Light Bent by Gravity: Precision Down to Billionths of a Degree express
A light ray passing by a massive body slightly changes its direction, like a stream of water bending around a rock in a creek. In the solar system, there are many such "rocks": planets, their moons, asteroids. And each one, with its own gravity, bends the path of light coming from distant stars.
The authors of the article calculated this bending for the case when the light source and the observer are at finite distances from the massive body. The observer, for instance, is located at the Lagrange point L2 of the Sun–Earth system (a special point in space where a telescope can "hover" stationary relative to Earth and the Sun). The bodies can be of any shape, with any internal structure and rotation: their gravity is described by a full set of time-varying mass and spin multipoles (a mathematical description of mass distribution and rotation).
For axisymmetric bodies (those that look the same when rotated around an axis), the deflection is expressed through Chebyshev polynomials — a set of mathematical functions that allow a compact description of complex dependencies. This makes it possible to compute the deflection to any accuracy, taking into account all multipoles.
It is shown that the total light deflection, when the source and observer are infinitely far from the body, serves as an upper limit for any real case. It's as if the maximum bend of a water stream around a rock occurs when the rock sits in an infinitely long and calm creek.
These calculations are needed for future astrometric measurements with accuracy down to the sub-microarcsecond level. Such precision would allow, for example, more accurate determination of star positions and testing of fundamental laws of physics.
Original abstract
as written by the authors arXiv:2608.15658v1
The effect of deflection of a light signal that propagates through the gravitational field of a solar system body at rest is considered in the case that the source and the observer are at a finite distance from the body. The observer is assumed to be located somewhere nearby the Earth, for instance at Lagrange point L2 of the Sun-Earth system, while the spatial position of the celestial light source is arbitrary. The gravitational fields in the exterior of the bodies are described by the full set of time-dependent mass-multipoles and spin-multipoles of these bodies. So the gravitating bodies can be of arbitrary shape, inner structure and rotational motion. The unit tangent vector of the light trajectory at the position of the observer is determined in the 1PN and 1.5PN approximation of the post-Newtonian (PN) scheme. A simplified expression for the unit tangent vector is obtained, where all terms are neglected that together contribute less than 10 nano-arcseconds in light deflection for all astrometric configurations between source, body and observer. It is shown that in the case of an axi-symmetric body the unit tangent vector of the light ray as well as the light deflection are given in terms of Chebyshev polynomials. This fact allows for determining the effect of light deflection in the gravitational fields of the bodies up to any order of the mass-multipoles and spin-multipoles. It is shown that the total light deflection, that is the angle of light deflection where source and observer are located at infinite spatial distance from the body, represents an upper limit of the effect of light deflection. These investigations are aiming at astrometric measurements on the sub-micro-arcsecond level of accuracy.
The deflection of light by the Sun's gravity is only 1.75 arcseconds: that's the angle at which a finger is seen at arm's length, divided by a thousand.