Relativistic chaos is coordinate invariant.

Area of Science:

• Relativity and Gravitation
• Dynamical Systems and Chaos Theory

Background:

• Previous studies suggested Lyapunov exponents in general relativity are non-invariant, implying chaos depends on spacetime coordinates.
• This led to conclusions that chaos in cosmology is coordinate-dependent, a significant claim in general relativity research.

Purpose of the Study:

• To investigate the coordinate dependence of chaos in general relativity.
• To determine the transformation laws of Lyapunov exponents under general spacetime transformations.
• To resolve the apparent noninvariance of chaos in cosmological contexts.

Main Methods:

• Derivation of transformation laws for Lyapunov exponents under general spacetime coordinate transformations.
• Analysis of the behavior of Lyapunov exponents in relativistic systems.
• Re-evaluation of previous claims regarding chaos in cosmology based on derived transformation laws.

Main Results:

• Lyapunov exponents exhibit specific transformation laws under general spacetime transformations.
• Chaos, defined by positive Lyapunov exponents, is demonstrated to be coordinate invariant.
• The noninvariance of chaos previously claimed in cosmology is shown to violate the hypotheses necessary for defining Lyapunov exponents.

Conclusions:

• Chaos in general relativity is fundamentally coordinate invariant.
• The coordinate dependence of chaos in cosmology is an artifact of improper application of Lyapunov exponent definitions.
• This finding reconciles chaos theory with general relativity, establishing a robust understanding of chaotic dynamics in curved spacetimes.