arXiv subjects

Masumi Kasai

Publications and source records attributed to Masumi Kasai.

At least 19 recordsLinked to original sources

arXiv2023

A unified treatment of the redshift, the Doppler effect, and the time dilation in general relativity

We present a unified treatment of the gravitational and cosmological redshift, the Doppler effect due to the moving observer or light source, and the time dilation in the gravitational field in the framework of general relativity. The primary purpose of this paper is to extend the description of Narlikar (1994) on the unified approach towards the redshifts and the Doppler effect in a more generalized form, with the help of the four facts extracted from the comprehensive review article by Ellis (1971). We apply it to the cases of moving observer or light source in the gravitational field and obtain the Doppler effect term, in addition to the standard gravitational or cosmological redshift. The secondary purpose is to explicitly show that the time dilation of a moving clock in the gravitational field can also be understood within the same framework of the unified treatment. We examine the time dilation of the moving clock on geodesic in the gravitational field. We also derive the time dilation of the moving clock on elliptical orbit, based on the same unified treatment. The tertiary purpose is to show that we can understand special-relativistic effects without using the Lorentz transformation. We derive the special-relativistic formulae such as the Doppler effect and aberration of light, the kinetic time dilation, and the Lorentz contraction in the general-relativistic framework.

gr-qc↗

arXiv2023

The gauge-invariant formulation of the local expansion rate driven by the local average density in an inhomogeneous universe

The Hubble tension casts a blight on the standard cosmology. As a possible solution to the problem, the local variation of the expansion rate has been proposed where the spatial averaging over a finite domain was introduced in order to restore the local Friedmannian behavior in an inhomogeneous cosmology. So far, however, the approaches are limited to the particular choices of the gauges, and it has been unclear whether the results are gauge-invariant. In this paper, we present the gauge-invariant formulation of the local expansion rate which is driven by the spatial average of the gauge-invariant inhomogeneous density. We show that the local cosmological parameters in the finite domain may change from the global parameters, and the relations between them are expressed by the gauge-invariant averaged density.

gr-qc↗

arXiv2019

A possible solution to the Hubble constant discrepancy -- Cosmology where the local volume expansion is driven by the domain average density

The Hubble constant problem is the discrepancy between different measurements of the Hubble constant in different scales. We show that this problem can be resolved within the general relativistic framework of the perturbation theory in the inhomogeneous universe, with the help of spatial averaging procedure over a finite local domain in the $t=\mbox{const.}$ hypersurface. The idea presented in this paper is unique in the sense that it has all of the following properties. a) It is based on the general relativistic perturbation theory, with ordinary dust matter only. No strange matter nor energy components are required. b) The employment of the spatially invariant averaging procedure on the finite domain is essential. c) The key is the first-order effect of the inhomogeneities in the linear perturbation theory. No non-linear effects are required.

gr-qc↗

arXiv2012

Effect of the cosmological constant on the bending of light and the cosmological lens equation

We revisit the effect of cosmological constant $Λ$ on the light deflection and its role in the cosmological lens equation. First, we re-examine the motion of photon in the Schwarzschild spacetime, and explicitly describe the trajectory of photon and deflection angle $α$ up to the second-order in $G$. Then the discussion is extended to the contribution of the cosmological constant $Λ$ in the Schwarzschild-de Sitter or Kottler spacetime. Contrary to the previous arguments, we emphasize the following points: (a) the cosmological constant $Λ$ does appear in the orbital equation of light, (b) nevertheless the bending angle of light $α$ does not change its form even if $Λ\neq 0$ since the contribution of $Λ$ is thoroughly absorbed into the definition of the impact parameter, and (c) the effect of $Λ$ is completely involved in the angular diameter distance $D_A$.

gr-qc↗

arXiv2011

An analytical approximation of the luminosity distance in flat cosmologies with dark energy

We present an analytical approximation formula for the luminosity distance in spatially flat cosmologies with dust and a cosmological constant. We also show the approximate formulae for the so-called Dyer-Roeder distance (empty beam case) and the generalised angular diameter distance from redshift $z=z_1$ to $z=z_2$, which are particularly useful in analysing the gravitational lens effects. Our formulae are widely applicable over the range of the density parameter and the redshift with sufficiently small uncertainties. In particular, in the range of density parameter $0.3 \lid Ω_{\rmn m} \lid 1$ and redshift $0.03 \lid z \lid 1000$, the relative error for the luminosity distance by our formula is always smaller than that of the recent work by \cite{wu}.

astro-ph.CO↗

arXiv2011

An analytical approximation of the luminosity distance in flat cosmologies with a cosmological constant

