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Uniqueness and nonuniqueness in the Einstein constraints.

Harald P Pfeiffer1, James W York

  • 1Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, CA 91125, USA.

Physical Review Letters
|October 4, 2005
PubMed
Summary
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Researchers numerically solved the conformal thin-sandwich (CTS) equations, finding one solution for perturbed Minkowski space. However, specifying mean curvature yielded two distinct solutions, including one with a black hole, revealing nonunique solutions in general relativity.

Area of Science:

  • General Relativity
  • Mathematical Physics
  • Computational Astrophysics

Background:

  • The conformal thin-sandwich (CTS) equations are essential for numerical relativity, generalizing Newton's Laplace-Poisson equation.
  • Understanding solutions to these equations is crucial for simulating black holes and gravitational waves.

Purpose of the Study:

  • To numerically investigate solutions of the CTS equations for perturbed Minkowski space.
  • To explore the impact of specifying mean curvature on the number of solutions.

Main Methods:

  • Numerical examination of the CTS equations.
  • Analysis of elliptic systems with nonunique solutions.

Main Results:

  • A single solution was found for perturbed Minkowski space under standard CTS formulations.

Related Experiment Videos

  • Two distinct solutions emerged when mean curvature was specified via a fifth elliptic equation, one solution featuring a black hole.

    • Conclusions:

    • The study highlights the existence of nonunique solutions in general relativity, even in simplified systems.
    • This finding has implications for both fundamental general relativity and the broader study of elliptic systems.