scholarly journals General relativistic aberration equation and measurable angle of light ray in Kerr spacetime

2021 ◽  
pp. 2150045
Author(s):  
Hideyoshi Arakida
Keyword(s):  
Deflection Angle ◽  
Transverse Velocity ◽  
Null Geodesic ◽  
Transverse Motion ◽  
Static Case ◽  
Kerr Spacetime ◽  
Frame Dragging ◽  
Finite Distance ◽  
General Relativistic ◽  
The Impact

We will mainly discuss the measurable angle (local angle) of the light ray [Formula: see text] at the position of the observer [Formula: see text] instead of the total deflection angle (global angle) [Formula: see text] in Kerr spacetime. We will investigate not only the effect of the gravito-magnetic field or frame dragging due to the spin of the central object but also the contribution of the motion of the observer with a coordinate radial velocity [Formula: see text] and a coordinate transverse velocity [Formula: see text] where [Formula: see text] is the impact parameter ([Formula: see text] and [Formula: see text] are the angular momentum and the energy of the light ray, respectively) and [Formula: see text] is a coordinate angular velocity. [Formula: see text] and [Formula: see text] are computed from the components of the four-velocity of the observer [Formula: see text] and [Formula: see text], respectively. Because the motion of observer causes an aberration, we will employ the general relativistic aberration equation to obtain the measurable angle [Formula: see text] which is determined by the four-momentum of the light ray [Formula: see text] and the four-momentum of the radial null geodesic [Formula: see text] as well as the four-velocity of the observer [Formula: see text]. The measurable angle [Formula: see text] given in this paper can be applied not only to the case of the observer located in an asymptotically flat region but also to the case of the observer placed within the curved and finite-distance region. Moreover, when the observer is in radial motion, the total deflection angle [Formula: see text] can be expressed by [Formula: see text]; this is consistent with the overall scaling factor [Formula: see text] instead of [Formula: see text] with respect to the total deflection angle [Formula: see text] in the static case ([Formula: see text] is the velocity of the lens object). On the other hand, when the observer is in transverse motion, the total deflection angle is given by the form [Formula: see text] if we define the transverse velocity as having the form [Formula: see text].

scholarly journals General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr–de Sitter Spacetime

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 173
Author(s):  
Hideyoshi Arakida
Keyword(s):  
Cosmological Constant ◽  
Gravitational Lensing ◽  
Transverse Velocity ◽  
Second Order ◽  
Transverse Motion ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Sitter Spacetime ◽  
General Relativistic ◽  
The Impact

As an extension of our previous paper, instead of the total deflection angle α, we will mainly focus on the discussion of measurable angle of the light ray ψP at the position of observer P in Kerr–de Sitter spacetime, which includes the cosmological constant Λ. We will investigate the contribution of the radial and transverse motion of the observer which are connected with radial velocity vr and transverse velocity bvϕ (b is the impact parameter) as well as the spin parameter a of the central object which induces the gravito-magnetic field or frame dragging and the cosmological constant Λ. The general relativistic aberration equation is employed to take into account the influence of motion of the observer on the measurable angle ψP. The measurable angle ψP derived in this paper can be applicable to the observer placed within the curved and finite-distance region in the spacetime. The equation of light trajectory will be obtained in such a sense that the background is de Sitter spacetime instead of Minkowski one. As an example, supposing the cosmological gravitational lensing effect, we assume that the lens object is the typical galaxy and the observer is in motion with respect to the lensing object at a recession velocity vr=bvϕ=vH=H0D (where H0 is a Hubble constant and D means the distance between the observer and the lens object). The static terms O(Λbm,Λba) are basically comparable with the second order deflection term O(m2), and they are almost one order smaller that the Kerr deflection −4ma/b2. The velocity-dependent terms O(Λbmvr,Λbavr) for radial motion and O(Λb2mvϕ,Λb2avϕ) for transverse motion are at most two orders of magnitude smaller than the second order deflection O(m2). We also find that even when the radial and transverse velocity have the same sign, asymptotic behavior as ϕ approaches 0 is different from each other, and each diverges to opposite infinity.

scholarly journals Waves and null congruences in a draining bathtub

2016 ◽  
Vol 25 (09) ◽  
pp. 1641004 ◽  
Author(s):  
David Dempsey ◽  
Sam R. Dolan
Keyword(s):  
Light Cone ◽  
Null Geodesic ◽  
Method Of Lines ◽  
Eikonal Approximation ◽  
Lorentzian Geometry ◽  
Circular Orbits ◽  
Kerr Spacetime ◽  
Frame Dragging ◽  
Klein Gordon ◽  
The Time Domain

