Polar Gravitational Wave Research Articles

Imprints of a gravitational wave through the weak field deflection of photons* *A. Ö. and R. P. acknowledge the networking support of the COST Action CA18108 - COST Action CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and experiments (CosmicWISPers), Quantum gravity phenomenology in the multi-messenger approach (QG-MM), the COST Action CA21136 - Addressing observational tensions in cosmology

In this study, we investigate the novel phenomenon of gravitational lensing experienced by gravitational waves traveling past a Schwarzschild black hole perturbed by a specific, first-order, polar gravitational wave. We apply the Gauss-Bonnet theorem, finding a topological contribution to the deflection of light rays passing near the black hole. We demonstrate that the deflection angle can be determined by analyzing a region entirely outside the path of the light ray, leading to a calculation based solely on the parameters of the perturbing wave (Legendre polynomial order, l; frequency, σ). This approach offers a unique perspective on gravitational lensing and expands our understanding of black hole interactions with gravitational waves.

Polar gravitational waves in f(R, T ϕ ) framework

This work analyzes the propagation of polar gravitational waves for flat FRW cosmic background and in view of f(R, T ϕ ) theory. In what follows, we opt for the Regge-Wheeler polar perturbation scheme in flat geometry of background spacetime and corresponding scalar, field source is also perturbed. The two sets of field equations for flat and perturbed geometry are constructed and furthermore used to evaluate perturbation parameters. The obtained expressions suggest that polar fluctuations affect both geometry and scalar field. The radial and temporal parts of perturbation parameters χ and σ are plotted explicitly while the behavior of ψ is observed via a numerical solution plotted as a 3D graph. The change in these fluctuations for different values of model parameter λ is noticed through these plots.

Gravitational waves in f(R, T)-rainbow gravity: even modes and the Huygens principle

In the context of f(R, T)-gravity, propagation of gravitational waves (GWs) for even (or polar) modes is explored by using the Regge-Wheeler gauge in the conformally flat Friedman-Lemaitre-Robertson-Walker type rainbow (CFR) universe. Writing the perturbed field equations for the polar GWs in the CFR spacetime, we first acquire a second-order differential equation for one of the unknown perturbation factors and then get all other unknown perturbation functions. Withal, we reach a conclusion that both the four-velocity vector components except the third one and the corresponding matter distribution are affected by the polar perturbation. Furthermore, the effect of rainbow functions, which can change the geometry of space-time, on the polar GWs is also analyzed graphically. We achieve that the shape (wavelength and amplitude) of polar GWs is dramatically impressed by the alteration of rainbow functions. Lastly, we investigate whether the polar GWs satisfy the Huygens principle.

Evolution of axial gravitational waves in rainbow universe

In view of the Regge-Wheeler formalism (RWF) causing axial and polar gravitational waves (GWs), axial modes of the GWs are investigated by considering the spatially flat conformal Friedmann-Lemaitre-Robertson-Walker (FLRW) metric in the rainbow gravity (RBG) that is one of the important nominees for the theory of quantum gravity. We use the zeroth-order and first-order field equations for the rainbow FLRW background with respect to the Regge-Wheeler gauge to obtain unknown perturbation functions that elicit how perturbation influences the metric, material content and the four-velocity vector. In this context, we show that (i) not only the background metric but also the first and third components of four-velocity vector are affected from axial modes and (ii) the background matter distribution is not affected by the perturbation. Furthermore, impacts of the rainbow functions, which are of great importance in describing the physical system, on axial modes are clearly revealed by graphical analysis. We also discuss whether the axial GWs satisfy the Huygens principle.

