Sven Zschocke, Sergei A. Klioner
The transformation n to k in post-post-Newtonian order is simplified. All post-post-Newtonian terms of the order m^2/d^2 are neglected and we show that the total sum of these terms is smaller than 15/4 pi m^2/d^2. This simpler transformation will improve the efficiency of Gaia data reduction.
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Total light deflection in the gravitational field of solar system bodies
本文提出总光线偏折概念，利用切比雪夫多项式给出光线轨迹在未来无穷远处的切向量，从而确定各多极矩偏折的严格上限，并应用于木星和土星。
Light deflection in the gravitational field of a solar system body with finite distance of source and observer for sub-micro-arcsecond astrometry
该研究针对源与观测者距太阳系天体有限距离的情况，在1PN和1.5PN近似下推导光线偏转的简化表达式，可用于亚微角秒精度的天体测量。
In this paper, we study the motion of photons around a Kehagias-Sfetsos (KS) black hole and obtain constraints on IR modified Ho$\check{r}$ava gravity without cosmological constant ($\sim Λ_{W}$). An analytic formula for the light deflection angle is obtained. For a propagating photon, the deflection angle $δϕ$ increases with large values of the Ho$\check{r}$ava gravity parameter $ω$. Under the UV limit $ω\longrightarrow \infty$, deflection angle reduces to the result of usual Schwarzschild case, $4GM/R$. It is also found that with increasing scale of astronomical observation system the Ho$\ch…
Precise starlight positions near the sun were measured during the 21 August 2017 total solar eclipse in order to measure their gravitational deflections. The equipment, procedures, and analysis are described in detail. A portable refractor, a CCD camera, and a computerized mount were set up in Wyoming. Detailed calibrations were necessary to improve accuracy and precision. Nighttime measurements taken just before the eclipse provided cubic optical distortion corrections. Calibrations based on star field images 7.4 deg on both sides of the sun taken during totality gave linear and quadratic pla…
We estimate the changes in solar system tests of general relativity when "dark energy" arises from a Weyl scaling invariant action, using the modified Schwarzschild-like spherically symmetric solution obtained in this case by Adler and Ramazano\vglu. We show that the standard results of general relativity are modified by amounts that are far below current experimental errors. However, because the metric is not conformally flat when the central mass $M$ vanishes, there is a contribution to light deflection proportional to the cosmological constant $Λ$, with opposite sign to the one proportional…
We study the impact of relativistic gravitational deflection of light on the accuracy of future Space Interferometry Mission (SIM). We estimate the deflection angles caused by the monopole, quadrupole and octupole components of gravitational fields for a number of celestial bodies in the solar system. We observe that, in many cases, the magnitude of the corresponding effects is significantly larger than the 1 uas accuracy expected from SIM. This fact argues for the development of a relativistic observational model for the mission that would account for the influence of both static and time-var…
We have computed the deflection angles caused by 195 objects in the solar system, including 177 satellites and eight asteroids. Twenty-one satellites and six asteroids can bend light from distant compact extragalactic sources by more than 0.1 $μ$as, and fourteen satellites and the asteroid Ceres can deflect light by more than 1.0 $μ$as. We calculated the zones and durations of perturbations posed by the gravitational fields of five planets (excluding Earth, Jupiter, and Saturn), Pluto, and Ceres, where the perturbations would affect astrometry measured with the Squared Kilometre Array (SKA). P…
Compact objects with a light sphere such as black holes and wormholes can reflect light rays like a mirror. This gravitational lensing phenomenon is called retrolensing and it is an interesting tool to survey dark and compact objects with a light sphere near the solar system. In this paper, we calculate the deflection angle analytically in the strong deflection limit in the Reissner-Nordstrom spacetime without Taylor expanding it in the power of the electric charge. Using the obtained deflection angle in the strong deflection limit, we investigate the retrolensing light curves and the separati…
Based on the 4D momentum, the influence of quintessence on the gravitational frequency shift and the deflection of light are examined in modified Schwarzschild space. We find that the frequency of photon depends on the state parameter of quintessence $w_q$: the frequency increases for $-1<w_q<-1/3$ and decreases for $-1/3<w_q<0$. Meanwhile, we adopt an integral power number $a$ ($a = 3ω_q + 2$) to solve the orbital equation of photon. The photon's potentials become higher with the decrease of $ω_q$. The behavior of bending light depends on the state parameter $ω_q$ sensitively. In particular, …
Total light deflection in the gravitational field of an axisymmetric body at rest with full mass and spin multipole structure
本文在1PN和1.5PN近似下推导了具有完整质量与自旋多极结构的静止轴对称体引力场中的总光线偏折，发现其与切比雪夫多项式相关，并据此给出上限判据，用于确定太阳和巨行星在纳角秒精度下所需的多极阶数。
An Abel-Inversion Formalism for Spacetime Metric Reconstruction from Light Deflection
本文提出基于Abel变换的逆透镜形式，从光偏折角直接重建静态球对称时空度规，无需预设度规，并在太阳偏折案例中恢复Schwarzschild解，为观测与时空几何建立新桥梁。