We study wave propagation in a draining bathtub: a black hole analogue in fluid mechanics whose perturbations are governed by a Klein–Gordon equation on an effective Lorentzian geometry. Like the Kerr spacetime, the draining bathtub geometry possesses an (effective) horizon, an ergosphere and null circular orbits. We propose here that a ‘pulse’ disturbance may be used to map out the light-cone of the effective geometry. First, we apply the eikonal approximation to elucidate the link between wavefronts, null geodesic congruences and the Raychaudhuri equation. Next, we solve the wave equation numerically in the time domain using the method of lines. Starting with Gaussian initial data, we demonstrate that a pulse will propagate along a null congruence and thus trace out the light-cone of the effective geometry. Our new results reveal features, such as wavefront intersections, frame-dragging, winding and interference effects, that are closely associated with the presence of null circular orbits and the ergosphere.

scholarly journals Null geodesics and repulsive behavior of gravity in $(2+1)$D massive gravity

2019 ◽  
Vol 2019 (7) ◽  
Author(s):  
Keisuke Nakashi ◽  
Shinpei Kobayashi ◽  
Shu Ueda ◽  
Hiromi Saida
Keyword(s):  
Black Hole ◽  
Deflection Angle ◽  
Null Geodesic ◽  
Analytic Solutions ◽  
Massive Gravity ◽  
Geodesic Equation ◽  
Null Geodesics ◽  
Black Hole Background ◽  
Black Hole Spacetime ◽  
The Impact

Abstract We study the null geodesics in a static circularly symmetric (SCS) black hole spacetime, which is a solution in the $(2+1)$D massive gravity proposed by Bergshoeff, Hohm, and Townsend (BHT massive gravity). We obtain analytic solutions for the null geodesic equation in the SCS black hole background and find the explicit form of deflection angles. We see that, for various values of the impact parameter, the deflection angle can be positive, negative, or even zero in this black hole spacetime. The negative deflection angle indicates the repulsive behavior of the gravity that comes from the gravitational hair parameter that is the most characteristic quantity of the BHT massive gravity.

scholarly journals Strong Azimuthal Combustion Instabilities in a Spray Annular Chamber With Intermittent Partial Blow-Off

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Kevin Prieur ◽  
Daniel Durox ◽  
Thierry Schuller ◽  
Sébastien Candel
Keyword(s):  
Light Intensity ◽  
High Speed ◽  
Transverse Velocity ◽  
Pressure Fluctuations ◽  
Pressure Sensors ◽  
Nodal Line ◽  
Operating Conditions ◽  
Acoustic Energy ◽  
Transverse Motion ◽  
Azimuthal Mode

This article reports experiments carried out in the MICCA-spray combustor developed at EM2C laboratory. This system comprises 16 swirl spray injectors. Liquid n-heptane is injected by simplex atomizers. The combustion chamber is formed by two cylindrical quartz tubes allowing full optical access to the flame region and it is equipped with 12 pressure sensors recording signals in the plenum and chamber. A high-speed camera provides images of the flames and photomultipliers record the light intensity from different flames. For certain operating conditions, the system exhibits well defined instabilities coupled by the first azimuthal mode of the chamber at a frequency of 750 Hz. These instabilities occur in the form of bursts. Examination of the pressure and the light intensity signals gives access to the acoustic energy source term. Analysis of the phase fluctuations between the two signals is carried out using cross-spectral analysis. At limit cycle, large pressure fluctuations of 5000 Pa are reached, and these levels persist over a finite period of time. Analysis of the signals using the spin ratio indicates that the standing mode is predominant. Flame dynamics at the pressure antinodal line reveals a strong longitudinal pulsation with heat release rate oscillations in phase and increasing linearly with the acoustic pressure for every oscillation levels. At the pressure nodal line, the flames are subjected to large transverse velocity fluctuations leading to a transverse motion of the flames and partial blow-off. Scenarios and modeling elements are developed to interpret these features.

scholarly journals Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Junji Jia ◽  
Ke Huang
Keyword(s):  
Gravitational Lensing ◽  
Deflection Angle ◽  
Strong Field ◽  
Distance Effect ◽  
Critical Value ◽  
Field Limit ◽  
Perturbative Method ◽  
The Deflection Angle ◽  
The Impact ◽  
Strong Field Limit

AbstractA perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $$(1-b_c/b)$$ ( 1 - b c / b ) where b is the impact parameter and $$b_c$$ b c is its critical value, with coefficients of the series explicitly given. This result also naturally takes into account the finite distance effect of both the source and detector, and allows to solve the apparent angles of the relativistic images in a more precise way. From this, the BH angular shadow size is expressed as a simple formula containing metric functions and particle/photon sphere radius. The magnification of the relativistic images were shown to diverge at different values of the source-detector angular coordinate difference, depending on the relation between the source and detector distance from the lens. To verify all these results, we then applied them to the Hayward BH spacetime, concentrating on the effects of its charge parameter l and the asymptotic velocity v of the signal. The BH shadow size were found to decrease slightly as l increases to its critical value, and increase as v decreases from light speed. For the deflection angle and the magnification of the images however, both the increase of l and decrease of v will increase their values.

