induced by the graviton, there is a minimum value of the measurement time below which it is impossible to detect any gravitational wave using a Bose-Einstein condensate. Finally, we consider interaction between the phonon modes of the Bose-Einstein condensate which results in a decoherence. We observe that the decoherence effect becomes significant for gravitons with minimal squeezing. Published by the American Physical Society 2024

We consider colliding wave packets consisting of hybrid mixtures of electromagnetic, gravitational, and scalar waves. Irrespective of the scalar field, the electromagnetic wave still reflects from the gravitational wave. Some reflection processes are given for different choice of packets in which the Coulomb-like component {psi}{sub 2} vanishes. Exact solution for multiple reflection of an electromagnetic wave from successive impulsive gravitational waves is obtained in a closed form. It is shown that a successive sign flip in the Maxwell spinor arises as a result of encountering with an impulsive train (i.e. the Dirac's comb curvature) of gravitational waves. Such an observable effect may be helpful in the detection of gravitational wave bursts.

Generalized Killing tensors (GKT) and generalized conformal Killing tensors (GCKT) are defined as the class of totally symmetric tensor fields that generate solutions of a suitable inhomogeneous equation of geodesic deviation along arbitrary and null geodesics, respectively. A geometric interpretation of these fields as generators of Jacobi fields along arbitrary and null geodesics is also given. It is shown that well known fields such as Killing tensors, conformal Killing tensors, and geodesic collineations belong to the above classes. Finally, first integrals of geodesic motion concomitant with the existence of GKT’s and GCKT’s are determined.

We generalize our recent method for constructing Killing tensors of the second rank to conformal Killing tensors. The method is intended for foliated spacetimes of arbitrary dimension $m$, which have a set of conformal Killing vectors. It applies to foliations of a more general structure than in previous literature. The primary idea is to start with a Killing tensor reducible in the foliation slices and lift it to the whole manifold using a system of differential equations. The integrability conditions permitting this lift are derived, and a constructive lifting procedure is presented. The resulting conformal Killing tensor may be irreducible. It is shown that subdomains of foliation slices suitable for the method are fundamental photon surfaces if some additional photon region inequality is satisfied. Thus our procedure also opens the way to obtain a simple general analytical expression for the boundary of the gravitational shadow. We apply this technique to electrovacuum, and $\mathcal{N}=2$, 4, 8 supergravity black holes, providing a new easy way to establish the existence of exact and conformal Killing tensors.

We report on a program, written in the computer algebra system SHEEP, for verifying the components of Killing tensors and conformal Killing tensors. We give some examples, including the components of the Killing tensor admitted by the Kerr metric. We also note that the explicit form of all conformal Killing tensors for a subclass of the Petrov typeD solutions is known.

The Riemann tensor is the most important geometric quantity that one can compute at any given point on a manifold of interest. It measures the extent to which the metric at the given point deviates from flatness. The Riemann tensor can be decomposed in two parts: a tracefree part that is given by the Weyl tensor and a part that is related to the Ricci curvature and the scalar associated with this. Of course, in Einstein’s theory of gravity, the Ricci part of the Riemann tensor is constrained to be equal to the energy-momentum tensor of the matter fields via the Einstein equation. Thus, in general relativity, the Weyl tensor is the geometric quantity that encodes the propagating degrees of freedom in a solution of the theory. The existence of gravitational radiation is an important prediction of general relativity and, as explained above, it is the Weyl tensor that contains information about the specific characteristics of this radiation. In general, a source of gravitational radiation has non-trivial dynamics, Thus, the emitted gravitational radiation, and by implication, the Weyl tensor will be complicated. However, in four spacetime dimensions, the Weyl tensor exhibits a “peeling” property, which ensures that at distances far enough away from the isolated source, complicated components of the radiation decay away. More precisely, in 4d, the Weyl tensor of an asymptotically flat spacetime is of the form:

We study Killing tensors in the context of warped products and apply the results to the problem of orthogonal separation of the Hamilton-Jacobi equation. This work is motivated primarily by the case of spaces of constant curvature where warped products are abundant. We first characterize Killing tensors which have a natural algebraic decomposition in warped products. We then apply this result to show how one can obtain the Killing-Stäckel space (KS-space) for separable coordinate systems decomposable in warped products. This result in combination with Benenti's theory for constructing the KS-space of certain special separable coordinates can be used to obtain the KS-space for all orthogonal separable coordinates found by Kalnins and Miller in Riemannian spaces of constant curvature. Next we characterize when a natural Hamiltonian is separable in coordinates decomposable in a warped product by showing that the conditions originally given by Benenti can be reduced. Finally, we use this characterization and concircular tensors (a special type of torsionless conformal Killing tensor) to develop a general algorithm to determine when a natural Hamiltonian is separable in a special class of separable coordinates which include all orthogonal separable coordinates in spaces of constant curvature.