Cut-and-paste for impulsive gravitational waves with : the mathematical analysis

Area of Science:

• Theoretical physics
• General relativity
• Gravitational wave physics

Background:

• Impulsive gravitational waves are short, violent bursts of gravitational radiation.
• They are described by distinct spacetime metrics (Lipschitz and distributional).
• These metrics are considered 'physically equivalent' via discontinuous coordinate transformations.

Purpose of the Study:

• To provide a mathematical analysis of the physical equivalence of different spacetime metrics for impulsive gravitational waves.
• To investigate nonexpanding impulsive gravitational waves in constant curvature spacetimes.
• To clarify the relationship between Lipschitz and distributional metrics.

Main Methods:

• Geometric regularization procedure.
• Analysis of coordinate transformations.
• Distributional limits of smooth families of spacetimes.

Main Results:

• The discontinuous coordinate transformation arises as a distributional limit.
• Both Lipschitz and distributional spacetimes are distributional limits of a smooth sandwich wave.
• A natural geometric regularization procedure is devised.

Conclusions:

• The mathematical framework confirms the physical equivalence of distinct spacetime metrics for impulsive gravitational waves.
• The study establishes a rigorous method to relate smooth and distributional descriptions of these waves.
• This work provides a deeper understanding of the mathematical structure of gravitational waves.