# Modal series expansions for plane gravitational waves

## TL;DR

This paper introduces a modal series expansion method for plane gravitational waves, providing a convergent solution framework with finite curvature and energy, advancing mathematical understanding of gravitational wave solutions.

## Contribution

It develops a new modal series expansion technique for plane gravitational waves with proven convergence and finite physical quantities, enhancing mathematical modeling capabilities.

## Key findings

- Series solutions are convergent under Raabe-Duhamel criteria.
- Solutions have well-defined, finite curvature tensors.
- Solutions possess finite energy content.

## Abstract

Propagation of gravitational disturbances at the speed of light is one of the key predictions of the General Theory of Relativity. This result is now backed indirectly by the observations of the behaviour of the ephemeris of binary pulsar systems. These new results have in- creased the interest in the mathematical theory of gravitational waves in the last decades and several mathematical approaches have been developed for a better understanding of the solutions. In this paper we develop a modal series expansion technique in which solutions can be built for plane waves from a seed integrable function. The convergence of these series is proven by the Raabe- Duhamel criteria and we show that these solutions are characterized by a well-defined and finite curvature tensor and also a finite energy content.