We present an analytical approximation formula for the luminosity distance in spatially flat cosmologies with dust and a cosmological constant. Apart from the overall factor, the effect of non-zero cosmological constant in our formula is written simply in terms of a rational function. We also show the approximate formulae for the Dyer-Roeder distance (empty beam case) and the generalized angular diameter distance from redshift $z_1$ to $z_2$, which are particularly useful in analyzing the gravitational lens effects. Our formulae are widely applicable over the range of the density parameter and the redshift with sufficiently small uncertainties. In particular, in the range of density parameter $0.3 \leq Ω_{\rm m} \leq 1$ and redshift $0.03 \leq z \leq 1000$, the relative error for the luminosity distance by our formula is always smaller than that of the recent work by Wickramasinghe and Ukwatta (2010). Hence, we hope that our formulae will be an efficient and useful tool for exploring various problems in observational cosmology.

astro-ph.CO↗

arXiv2011

Perturbative Approach to the Gravitational Lensing by a Non-spherically Distorted Compact Object

We investigate the gravitational lens effect caused by a non-spherically distorted compact object. The non-spherical property of the gravitational potential is modeled by a quadrupole moment. Under the assumption that the quadrupole contribution is small, we solve perturbatively the lens equation and obtain the image positions and the amplification factors. We show that the separation angle of two major images is only slightly changed by the existence of the quadrupole contribution, whereas the difference of the amplification factors may be significantly modified. Our results indicate that even a tiny non-spherical distortion of the lens potential may cause significant amount of flux anomalies in the lensed images.

astro-ph.CO↗

arXiv2010

An analytical approximation of the growth function in Friedmann-Lemaître universes

We present an analytical approximation formula for the growth function in a spatially flat cosmology with dust and a cosmological constant. Our approximate formula is written simply in terms of a rational function. We also show the approximate formula in a dust cosmology without a cosmological constant, directly as a function of the scale factor in terms of a rational function. The single rational function applies for all, open, closed and flat universes. Our results involve no elliptic functions, and have very small relative error of less than 0.2 per cent over the range of the scale factor $1/1000 \la a \lid 1$ and the density parameter $0.2 \la Ω_{\rmn{m}} \lid 1$ for a flat cosmology, and less than $0.4$ per cent over the range $0.2 \la Ω_{\rmn{m}} \la 4$ for a cosmology without a cosmological constant.

astro-ph.CO↗

arXiv2009

Secular increase of the Astronomical Unit: a possible explanation in terms of the total angular momentum conservation law

We give an idea and the order-of-magnitude estimations to explain the recently reported secular increase of the Astronomical Unit (AU) by Krasinsky and Brumberg (2004). The idea proposed is analogous to the tidal acceleration in the Earth-Moon system, which is based on the conservation of the total angular momentum and we apply this scenario to the Sun-planets system. Assuming the existence of some tidal interactions that transfer the rotational angular momentum of the Sun and using reported value of the positive secular trend in the astronomical unit, $\frac{d}{dt}{AU} = 15 \pm 4 {(m/cy)}$, the suggested change in the period of rotation of the Sun is about $21 {ms/cy}$ in the case that the orbits of the eight planets have the same "expansion rate." This value is sufficiently small, and at present it seems there are no observational data which exclude this possibility. Effects of the change in the Sun's moment of inertia is also investigated. It is pointed out that the change in the moment of inertia due to the radiative mass loss by the Sun may be responsible for the secular increase of AU, if the orbital "expansion" is happening only in the inner planets system. Although the existence of some tidal interactions is assumed between the Sun and planets, concrete mechanisms of the angular momentum transfer are not discussed in this paper, which remain to be done as future investigations.

astro-ph.EP↗

arXiv2007

Apparent Acceleration through Large-scale Inhomogeneities --Post-Friedmannian Effects of Inhomogeneities on the Luminosity Distance--

We re-analyze the observed magnitude-redshift relation of type Ia supernovae (SNe Ia) and examine the possibility that the apparent acceleration of the cosmic expansion is not caused by dark energy but is instead a consequence of the large-scale inhomogeneities in the universe. We propose a method to phenomenologically describe the effects of the large-scale inhomogeneities without relying on the specific toy models of the inhomogeneous universe. This method clearly illustrates how the post-Friedmannian effects of inhomogeneities, i.e. the effects due to the deviation from a perfectly homogeneous and isotropic model, act as an effective cosmological constant in the magnitude-redshift relation of SNe Ia.

astro-ph↗

arXiv2006

Toward a No-Go Theorem for an Accelerating Universe through a Nonlinear Backreaction

The backreaction of nonlinear inhomogeneities to the cosmic expansion is re-analyzed in the framework of general relativity. Apparent discrepancies regarding the effect of the nonlinear backreaction, which exist among the results of previous works in different gauges, are resolved. By defining the spatially averaged matter energy density as a conserved quantity in the large comoving volume, it is shown that the nonlinear backreaction neither accelerates nor decelerates the cosmic expansion in a matter-dominated universe. The present result in the Newtonian gauge is consistent with the previous results obtained in the comoving synchronous gauge. Although our work does not give a complete proof, it strongly suggests the following no-go theorem: No cosmic acceleration occurs as a result of the nonlinear backreaction via averaging.

