Weyl Scalar Research Articles

Petrov type, principal null directions, and Killing tensors of slowly rotating black holes in quadratic gravity

The ability to test general relativity in extreme gravity regimes using gravitational wave observations from current ground-based or future space-based detectors motivates the mathematical study of the symmetries of black holes in modified theories of gravity. In this paper we focus on spinning black hole solutions in two quadratic gravity theories: dynamical Chern-Simons and scalar Gauss-Bonnet gravity. We compute the principal null directions, Weyl scalars, and complex null tetrad in the small-coupling, slow rotation approximation for both theories, confirming that both spacetimes are Petrov type I. Additionally, we solve the Killing equation through rank 6 in dynamical Chern-Simons gravity and rank 2 in scalar Gauss-Bonnet gravity, showing that there is no nontrivial Killing tensor through those ranks for each theory. We therefore conjecture that the still-unknown, exact, quadratic-gravity, black-hole solutions do not possess a fourth constant of motion.

On the stability of pressure isotropy condition in Palatini f(R) gravity

This paper copes with the issue of analyzing the conditions to check the stability of the pressure isotropy condition by taking into account a spherically symmetric dissipative astrophysical configuration with the Palatini [Formula: see text] gravity theory. We work out a differential equation in terms of the Weyl scalar that has a crucial part in the analysis of evolution of considered system. Using this equation, we devise another stellar equation that characterizes the evolution of anisotropic factor. Later, we assume an axially symmetric configuration and extend our analysis for that particular symmetry. It is worth-observing that the physical factors responsible for compelling an initially isotropic object to trigger pressure anisotropy incorporate energy density, dissipative flux and shear in fluid flow.

The Rotating Metrics with the Mass Function ˆM(u; r) in reference to the Theory of General Relativity

Background/Objectives: With reference to the theory of general relativity, the rotating metrics can be designed with mass function ˆM (u; r) . Methods/Statistical analysis: The methods/analysis adapted is the theoretical and mathematical analysis on the theory of general relativity. Findings: The line element can be found out with the help of mass function ˆM(u; r) following the find of the covariant complex null tetrad vectors for the rotating metrics. Then, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars for the rotating metrics can be found out. The expanded form of the mass function ˆM(u; r) with (a=0) can be shown from Wang and Wu (1999). From the expended form of the mass function ˆM(u; r)with (a ̸= 0) , the NP coefficients can be derived. With the help of the scalar k, the surface gravity of the black hole is derived. Novelty/Applications: The findings of covariant complex null tetrad vectors for the rotating metrics, the NP spin — coefficients, the Ricci scalars, and the Weyl scalars are new analysis towards the theory of general relativity. Specific applications are the deep studies on the black hole and its surface gravity. It can be concluded that generally the rotating metric possesses a geodesic (k =e = 0), actually shear free (s = 0) , purely expanding (ˆq ̸ ( = 0) and a non-zero twistw2 ̸= 0) null vector la (Chandrasekhar, 1983). With the help of a scalar k, on a horizon of a Black hole, the surface gravity of the black hole is derived. Keywords: The Mass Function; NP Spin – Coefficients; The Ricci Scalars; The Weyl Scalars; The Surface Gravity of the Black Hole

Dynamics of charged anisotropic spherical collapse in energy-momentum squared gravity

This paper investigates the dynamics of charged spherical collapse with anisotropic matter configuration in the context of energy-momentum squared gravity. This newly developed proposal resolves the big-bang singularity and yields the physically viable cosmological results in the early time universe. We establish dynamical equations through Misner-Sharp technique and analyze the effects of charge, anisotropy, effective matter variables and dark source terms on the collapse rate. A relation between Weyl scalar, fluid parameters and dark source terms is also established. The spacetime is not conformally flat due to the presence of anisotropic pressure, multivariate functions and their derivatives. In order to obtain conformally flat spacetime, we consider a specific model of this gravity, neglect the impact of charge and assume the isotropic matter distribution which yields homogeneity of the energy density and conformally flat spacetime. We conclude that positive dark source terms, anisotropy and charge yield the action of a repulsive force which enhances the stability of the system and hence diminishes the collapse rate.

