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

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This study mathematically analyzes impulsive gravitational waves, showing that distinct spacetime metrics are equivalent. We demonstrate how smooth transformations bridge these metrics, clarifying their physical equivalence in gravitational wave theory.

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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.