Propagation of axial and polar gravitational waves in Kantowski–Sachs universe

We apply the Regge-Wheeler formalism to study axial and polar gravitational waves in Kantowski-Sachs universe. The background field equations and the linearised perturbation equations for the modes are derived in presence of matter. To find analytical solutions, we analyze the propagation of waves in vacuum spacetime. The background field equations in absence of matter are solved by assuming the expansion scalar to be proportional to the shear scalar. Using the method of separation of variables, the axial perturbation parameter $ h_0(t, r) $ is obtained from its wave equation. The other perturbation $ h_1(t, r) $ is then determined from $ h_0(t, r) $. The anisotropy of the background spacetime is responsible for the damping of the axial waves. The polar perturbation equations are much more involved compared to their FLRW counterparts, as well as to the axial perturbations in Kantowski-Sachs background. In both the axial and polar cases, the radial and temporal solutions separate out as products. The temporal part of the polar perturbation solutions are plotted against time to obtain an order of magnitude estimate of the frequency of the propagating GWs and lies in the range 1000-2000 Hz. Using standard observational data for the GW strain we have placed constraints on the parameters in the polar perturbation solutions. The perturbation equations in presence of matter show that the axial waves can cause perturbations only in the azimuthal velocity of the fluid without deforming the matter field. But the polar waves must perturb the energy density, the pressure and also the non-azimuthal components of the fluid velocity.

Polar modes of gravitational waves in Rastall cosmology

In this article, we discuss propagation expressions for polar gravitational waves in the spatially flat Friedmann–Lemaitre–Robertson–Walker spacetime dominated by a perfect fluid in the Rastall theory. We perturb the spatially flat spacetime description by making use of Regge–Wheeler perturbations inducing the polar gravitational waves and formulate the corresponding field equations for both unperturbed and perturbed cases. Then, we focus on these field equations simultaneously to find out the unknown perturbation functions. We attain that the assumed perturbations affect the background matter distribution as well as the four-velocity components. We also investigate the impact of model parameters on the amplitude of the polar gravitational waves.

Propagation of polar gravitational waves in f(R, T) scenario

This paper investigates the propagation of polar gravitational waves in the spatially flat FRW universe consisting of a perfect fluid in the scenario of $R+2\lambda T$ model of $f(R,T)$ gravity ($\lambda$ being the model parameter). The spatially flat universe model is perturbed via Regge-Wheeler perturbations inducing polar gravitational waves and the field equations are formulated for both unperturbed as well as perturbed spacetimes. We solve these field equations simultaneously for the perturbation parameters introduced in the metric, matter, and velocity in the radiation, as well as dark energy, dominated phases. It is found that the polar gravitational waves can produce changes in the background matter distribution as well as velocity components in the radiation era similar to general relativity case. Moreover, we have discussed the impact of model parameter on the amplitude of gravitational waves.

Gravitational waves in Friedman–Lemaitre–Robertson–Walker cosmology, material perturbations and cosmological rotation, and the Huygens principle

We analyze propagation equations for the polar modes of gravitational waves in cosmological space-times. We prove that polar gravitational waves must perturb the density and non-azimuthal components of the velocity of material medium of the Friedman–Lemaitre–Robertson–Walker spacetimes. Axial gravitational waves can influence only the azimuthal velocity, leading to local cosmological rotation. The whole gravitational dynamics reduces to the single ‘master equation’ that has the same form for polar and axial modes. That allows us to conclude that the status of the Huygens principle is the same for axial and polar gravitational waves. In particular, this principle is valid exactly in radiation spacetimes with the vanishing cosmological constant, and it is broken otherwise.

Axial and polar gravitational wave equations in a de Sitter expanding universe by Laplace transform

In this paper we study the propagation in a de Sitter universe of gravitational waves generated by perturbating some unspecified spherical astrophysical object in the frequencies domain. We obtain the axial and polar perturbation equations in a cosmological de Sitter universe in the usual comoving coordinates, the coordinates we occupy in our galaxy. We write down the relevant equations in terms of Laplace transform with respect to the comoving time t instead of the usual Fourier one that is no longer available in a cosmological context. Both axial and polar perturbation equations are expressed in terms of a non trivial mixture of retarded-advanced metric coefficients with respect to the Laplace parameter s (complex translation). The axial case is studied in more detail. In particular, the axial perturbations can be reduced to a master linear second-order differential equation in terms of the Regge–Wheeler function Z where a coupling with a retarded Z with respect to the cosmological time t is present. It is shown that a de Sitter expanding universe can change the frequency ω of a gravitational wave as perceived by a comoving observer. The polar equations are much more involved. Nevertheless, we show that the polar perturbations can also be expressed in terms of four independent integrable differential equations.