Original Article The Evaluation of the Rotating Metrics with a ̸= 0 from the Mass Function ˆM (u; r) with the reference to the Theory of General Relativity

Objectives: With the reference to the Einstein’s theory of general relativity (1915), the evaluation of the rotating metrics such as Newman – Penrose Spin – Coefficients or NP Spin – Coefficients, the Ricci Scalars, the Weyl Scalars are designed with a ̸=0from the mass function or metric function ˆM (u; r). Methods: The methods / analysis adapted are the theoretical and mathematical analysis on the Einstein’s theory of general relativity. Findings: The Newman – Penrose Spin Coefficients (NP Spin – Coefficients), the Ricci Scalars, and the Weyl Scalars for the rotating metrics ˆM(u; r) with a ̸= 0 has been evaluated. Given by Wang and Wu (1999), the expanded form of the mass function or metric function with a ̸=0 has been used to evaluate the rotating metrics – NP Spin – Coefficients, the Ricci Scalars and the Weyl Scalars for a ̸= 0. The outcome is that the evaluation of rotating metrics with a ̸= 0 i.e. all the Newman – Penrose Spin Coefficients (NP Spin – Coefficients), the Ricci Scalars and the Weyl Scalars greatly simplifies the analysis of the theory of general relativity. Novelty: From the expended form of the mass function or metric function ˆM(u; r) with a ̸= 0 given by Wang and Wu (1999), all NP Spin – Coefficients, the Ricci Scalars, the Weyl Scalars has been derived for the option a ̸= 0. This paper evaluates the rotating metrics with all NP Spin – Coefficients, the Ricci Scalars and the Weyl Scalars with which greatly simplifies the analysis of the theory of general relativity. Also it is new way of formulation of the theory of general relativity with a ̸= 0. Keywords: The Einstein’s theory of general relativity; The mass function or metric function; The Newman – Penrose Spin - Coefficients (NP Spin – Coefficients); The Ricci Scalars; The Weyl Scalars.

Extending gravitational wave extraction using Weyl characteristic fields

We present a detailed methodology for extracting the full set of Newman-Penrose Weyl scalars from numerically generated spacetimes without requiring a tetrad that is completely orthonormal or perfectly aligned to the principal null directions. We also describe how to implement an extrapolation technique for computing the Weyl scalars' contribution at asymptotic null infinity in postprocessing. These methods have been used to produce $\Psi_4$ and $h$ waveforms for the Simulating eXtreme Spacetimes (SXS) waveform catalog and now have been expanded to produce the entire set of Weyl scalars. These new waveform quantities are critical for the future of gravitational wave astronomy in order to understand the finite-amplitude gauge differences that can occur in numerical waveforms. We also present a new analysis of the accuracy of waveforms produced by the Spectral Einstein Code. While ultimately we expect Cauchy characteristic extraction to yield more accurate waveforms, the extraction techniques described here are far easier to implement and have already proven to be a viable way to produce production-level waveforms that can meet the demands of current gravitational-wave detectors.

Computation of displacement and spin gravitational memory in numerical relativity

We present the first numerical relativity waveforms for binary black hole mergers produced using spectral methods that show both the displacement and the spin memory effects. Explicitly, we use the SXS Collaboration's $\texttt{SpEC}$ code to run a Cauchy evolution of a binary black hole merger and then extract the gravitational wave strain using $\texttt{SpECTRE}$'s version of a Cauchy-characteristic extraction. We find that we can accurately resolve the strain's traditional $m=0$ memory modes and some of the $m\not=0$ oscillatory memory modes that have previously only been theorized. We also perform a separate calculation of the memory using equations for the Bondi-Metzner-Sachs charges as well as the energy and angular momentum fluxes at asymptotic infinity. Our new calculation uses only the gravitational wave strain and two of the Weyl scalars at infinity. Also, this computation shows that the memory modes can be understood as a combination of a memory signal throughout the binary's inspiral and merger phases, and a quasinormal mode signal near the ringdown phase. Additionally, we find that the magnetic memory, up to numerical error, is indeed zero as previously conjectured. Lastly, we find that signal-to-noise ratios of memory for LIGO, the Einstein Telescope (ET), and the Laser Interferometer Space Antenna (LISA) with these new waveforms and new memory calculation are larger than previous expectations based on post-Newtonian or Minimal Waveform models.

Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms

We present several improvements to the Cauchy-characteristic evolution procedure that generates high-fidelity gravitational waveforms at $\mathcal{I}^+$ from numerical relativity simulations. Cauchy-characteristic evolution combines an interior solution of the Einstein field equations based on Cauchy slices with an exterior solution based on null slices that extend to $\mathcal{I}^+$. The foundation of our improved algorithm is a comprehensive method of handling the gauge transformations between the arbitrarily specified coordinates of the interior Cauchy evolution and the unique (up to BMS transformations) Bondi-Sachs coordinate system of the exterior characteristic evolution. We present a reformulated set of characteristic evolution equations better adapted to numerical implementation. In addition, we develop a method to ensure that the angular coordinates used in the volume during the characteristic evolution are asymptotically inertial. This provides a direct route to an expanded set of waveform outputs and is guaranteed to avoid pure-gauge logarithmic dependence that has caused trouble for previous spectral implementations of the characteristic evolution equations. We construct a set of Weyl scalars compatible with the Bondi-like coordinate systems used in characteristic evolution, and determine simple, easily implemented forms for the asymptotic Weyl scalars in our suggested set of coordinates.