Nonlinear Inversion of Ultrasonic Guided Waves for in Vivo Evaluation of Cortical Bone Properties

2021 ◽  
Author(s):  
Xiaojun Song ◽  
Tiandi Fan ◽  
Jundong Zeng ◽  
QinZhen Shi ◽  
Qiong Huang ◽  
...  
Keyword(s):  
Soft Tissue ◽  
Cortical Bone ◽  
Cortical Thickness ◽  
Guided Waves ◽  
Transverse Velocity ◽  
Ultrasonic Guided Waves ◽  
Bulk Wave ◽  
True Value ◽  
The Impact

Abstract Ultrasonic guided waves (UGW), which propagate throughout the whole thickness of cortical bone, are attractive in the early diagnosis of osteoporosis. However, it is challenging due to the impact of soft tissue and the inherent difficulties related to the multiparametric inversion of cortical bone quality factors, such as cortical thickness and bulk wave velocities. Therefore, an UGW based multiple-parameter inversion algorithm is developed to predict strength-related factors in this research. In simulation, a free plate (cortical bone) and a bilayer plate (soft tissue and cortical bone) are used to validate the proposed method. The inverted cortical thickness (CTh), longitudinal velocity (V L ) and transverse velocity (V T ) are in accordance with the true value. Then four bovine cortical bone plates are used in the in vitro experiments. Compared with the reference values, the relative errors for cortical thicknesses are 3.96%, 0.83%, 2.87% and 4.25% respectively. In the in vivo measurements, ultrasonic guided waves are collected from ten volunteers’ tibia. The theoretical dispersion curves depicted by the estimated parameters (V T , V L , CTh) match well with the extracted experimental ones. In comparison to the dual-energy x-ray absorptiometry (DXA), the results show that the estimated transverse velocity and cortical thickness are highly sensitive to the osteoporosis. Therefore, these two parameters (CTh and V T ) of long bones have potential to diagnose bone status in clinical applications.

The effect analysis of an engine jet on an aircraft blast deflector

2018 ◽  
Vol 41 (4) ◽  
pp. 990-1001
Author(s):  
Song Ma ◽  
Jianguo Tan ◽  
Xiankai Li ◽  
Jiang Hao
Keyword(s):  
High Temperature ◽  
Flow Field ◽  
High Speed ◽  
Deflection Angle ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Aircraft Engine ◽  
Experimental Results ◽  
Exhaust Gases ◽  
The Impact

This paper establishes a novel mathematical model for computing the plume flow field of a carrier-based aircraft engine. Its objective is to study the impact of jet exhaust gases with high temperature, high speed and high pressure on the jet blast deflector. The working condition of the nozzle of a fully powered on engine is first determined. The flow field of the exhaust jet is then numerically simulated at different deflection angle using the three-dimensional Reynolds averaged Navier–Stokes equations and the standard [Formula: see text]-[Formula: see text] turbulence method. Moreover, infra-red temperature tests are further carried out to test the temperature field when the jet blast deflector is at the [Formula: see text] deflection angle. The comparison between the simulation results and the experimental results show that the proposed computation model can perfectly describe the system. There is only 8–10% variation between them. A good verification is achieved. Moreover, the experimental results show that the jet blast deflector plays an outstanding role in driving the high-temperature exhaust gases. It is found that [Formula: see text] may be the best deflection angle to protect the deck and the surrounding equipment effectively. These data results provide a valuable basis for the design and layout optimization of the jet blast deflector and deck.

scholarly journals An improved test of the general relativistic effect of frame-dragging using the LARES and LAGEOS satellites

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
Ignazio Ciufolini ◽  
Antonio Paolozzi ◽  
Erricos C. Pavlis ◽  
Giampiero Sindoni ◽  
John Ries ◽  
...  
Keyword(s):  
General Relativity ◽  
Gravity Field ◽  
Temporal Variations ◽  
Gravity Field Model ◽  
Earth’S Gravity Field ◽  
Frame Dragging ◽  
Earth Gravity ◽  
Gravity Field Determination ◽  
General Relativistic ◽  
Static Part

Abstract We report the improved test of frame-dragging, an intriguing phenomenon predicted by Einstein’s General Relativity, obtained using 7 years of Satellite Laser Ranging (SLR) data of the satellite LARES (ASI, 2012) and 26 years of SLR data of LAGEOS (NASA, 1976) and LAGEOS 2 (ASI and NASA, 1992). We used the static part and temporal variations of the Earth gravity field obtained by the space geodesy mission GRACE (NASA and DLR) and in particular the static Earth’s gravity field model GGM05S augmented by a model for the 7-day temporal variations of the lowest degree Earth spherical harmonics. We used the orbital estimator GEODYN (NASA). We measured frame-dragging to be equal to $$0.9910 \pm 0.02$$0.9910±0.02, where 1 is the theoretical prediction of General Relativity normalized to its frame-dragging value and $$\pm 0.02$$±0.02 is the estimated systematic error due to modelling errors in the orbital perturbations, mainly due to the errors in the Earth’s gravity field determination. Therefore, our measurement confirms the prediction of General Relativity for frame-dragging with a few percent uncertainty.

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