We derive angle differences between two null geodesics, propagating from light sources to an observer, on the de Sitter spacetime with multi-lensing objects. Assuming the lensing objects are mass monopoles on the de Sitter background, we derive the metric tensor by solving the Einstein equation perturbatively. On that spacetime, we solve a null geodesic parametrized by the coordinate time. Using the null geodesics, we define the angle differences in a coordinate invariant way. We take in the relativistic effects up to the first order of perturbation and clarify the magnitude of approximation e…
Two different ways of generalizing Einstein's general theory of relativity with a cosmological constant to Brans-Dicke type scalar-tensor theories are investigated in the linearized field approximation. In the first case a cosmological constant term is coupled to a scalar field linearly whereas in the second case an arbitrary potential plays the role of a variable cosmological term. We see that the former configuration leads to a massless scalar field whereas the latter leads to a massive scalar field. General solutions of these linearized field equations for both cases are obtained correspond…
The measurement of the deflection of starlight during a total solar eclipse on May 29, 1919 was the first verification of general relativity by an external team of scientists, brought Einstein and his theory to the attention of the general public, and left a legacy of experimental testing that continues today. The discovery of gravitational lenses turned Einstein's deflection into an important tool for astronomy and cosmology. This article reviews the history of the 1919 measurement and other eclipse measurements, describes modern measurements of the effect using radio astronomy, and of its co…
We consider the parametrized post-Newtonian (PPN) limit of ghost-free massive bimetric gravity with two mutually non-interacting matter sectors coupled to the two metrics. Making use of a gauge-invariant differential decomposition of the metric perturbations, we solve the field equations up to the linear PPN order for a static, point-like mass source. From the result we derive the PPN parameter $γ$ for spherically symmetric systems, which describes the gravitational deflection of light by visible matter. By a comparison to its value measured in the solar system we obtain bounds on the paramete…
In this paper, the perihelion precession and deflection of light have been investigated in the 4-dimensional general spherically symmetric spacetime, and the main equation is obtained. As the application of this main equation, the Reissner-Nordstorm-AdS solution and Clifton-Barrow solution in $f(R)$ gravity have been taken for the examples. We find that both the electric charge and $f(R)$ gravity can affect the perihelion precession and deflection of light, while the cosmological constant can only effect the perihelion precession. Moreover, we clarify a subtlety in the deflection of light in t…
Light deflection offers an unbiased test of Weyl's gravity since no assumption on the conformal factor needs to be made. In this second paper of our series ``Light deflection in Weyl gravity'', we analyze the constraints imposed by light deflection experiments on the linear parameter of Weyl's theory. Regarding solar system experiments, the recent CASSINI Doppler measurements are used to infer an upper bound, $\sim 10^{-19} $m$^{-1}$, on the absolute value of the above Weyl parameter. In non-solar system experiments, a condition for unbound orbits together with gravitational mirage observation…
The Sun's gravitational field deflects the apparent positions of close objects in accordance with the formulae of general relativity. Optical astrometry is used to test the prediction, but only with the stars close to the Sun and only during total Solar eclipses. Geodetic Very Long Baseline Interferometry (VLBI) is capable of measuring the deflection of the light from distant radio sources anytime and across the whole sky. We show that the effect of light deflection is equivalent to the gravitational delay calculated during the reduction of VLBI data. All reference radio sources display an ann…
We derive the deflection angle of light rays passing near a black hole with mass m and tidal charge q, confined to a generalized Randall-Sundrum brane with codimension one. We employ the weak lensing approach, up to the second order in perturbation theory by two distinct methods. First we adopt a Lagrangian approach and derive the deflection angle from the analysis of the geodesic equations. Then we adopt a Hamiltonian approach and we recover the same result from the analysis of the eikonal. With this we re-establish the unicity of the result as given by the two methods. Our results in turn im…
General relativistic deflection of light by mass, dipole, and quadrupole moments of gravitational field of a moving massive planet in the Solar system is derived in the approximation of the linearized Einstein equations. All terms of order 1 microarcsecond are taken into account, parametrized, and classified in accordance with their physical origin. We discuss the observational capabilities of the near-future optical and radio interferometers for detecting the Doppler modulation of the radial deflection, and the dipolar and quadrupolar light-ray bendings by Jupiter and the Saturn.
