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Gravitational lensing observables in stationary and axisymmetric solutions in general relativity

Matteo Luca Ruggiero, Davide Astesiano

TL;DR

The paper investigates light propagation in stationary, axially symmetric rotating dust configurations within General Relativity, where rotation introduces frame-dragging through off-diagonal metric terms. By deriving a weak-field metric that includes a dragging potential ψ (solving the Grad-Shafranov equation) and a modified Poisson equation for Φ, the authors formulate a Fermat-based lensing framework and compute a leading gravitomagnetic deflection term. They show that the dragging contribution yields an effective mass $M_{\ ho}$, and that the gravitomagnetic deflection is of order $O(1/c)$, enabling observable image asymmetries that depend on the Grad-Shafranov solution. The work highlights non-Newtonian, rotation-driven lensing signatures that could inform interpretations of dark matter tracers and motivate high-resolution gravitational lensing observations.

Abstract

We investigate light propagation in self-gravitating systems composed of an axially symmetric, stationary, rotating dust fluid. These configurations are intrinsically relativistic, sustained entirely by their rotation, since no compact or finite dust distribution can exist under the same symmetry conditions in Newtonian gravity. In such systems, rotational effects arise from off-diagonal components of the spacetime metric, which are not negligible compared to their Newtonian counterparts. We analyze how these components affect the deflection angle of light, showing that they can be interpreted as contributing an additional effective mass. Moreover, their presence can, in principle, be detected through the characteristic asymmetry they induce in the images of background sources.

Gravitational lensing observables in stationary and axisymmetric solutions in general relativity

TL;DR

The paper investigates light propagation in stationary, axially symmetric rotating dust configurations within General Relativity, where rotation introduces frame-dragging through off-diagonal metric terms. By deriving a weak-field metric that includes a dragging potential ψ (solving the Grad-Shafranov equation) and a modified Poisson equation for Φ, the authors formulate a Fermat-based lensing framework and compute a leading gravitomagnetic deflection term. They show that the dragging contribution yields an effective mass , and that the gravitomagnetic deflection is of order , enabling observable image asymmetries that depend on the Grad-Shafranov solution. The work highlights non-Newtonian, rotation-driven lensing signatures that could inform interpretations of dark matter tracers and motivate high-resolution gravitational lensing observations.

Abstract

We investigate light propagation in self-gravitating systems composed of an axially symmetric, stationary, rotating dust fluid. These configurations are intrinsically relativistic, sustained entirely by their rotation, since no compact or finite dust distribution can exist under the same symmetry conditions in Newtonian gravity. In such systems, rotational effects arise from off-diagonal components of the spacetime metric, which are not negligible compared to their Newtonian counterparts. We analyze how these components affect the deflection angle of light, showing that they can be interpreted as contributing an additional effective mass. Moreover, their presence can, in principle, be detected through the characteristic asymmetry they induce in the images of background sources.

Paper Structure

This paper contains 5 sections, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Gravitational Lensing in GR
  3. The Weak Field Solution
  4. Light deflection in the axially symmetric and stationary solution
  5. Discussion and conclusions

Figures (2)

  • Figure 1: The unperturbed path lies in the $XY$ plane. We denote with $\Omega$ the angle between the $x$ axis and $l$, which is the intersection between the propagation plane and the reference plane $xy$, while the angle between the $z$ and $Z$ axes is $i$; eventually, we denote by $\omega$ the angle between $l$ and the $X$ axis.
  • Figure 2: The sources of light rays are distributed along a circle in the plane of the lens; the dust system rotates about the $z$ axis; the resulting image exhibits a deformation determined by the gravitomagnetic term.