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Light deflection in axially symmetric stationary spacetimes filled with a moving medium

Christian Pfeifer, Barbora Bezděková, Oleg Yu. Tsupko

TL;DR

This work extends gravitational light deflection studies to axially symmetric spacetimes filled with a moving dispersive medium, beyond the cold-plasma limit. Using Synge's geometrical-optics formalism and a tractable refractive-index form $n^2(\omega,r)= a_0(r)+ a_1(r)/\omega + a_2(r)/\omega^2$, the authors derive analytic orbit equations in the equatorial plane for radial, rotational, and mixed medium motions, and obtain explicit deflection-angle integrals. In a slow-velocity regime, they show that radial medium motion does not contribute at linear order and that spacetime rotation and medium rotation can cancel under a balance condition, with a Kerr example making the degeneracy explicit. These results illuminate potential observational degeneracies between gravity and medium dynamics in gravitational lensing and point to future work on lensing images and alternative refractive-index models for more realistic astrophysical environments.

Abstract

The deflection of light rays near gravitating objects can be influenced not only by gravity itself but also by the surrounding medium. Analytical studies of such effects are possible within the geometrical optics approximation, where the medium introduces additional light bending due to refraction. These studies typically assume a cold non-magnetized plasma, for which light propagation is independent of the medium's velocity. In this paper, we extend the analysis to the general case of dispersive refractive media in motion and study its influence on light deflection. We consider an axially symmetric stationary spacetime filled with a moving medium, motivated by the interplay between rotational effects originating from the spacetime and those induced by the medium's motion. We begin by analyzing light deflection in the equatorial plane of a rotating object in the presence of a radially moving and rotating medium. Assuming a specific form of the refractive index enables a fully analytic treatment. In the particular cases of either pure radial or pure rotational motion, we obtain explicit expressions for the deflection angle. Next, we analyze the case of a slowly moving medium and identify two particularly interesting results. First, we show that, to the first order in the medium's velocity, the radial motion does not affect the light deflection. Second, assuming slow rotation of the gravitating object, we demonstrate that the black hole rotation and the medium motion can produce equivalent observational signatures. We find the quantitative condition under which these effects compensate each other. This relation becomes particularly clear for a Kerr black hole, discussed as an example.

Light deflection in axially symmetric stationary spacetimes filled with a moving medium

TL;DR

This work extends gravitational light deflection studies to axially symmetric spacetimes filled with a moving dispersive medium, beyond the cold-plasma limit. Using Synge's geometrical-optics formalism and a tractable refractive-index form , the authors derive analytic orbit equations in the equatorial plane for radial, rotational, and mixed medium motions, and obtain explicit deflection-angle integrals. In a slow-velocity regime, they show that radial medium motion does not contribute at linear order and that spacetime rotation and medium rotation can cancel under a balance condition, with a Kerr example making the degeneracy explicit. These results illuminate potential observational degeneracies between gravity and medium dynamics in gravitational lensing and point to future work on lensing images and alternative refractive-index models for more realistic astrophysical environments.

Abstract

The deflection of light rays near gravitating objects can be influenced not only by gravity itself but also by the surrounding medium. Analytical studies of such effects are possible within the geometrical optics approximation, where the medium introduces additional light bending due to refraction. These studies typically assume a cold non-magnetized plasma, for which light propagation is independent of the medium's velocity. In this paper, we extend the analysis to the general case of dispersive refractive media in motion and study its influence on light deflection. We consider an axially symmetric stationary spacetime filled with a moving medium, motivated by the interplay between rotational effects originating from the spacetime and those induced by the medium's motion. We begin by analyzing light deflection in the equatorial plane of a rotating object in the presence of a radially moving and rotating medium. Assuming a specific form of the refractive index enables a fully analytic treatment. In the particular cases of either pure radial or pure rotational motion, we obtain explicit expressions for the deflection angle. Next, we analyze the case of a slowly moving medium and identify two particularly interesting results. First, we show that, to the first order in the medium's velocity, the radial motion does not affect the light deflection. Second, assuming slow rotation of the gravitating object, we demonstrate that the black hole rotation and the medium motion can produce equivalent observational signatures. We find the quantitative condition under which these effects compensate each other. This relation becomes particularly clear for a Kerr black hole, discussed as an example.

Paper Structure

This paper contains 14 sections, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous studies of light propagation in axially symmetric spacetimes in presence of cold plasma
  3. Synge's formalism and equations of motion with general refractive index
  4. Deflection Angle of Light Ray for a Particular Choice of Refractive Index
  5. Orbit Equation for a Medium with Both Radial and Rotational Motion
  6. Deflection Angle for a Radially Moving Medium
  7. Deflection Angle for Rotating Medium
  8. Slowly moving medium in slowly rotating spacetimes
  9. The Synge Hamiltonian to linear order in the medium velocity on arbitrary spacetime
  10. Orbit equation in the equatorial plane
  11. Slowly rotating gravitating objects -- linearization in $P(r)$
  12. Example: Slowly rotating Kerr metric with slowly moving medium
  13. Discussion and Conclusions
  14. Alternative derivation of the orbit equation (\ref{['eq:dphi']}) in Section \ref{['sec:slowmedium']}

Figures (1)

  • Figure 1: Deflection angle $\hat{\alpha}$ of a light ray propagating in the equatorial plane of an axially symmetric spacetime filled with a moving medium of refractive index $n(\omega, r)$. The medium moves around the central object with both radial and azimuthal components of velocity. The light ray has radial coordinate $r$, travels from infinity, passes through the point of closest approach $r=R$ and continues back to infinity.