Gravitational lensing, memory and the Penrose limit
In this paper, we discuss the gravitational field of ultrarelativistic extended spinning objects. For this purpose, we use a solution of the linearized gravitational equations obtained in the frame where such an object is translationally at rest, and boost this solution close to the speed of light. In order to obtain a regular limiting metric for non-spinning matter, it is sufficient to keep the energy of the boosted body fixed. This process is known as the Penrose limit. We demonstrate that in the presence of rotation, an additional rescaling is required for the angular momentum density components in the directions orthogonal to the boost. As a result of the Lorentz contraction, the thickness of the body in the direction of the boost shrinks. The body takes the form of a pancake, and its gravitational field is localized in the null plane. We discuss light and particle scattering in this gravitational field, and calculate the scattering parameters associated with the gravitational memory effect. We also show that by taking the inverse of the Penrose transform, one can use the obtained scattering map to study the gravitational lensing effect in the rest frame of a massive spinning object.