Cauchycharacteristic

Cauchy-characteristic refers to a numerical relativity framework that combines a Cauchy (3+1) evolution in an interior region with a characteristic (null) evolution in an exterior region. The goal is to model spacetime in a way that allows accurate extraction of gravitational radiation at null infinity, where waves are unambiguously defined.

In this approach, the interior region is evolved on spacelike hypersurfaces using the standard Cauchy formulation of Einstein’s equations. The exterior region is evolved along outgoing null hypersurfaces using the characteristic, or Bondi-Sachs, formalism. An interface, often called a worldtube, exchanges data between the two grids: the Cauchy evolution provides boundary data for the characteristic evolution, while the characteristic evolution supplies boundary data back to the Cauchy region. In Cauchy-characteristic extraction (CCE), this setup is used to propagate information from a finite-radius extraction surface out to null infinity, yielding gravitational waveforms directly at future null infinity.

The method offers advantages such as more accurate and gauge-invariant gravitational-wave extraction, reduced outer-boundary reflections, and the ability to track radiation to infinity without extrapolation. It is particularly valuable in simulations of compact binary mergers, core-collapse, and other strong-field scenarios where precise waveform information is essential.

Challenges include the complexity of coupling two distinct evolution schemes, stability and accuracy considerations at the interface, and computational overhead. Despite these, Cauchy-characteristic methods have become a mature tool in numerical relativity, complementing purely Cauchy approaches and improving waveform modeling for gravitational-wave science.