The gravitational perturbation of a Morris–Thorne wormhole and the Newman–Penrose formalism

The gravitational perturbation of the Morris–Thorne wormhole has been derived by using the Newman–Penrose formalism. We apply the Teukolsky equation to the wormhole spacetime, compute the perturbed Weyl scalars, and obtain its master equation, decomposed in spin weighted spherical harmonics with spin weight −2. For simplicity, we consider the perturbation provoked by a single Gaussian pulse of pressureless dust matter.

New definition of complexity factor in f(R,T,RμνTμν) gravity

This paper is devoted to present new definition of complexity factor for static cylindrically symmetric matter configurations in $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity. For this purpose, we have considered irrotational static cylindrical spacetime coupled with a locally anisotropic relativistic fluid. After formulating gravitational field and conservation equations, we have performed orthogonal splitting of the Riemann curvature tensor. Unlike GR (for spherical case) the one of the structure scalars $X_{TF}$, has been identified to be a complexity factor. This factor contains effective forms of the energy density, and anisotropic pressure components. Few peculiar relations among complexity factor, Tolman mass and Weyl scalar are also analyzed with the modified $f(R,T,R_{\mu\nu}T^{\mu\nu})$ corrections.

A Galerkin-collocation domain decomposition method: application to the evolution of cylindrical gravitational waves

We present a Galerkin-collocation domain decomposition algorithm applied to the evolution of cylindrical unpolarized gravitational waves. We show the effectiveness of the algorithm in reproducing initial data with high localized gradients and in providing highly accurate dynamics. We characterize the gravitational radiation with the standard Newman–Penrose Weyl scalar . We generate wave templates for both polarization modes, and +, outgoing and ingoing, to show how they exchange energy nonlinearly. In particular, considering an initially ingoing wave, we were able to trace a possible imprint of the gravitational analog of the Faraday effect in the generated templates.

New results on the physical interpretation of black-brane gravitational perturbations

The linear perturbation theory applied to the study of black holes is a traditional and powerful tool to investigate some of the basic properties of these objects, such as the stability of the event horizon, the spectra of quasinormal modes, the scattering and the production of waves in a process of gravitational collapse. Since long ago, the physical interpretation of the linear fluctuations in the metric of spherically symmetric black holes has been established. In a multipolar expansion, it is known that polar perturbations of a monopole type ($l=0$) can only increase the black-hole mass, axial perturbations of a dipole type ($l=1$) induce a slow rotation in the system, and perturbations with $l\geq 2$ always lead to the production of gravitational waves. However, in relation to the planar Schwarzschild anti--de Sitter black holes (or black branes, for short), there is still no conclusive study on some aspects of the physical meaning of these perturbations. In particular, there is some controversy concerning the polar sector of fluctuations with zero wave number $(k=0)$. Some authors claim that this kind of perturbations causes only a variation in the black-brane mass parameter, while others obtained also evidence for the existence of gravitational waves associated to such modes. The present study aims to contribute to the resolution of this controversy by revealing the physical meaning of the gravitational perturbations of anti--de Sitter black branes. In this work we use the Chandrasekhar's gauge formalism to evaluate the linear variations in the complex Weyl scalars in terms of the Regge-Wheeler-Zerilli gauge-invariant quantities. Then we use the Szekeres' proposal for the meaning of the Weyl scalars and the Pirani's criterion for the existence of gravitational radiation in order to give a physical interpretation of the black-brane perturbations with arbitrary wave number value.

Marolf-Ori singularity inside fast spinning black holes

The effective shock wave singularity at the outgoing leg of the inner horizon of a linearly perturbed fast spinning black hole is studied numerically for either scalar field, or vacuum, gravitational perturbations. We demonstrate the occurrence of the Marolf-Ori singularity, including changes of order unity in the scalar field $\phi$ for the scalar field model, and in the Weyl scalars $\psi_0$ and $\psi_4$ (rescaled appropriately by the horizon function $\Delta$) and the Kretschmann curvature scalar $K$ for the vacuum, gravitational perturbations model for both null and timelike geodesic observers. We quantify the shock sharpening effect and show that in all cases its rate agrees with expectations.