Classical light bending is investigated for weak gravitational fields in the presence of hypothetical local Lorentz violation. Using an effective field theory framework that describes general deviations from local Lorentz invariance, we derive a modified deflection angle for light passing near a massive body. The results include anisotropic effects not present for spherical sources in General Relativity as well as Weak Equivalence Principle violation. We develop an expression for the relative deflection of two distant stars that can be used to analyze data in past and future solar-system obser…
A conformal coupling of the metric in the Jordan frame to the energy-momentum tensor, screens the scalar field gravitational coupling strength $G$ in modified gravity (MOG). The scalar field acquires a mass which depends on the local matter density: the scalar field particle is massive for the Sun and earth, where the density is high compared to low density environments in cosmology and astrophysics. Together with the screening of the vector field $ϕ_μ$, this guarantees that solar system tests of gravity are satisfied. The conformal metric is coupled to the electromagnetic matter field and ene…
We investigate the solution of Maxwell's equations in curved spacetime within the framework of Modified Gravity (MOG). We show that besides the null-geodesic treatment of photons in MOG, using Maxwell's equations and covariant coupling with the extra vector sector of gravitation in MOG, we can extract the equation for the propagation of light. We obtain Fermat's potential and calculate the deflection angle of light during lensing from a point-like star. Our results show that the deflection angle obtained from the solution of the wave equation in MOG for the large-scale structures with larger i…
General relativistic deflection of light by mass, dipole, and quadrupole moments of gravitational field of a moving massive planet in the Solar system is derived. All terms of order 1 microarcsecond are taken into account, parametrized, and classified in accordance with their physical origin. We calculate the instantaneous patterns of the light-ray deflections caused by the monopole, the dipole and the quadrupole moments, and derive equations describing apparent motion of the deflected position of the star in the sky plane as the impact parameter of the light ray with respect to the planet cha…
The classical tests of general relativity - light deflection, time delay and perihelion shift - are applied, along with the geodetic precession test, to the five-dimensional extension of the theory known as Kaluza-Klein gravity, using an analogue of the four-dimensional Schwarzschild metric. The perihelion advance and geodetic precession calculations are generalized for the first time to situations in which the components of momentum and spin along the extra coordinate do not vanish. Existing data on light- bending around the Sun using long- baseline radio interferometry, ranging to Mars using…
This paper analyses the relativistic stellar aberration requirements for the Space Interferometry Mission (SIM). We address the issue of general relativistic deflection of light by the massive self-gravitating bodies. Specifically, we present estimates for corresponding deflection angles due to the monopole components of the gravitational fields of a large number of celestial bodies in the solar system. We study the possibility of deriving an additional navigational constraints from the need to correct for the gravitational bending of light that is traversing the solar system. It turns out tha…
Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: The basic transformations
为满足亚微角秒天体测量需求，研究完善太阳系光传播后牛顿二阶理论，求解边值问题，通过基本变换确定观测者处光轨迹单位切向量，给出巨行星光偏转数值，并说明其在Gaia任务所用GREM模型中的实现方式。
Estimating the strength of Lorentzian distribution in non-commutative geometry by solar system tests
本文研究了非对易几何中洛伦兹分布下Schwarzschild时空的四个经典测试，计算了非对易参数对行星进动等的一阶修正，并利用水星轨道进动观测将参数限制在$Θ\leq0.067579~\mathrm{m}^{2}$内，证实其符合普朗克长度量级。
We measured the relative positions between two pairs of compact extragalactic sources (CESs), J1925-2219 \& J1923-2104 (C1--C2) and J1925-2219 \& J1928-2035 (C1--C3) on 2020 October 23--25 and 2021 February 5 (totaling four epochs), respectively, using the Very Long Baseline Array (VLBA) at 15 GHz. Accounting for the deflection angle dominated by Jupiter, as well as the contributions from the Sun, planets other than Earth, the Moon and Ganymede (the most massive of the solar system's moons), our theoretical calculations predict that the dynamical ranges of the relative positions across four ep…
High-precision astrometry on sub-micro-arcsecond level in angular resolution requires accurate determination of the trajectory of a light-signal from the celestial light source through the gravitational field of the Solar system toward the observer. In this investigation the light trajectory in the gravitational field of N moving bodies is determined in the 1.5 post-Newtonian approximation. In the approach presented two specific issues of particular importance are accounted for: (1) According to the recommendations of International Astronomical Union, the metric of the Solar system is expresse…
Light propagation in the 2PN approximation in the monopole and quadrupole field of a body at rest: Boundary value problem
本研究从2PN近似下光传播的初值问题推导出边值问题的解，为太阳系超高精度光偏转和时间延迟测试奠定基础。