Gravitational radiation from collapsing magnetized dust. II. Polar parity perturbation

Using gauge-invariant perturbation theory, we study the effects of stellar magnetic fields on polar gravitational waves emitted during the collapse of homogeneous dust. We found that the emitted energy in gravitational waves depends strongly on the initial stellar radius as well as on the ratio between the poloidal and toroidal magnetic components. The polar gravitational-wave output of such a collapse can easily be up to a few orders of magnitude larger than what we get from the nonmagnetized collapse. The changes due to the existence of a magnetic field could be helpful in extracting some information about inner magnetic profiles of progenitors from the detection of the gravitational waves radiated during black hole formation, which results from the stellar collapse.

Universality in quasi-normal modes of neutron stars

We study universality in gravitational waves emitted from non-rotating neutron stars characterized by different equations of state (EOSs). We find that the quasi-normal mode frequencies of such waves, including the w-modes and the f-mode, display similar universal scaling behaviours that hold for most EOSs. Such behaviours are shown to stem from the mathematical structure of the axial and the polar gravitational wave equations, and the fact that the mass distribution function can be approximated by a cubic-quintic polynomial in the radius. As a benchmark for other realistic neutron stars, a simple model of neutron stars is adopted here to reproduce the pulsation frequencies and the generic scaling behaviours mentioned above with good accuracy.

Coupled Pulsations between Polar and Axial Modes in a Slowly Rotating Relativistic Star

Rotational coupling between l-polar mode and (l + 1)-axial mode of axially symmetric pulsations is studied. Equations governing them in a slowly rotating relativistic star show that these oscillation modes couple together in first order of the rotation, while they are separated in the non-rotating case. The effect on the characteristic frequencies is examined. In the oscillations induced by the in-coming polar gravitational wave, the axial mode is always excited, so that there is no characteristic frequency, if the out-going wave condition is imposed on the axial gravitational wave. If the out-going wave condition is imposed on both modes, the coupled system shows discrete eigenfrequencies, which are exactly the same values as in the non-rotating star, that is, the eigenfrequencies for axisymmetric perturbations are not affected by the first-order rotational effect.

Coupled Pulsations between Polar and Axial Modes in a Slowly Rotating Relativistic Star

Rotational coupling between l-polar mode and (l + 1)-axial mode of axially symmetric pulsations is studied. Equations governing them in a slowly rotating relativistic star show that these oscillation modes couple together in first order of the rotation, while they are separated in the non-rotating case. The effect on the characteristic frequencies is examined. In the oscillations induced by the in-coming polar gravitational wave, the axial mode is always excited, so that there is no characteristic frequency, if the out-going wave condition is imposed on the axial gravitational wave. If the out-going wave condition is imposed on both modes, the coupled system shows discrete eigen-frequencies, which are exactly the same values as in the non-rotating star, that is, the eigenfrequencies for axisymmetric perturbations are not affected by the first-order rotational effect

On the non-radial oscillations of a star. IV An application of the theory of Regge poles

It is shown how the flow of energy in the form of gravitational radiation, through a star in non-radial oscillations, can be determined by a suitable adaptation of Regge’s theory of potential scattering in the quantum theory. While the resonant scattering of axial gravitational waves can be treated by Regge’s theory, essentially, in its original form, the treatment of the resonant scattering of polar gravitational waves requires a generalization of the theory beyond its standard range and the incorporation of a ‘flux integral’ derived from independent considerations. Illustrative examples of the applications of the theory are provided.