Electromagnetic field and dark dynamical scalars for spherical systems

This paper studies the effects of inhomogeneous fluid configurations on the evolution of relativistic spheres within the context of electromagnetic field and fourth order gravity. We assume a compact distribution of spherical structure that evolves due to the curvature influences coming from f (R) gravity model and non-ideal matter source. After calculating the corresponding f (R) equations of motion, an explicit formulation among Weyl scalar, dark dynamical scalars and material variables is developed. With f (R) curvature and charged fluid terms, all the non-zero dynamical variables are presented from the orthogonal decomposition the Riemann curvature tensor. The Raychaudhuri, Weyl scalar and shearing equations are formulated through modified scalar functions with and without varying Ricci scalar for perfect and imperfect spherical systems.

New identities for linearized gravity on the Kerr spacetime

In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor $\mathcal{M}_{ab}$ which solves the linearized Einstein equations. The identity makes this manifest by relating $\mathcal{M}_{ab}$ to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of $\mathcal{M}_{ab}$ gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity.

Dynamics of cylindrical collapse in [formula omitted] gravity

This paper investigates the dynamics of cylindrical collapse with perfect fluid distribution in the background of curvature-matter coupled gravity. Using Misner-Sharp technique, we construct dynamical equations which lead to analyze the effects of correction terms and fluid parameters on the collapse process. A correlation between correction terms, fluid parameters and the Weyl scalar is also formulated. In the case of constant f(G,T) model, it is concluded that homogeneity of energy density yields the conformal flatness and vice-versa. We find that the collapse rate is reduced due to anti-gravitational behavior of the correction terms.

Physical objects approaching the Cauchy horizon of a rapidly rotating Kerr black hole

We solve the 2+1-dimensional Teukolsky equation numerically for the Weyl scalars $\psi_0$ and $\psi_4$ along a time-like geodesic approaching the Cauchy horizon of a rapidly rotating perturbed Kerr black hole. We find that both the amplitude and frequency of the Weyl scalars agree with the results of linear perturbation analysis. We then model a physical object by a simple damped harmonic oscillator, which is driven by an external force that mimics the tidal force experienced by the infalling object. We use this model to find the total deformation of the object at the Cauchy horizon, and the resonant effect when the driving force's frequency matches the internal frequency of the oscillator that models the object.

Linearized stability of extreme black holes

Extreme black holes have been argued to be unstable, in the sense that under linearized gravitational perturbations of the extreme Kerr spacetime the Weyl scalar $\psi_4$ blows up along their event horizons at very late advanced times. We show numerically, by solving the Teukolsky equation in 2+1D, that all algebraically-independent curvature scalar polynomials approach limits that exist when advanced time along the event horizon approaches infinity. Therefore, the horizons of extreme black holes are stable against linearized gravitational perturbations. We argue that the divergence of $\psi_4$ is a consequence of the choice of a fixed tetrad, and that in a suitable dynamical tetrad all Weyl scalars, including $\psi_4$, approach their background extreme Kerr values. We make similar conclusions also for the case of scalar field perturbations of extreme Kerr.

Study of the charged spherical stellar model in f(R) gravity

This paper is devoted to studying the dynamics of the charged spherical stellar model in f ( R ) gravity. We match the interior region incorporating a perfect fluid with the exterior spacetime through junction conditions and we construct dynamical equations via the Misner-Sharp formalism. The coupling of dynamical equations with the energy of the system describe the rate of collapse for both constant as well as general curvature terms. Finally, we develop a relationship between matter variables, dark source terms and the Weyl scalar. It is concluded that the energy density homogeneity exists if and only if the metric is conformally flat while the rate of collapse is found to be diminishing.

Extracting gravitational waves induced by plasma turbulence in the early Universe through an averaging process

This work is a follow-up to the paper, ‘Numerical relativity as a tool for studying the early Universe’. In this article, we determine if cosmological gravitational waves can be accurately extracted from a dynamical spacetime using an averaging process as opposed to conventional methods of gravitational wave extraction using a complex Weyl scalar. We calculate the normalized energy density, strain and degree of polarization of gravitational waves produced by a simulated turbulent plasma similar to what was believed to have existed shortly after the electroweak scale. This calculation is completed using two numerical codes, one which utilizes full general relativity calculations based on modified BSSN equations while the other utilizes a linearized approximation of general relativity. Our results show that the spectrum of gravitational waves calculated from the nonlinear code using an averaging process is nearly indistinguishable from those calculated from the linear code. This result validates the use of the averaging process for gravitational wave extraction of cosmological systems.