Optics equation provides approximation of gravitational lenses for students

Gravitational lensing is the phenomenon of light bending around massive objects along curves in space-time, a major conclusion of the general theory of relativity. Typically a thorough working knowledge of Einstein’s theory is required to develop an intuition about how this process works, making the topic difficult for students.

A pair of researchers proposed a simple optics equation that mimics the basic behavior of gravitational lenses. Drawing on Fermat’s principle of least time and the Euler equation, Szafraniec and Harford provided a simple ray trajectory equation that illustrates how light rays bend, reflect and loop as well as fatal attraction near black holes.

“The solution described in terms of trigonometric or hyperbolic functions is so simple that with very little effort it is possible to consider very complex scenarios, such as imaging a galaxy located behind the gravitational lens,” said author Bogdan Szafraniec. “It is the simplicity of the model that makes it attractive.”

The model describes captured rays with hyperbolic functions and escaping rays with trigonometric functions. The closer to the center of the lens, the more the escaping rays bend.

The researchers used these findings to reproduce arcs and halo-like Einstein rings that have been observed in images from the Hubble and James Webb space telescopes.

The new model adds to other attempts to make general relativity more accessible, such as using the base of a wine glass to mimic the effects of a gravitational lens.

The authors hope their model will inspire more students to look for simple solutions as well as study general relativity.

Source: “A simple model of a gravitational lens from geometric optics,” by Bogdan Szafraniec and James Frederick Harford, American Journal of Physics (2024). The article can be accessed at https://doi.org/10.1063/5.0157513 .