The general relativistic treatment of gravitation can be extended by preserving the geometrical nature of the theory but modifying the form of the coupling between curvature and stress tensors. The gravitation constant is thus replaced by two running coupling constants which depend on scale and differ in the sectors of traceless and traced tensors. When calculated in the solar system in a linearized approximation, the metric is described by two gravitation potentials. This extends the parametrized post-Newtonian (PPN) phenomenological framework while allowing one to preserve compatibility with…
We describe our explicit Lorentz-invariant solution of the Einstein and null geodesic equations for the deflection experiment of 2002 September 8 when a massive moving body, Jupiter, passed within 3.7' of a line-of-sight to a distant quasar. We develop a general relativistic framework which shows that our measurement of the retarded position of a moving light-ray deflecting body (Jupiter) by making use of the gravitational time delay of quasar's radio wave is equivalent to comparison of the relativistic laws of the Lorentz transformation for gravity and light. Because, according to Einstein, t…
The classical tests of general relativity (perihelion precession, deflection of light, and the radar echo delay) are considered for the Dadhich, Maartens, Papadopoulos and Rezania (DMPR) solution of the spherically symmetric static vacuum field equations in brane world models. For this solution the metric in the vacuum exterior to a brane world star is similar to the Reissner-Nordstrom form of classical general relativity, with the role of the charge played by the tidal effects arising from projections of the fifth dimension. The existing observational solar system data on the perihelion shift…
We present an alternate formula for calculating gravitational time delay. We use this formula to reduce geodetic Very Long Baseline Interferometry (VLBI) data, taking into account gravitational effects within the solar system, and to test general relativity. The alternate formula was obtained by expanding the conventional formula in a Taylor series. We show that the gravitational delay can be split into several terms including a term due to the coordinate transformation and terms that are explicitly linked to the light deflection angle. Our formula is compared numerically with the conventional…
Recently, there has been an interest in inflation and modified gravity with a Weyl term added to the general-relativistic action (N. Derulle, M. Sasaki, Y. Sendouda and A. Youssef, JCAP, 3, 040 (2011)). In this paper we study empirical constraint on this modified gravity from solar-system experiments/observations. We first derive linearized equation of motion in the weak field limit and solve it for isolated system in the slow motion limit. We then use it to derive the light propagation equations, and obtain the relativistic Shapiro time delay and the light deflection in one-body central probl…
The perihelion precession, the deflection of light, and the radar echo delay are classical tests of General Relativity here used to probe brane world topologically charged black holes in a f(R) bulk and to constrain the parameter that arises from the Shiromizu-Maeda-Sasaki procedure applied to a f(R) bulk as well. The existing Solar system observational data constrain the possible values of the tidal charge parameter and the effective cosmological constant including f(R) brane world effects. We show that the observational/experimental data for both perihelion precession and radar echo delay ma…
We consider the observational effects of a deficit angle, w, in the topology of the solar system and in the 'double pulsar' system PSR J0737-3039A/B. Using observations of the perihelion precession of Mercury, and the gravitational deflection of light due to the Sun, we constrain the magnitude of such a deficit angle in the solar system to be 2*pi*(1-w), with 0<(1-w)<10^(-9) at 95% confidence. We calculate the effects of a deficit angle on the periastron advance, geodetic precession rate and inclination angle of the double pulsar system and use the observational data to obtain the constraint 0…
The accuracy of astrometric observations conducted via a space-borne optical interferometer orbiting the Earth is expected to approach a few microarcseconds. Data processing of such extremely high-precision measurements requires access to a rigorous relativistic model of light ray propagation developed in the framework of General Relativity. The data-processing of the space interferometric observations must rely upon the theory of general-relativistic transformations between the spacecraft, geocentric, and solar barycentric reference systems allowing unique and unambiguous interpretation of th…
The mysterious dark energy remains one of the greatest puzzles of modern science. Current detections for it are mostly indirect. The spacetime effects of dark energy can be locally described by the SdSw metric. Understanding these local effects exactly is an essential step towards the direct probe of dark energy. From first principles, we prove that dark energy can exert a repulsive dark force on astrophysical scales, different from the Newtonian attraction of both visible and dark matter. One way of measuring local effects of dark energy is through the gravitational deflection of light. We ge…
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