Transformation of Solid Angle in Gravitational Lenses by P.Schneider et al.

[center o bass]

When considering a small beam of null-geodesics in spacetime it is possible to define the solid angle spanned by two of the rays at the observer.

At page 111 in "Gravitational Lenses" by P.Schneider et. al. they state with reference to Figure (b) that:

"The dependence of this distance on the 4-velocity of the observer, given
the events 8 and 0, is due to the phenomenon of aberration. In fact, one
infers from [Figure (b)] that, if $k^\alpha$ is any vector tangent to the ray $SO$ at $O$ and $U^\rho$, $\tilde{U}^\rho$ are two 4-velocities at $O$ with corresponding solid angles $d\Omega$ and $d\tilde{\Omega}$, then
$$ \frac{d \tilde{ \Omega} }{d \Omega} = \frac{(k_\rho U^\rho )^2}{(k_\rho \tilde{U}^\rho)^2}."$$

No further explanations are given, so I guess the relation above must somehow be easily inferred from Figure (b), however I can not see how.

How is the relation above inferred from the figure?

 

[Ambitwistor]

Thank you for your forum post. I would like to clarify the concept of solid angle and aberration before explaining how the relation above can be inferred from the figure.

A solid angle is a measure of the amount of space spanned by a cone of rays from a point in three-dimensional space. It is usually measured in steradians (sr) and is analogous to the concept of angle in two-dimensional space. In the figure, the solid angle is represented by the shaded area.

Aberration is a phenomenon where the direction of a ray of light appears to be different when viewed from different points in space. This is due to the relative motion between the observer and the source of light. In the figure, the aberration is represented by the difference in the direction of the rays SO and S'O when viewed from the observer O.

Now, let us consider the two 4-velocities U and ̃U at point O in the figure. These 4-velocities represent the motion of the observer and the source of light respectively. The solid angles dΩ and d̃Ω represent the amount of space spanned by the cones of light from points S and S' respectively, as viewed by the observer at O.

The relation above states that the ratio of the solid angles d̃Ω and dΩ is equal to the square of the ratio of the dot products of the 4-velocities k and U, and k and ̃U. In other words, it is the ratio of the amount of space spanned by the cones of light from points S' and S respectively, as viewed by the observer at O.

This can be easily inferred from the figure by considering the angle between the rays SO and S'O. As the angle between the rays increases, the solid angle d̃Ω will also increase, while the solid angle dΩ remains the same. This is because the observer at O is seeing the light from point S' at a different angle due to aberration. Therefore, the ratio of the solid angles d̃Ω and dΩ will increase as the angle between the rays increases, resulting in the relation stated above.

I hope this explanation helps you understand how the relation can be inferred from the figure. If you have any further questions, please do not hesitate to ask. Thank you.