astro-ph↗

arXiv2006

General Relativistic Effects of Gravity in Quantum Mechanics -- A Case of Ultra-Relativistic, Spin 1/2 Particles --

We present a general relativistic framework for studying gravitational effects in quantum mechanical phenomena. We concentrate our attention on the case of ultra-relativistic, spin-1/2 particles propagating in Kerr spacetime. The two-component Weyl equation with general relativistic corrections is obtained in the case of a slowly rotating, weak gravitational field. Our approach is also applied to neutrino oscillations in the presence of a gravitational field. The relative phase of two different mass eigenstates is calculated in radial propagation, and the result is compared with those of previous works.

gr-qc↗

arXiv2004

Inversion formula for determining parameters of an astrometric binary

It is believed that some numerical technique must be employed for the determination of the system parameters of a visual binary or a star with a planet because the relevant equations are not only highly nonlinear but also transcendental owing to the Kepler's equation. Such a common sense, however, is not true; we discover an analytic inversion formula, in which the original orbital parameters are expressed as elementary functions of the observable quantities such as the location of four observed points and the time interval between these points. The key thing is that we use the time interval but not the time of each observation in order to avoid treating the Kepler's equation. The present formula can be applied even in cases where the observations cover a short arc of the orbit during less than one period. Thus the formula will be useful in the future astrometric missions such as SIM, GAIA and JASMINE.

astro-ph↗

arXiv2004

Euclidean Algorithm for a Gravitational Lens in a Polynomial Equation

The Euclidean algorithm in algebra is applied to a class of gravitational lenses for which the lens equation consists of any set of coupled polynomial equations in the image position. In general, this algorithm allows us to reduce an apparently coupled system to a single polynomial in one variable (say $x$ in Cartesian coordinates) without the other component (say $y$), which is expressed as a function of the first component. This reduction enables us to investigate the lensing properties in an algebraic manner: For instance, we can obtain an analytic expression of the caustics by computing the discriminant of the polynomial equation. To illustrate this Euclidean algorithm, we re-examine a binary gravitational lens and show that the lens equation is reduced to a single real fifth-order equation, in agreement with previous works. We apply this algorithm also to the linearlized Kerr lens and find that the lens equation is reduced to a single real fifth-order one.

astro-ph↗

arXiv2003

Separability of Rotational Effects on a Gravitational Lens

We derive the deflection angle up to $O(m^2a)$ due to a Kerr gravitational lens with mass $m$ and specific angular momentum $a$. It is known that at the linear order in $m$ and $a$ the Kerr lens is observationally equivalent to the Schwarzschild one because of the invariance under the global translation of the center of the lens mass. We show, however, nonlinear couplings break the degeneracy so that the rotational effect becomes in principle separable for multiple images of a single source. Furthermore, it is distinguishable also for each image of an extended source and/or a point source in orbital motion. In practice, the correction at $O(m^2a)$ becomes $O(10^{-10})$ for the supermassive black hole in our galactic center. Hence, these nonlinear gravitational lensing effects are too small to detect by near-future observations.

astro-ph↗

arXiv2002

Images for an Isothermal Ellipsoidal Gravitational Lens from a Single Real Algebraic Equation

We present explicit expressions for the lens equation for a cored isothermal ellipsoidal gravitational lens as a single real sixth-order algebraic equation in two approaches; 2-dimensional Cartesian coordinates and 3-dimensional polar ones. We find a condition for physical solutions which correspond to at most five images. For a singular isothermal ellipsoid, the sixth-order equation is reduced to fourth-order one for which analytic solutions are well-known. Furthermore, we derive analytic criteria for determining the number of images for the singular lens, which give us simple expressions for the caustics and critical curves. The present formulation offers a useful way for studying galaxy lenses frequently modeled as isothermal ellipsoids.

astro-ph↗

arXiv2002

Algebraic Properties of the Real Quintic Equation for a Binary Gravitational Lens

It has been recently shown that the lens equation for a binary gravitational lens, which is apparently a coupled system, can be reduced to a real fifth-order (quintic) algebraic equation. Some algebraic properties of the real quintic equation are revealed. We find that the number of images on each side of the separation axis is independent of the mass ratio and separation unless the source crosses the caustics. Furthermore, the discriminant of the quintic equation enables us to study changes in the number of solutions, namely in the number of images. It is shown that this discriminant can be factorized into two parts: One represents the condition that the lens equation can be reduced to a single quintic equation, while the other corresponds to the caustics.

astro-ph↗

arXiv2000

Do relativistic corrections affect microlensing amplification?

Studies of gravitational microlensing are usually based on the lens equation, which is valid only to first order in the gravitational constant $G$. However, the amplification factor of microlensing is a second-order quantity with respect to $G$. Despite this fact, conventional studies are still based on the lowest-order lens equation. Then the question naturally arises: Why are these conventional studies justified? We carefully study the relativistic correction to the amplification factor at $O(G^2)$. We show that the amplification factor for each image is corrected. However, the total amplification remains unchanged at this order.

astro-ph↗