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The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples.

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This study reveals that gravitational radiation near infinity exhibits non-smooth structures, challenging previous models. Logarithmic terms in asymptotic expansions indicate a complex dynamical behavior of spacetime at future null infinity.

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Area of Science:

  • General Relativity and Gravitational Physics
  • Mathematical Physics
  • Cosmology

Background:

  • Penrose's proposal of smooth conformal compactification of spacetime (smooth null infinity) may not fully capture gravitational radiation structure from infalling masses.
  • Previous models of gravitational radiation near infinity require a more dynamic understanding of spacetime structure.

Purpose of the Study:

  • To investigate the precise structure of gravitational radiation near infinity.
  • To provide a dynamical understanding of the non-smoothness of null infinity.
  • To analyze the asymptotic behavior of scalar radiation under specific boundary conditions.

Main Methods:

  • Construction of solutions to spherically symmetric Einstein-Scalar field equations.
  • Application of polynomially decaying boundary data and the no incoming radiation condition.
  • Analysis of asymptotic expansions of scalar radiation near future null infinity.

Main Results:

  • Demonstrated that non-zero initial Hawking mass leads to logarithmic terms in the asymptotic expansion of scalar radiation near future null infinity.
  • Showed that these logarithmic terms (proportional to r^-3 log r) appear in accordance with the non-smoothness of future null infinity.
  • Confirmed that similar logarithmic terms arise in the linear theory on a Schwarzschild background, appearing at second order (r^-4 log r) in scattering problems.

Conclusions:

  • The structure of gravitational radiation near infinity is inherently non-smooth, necessitating a dynamical approach.
  • Logarithmic terms in asymptotic expansions are a key indicator of this non-smoothness.
  • The findings have implications for understanding wave scattering on black hole backgrounds.