The initial value problem as it relates to numerical relativity.

Area of Science:

• Numerical relativity
• Gravitational physics

Background:

• The 3+1 split in general relativity foliates spacetime into spatial hypersurfaces.
• Initial value problem requires specifying field data on a hypersurface for time evolution.
• Geometric and matter fields constitute initial data, constrained by Einstein's equations.

Purpose of the Study:

• To discuss methods for constructing constraint-satisfying initial data in general relativity.
• To explore the application of conformal decompositions for black hole and neutron star binaries.
• To review quasi-equilibrium approximations for simplifying calculations.

Main Methods:

• Conformal transverse traceless and conformal thin sandwich decompositions are used.
• Free data specifying physical properties are chosen.
• Elliptic equations derived from constraint equations are solved to determine metric fields.

Main Results:

• Initial data for single black holes and neutron stars are described.
• Conformal decompositions are applied to construct initial data for binary systems.
• Quasi-equilibrium assumptions simplify elliptic equations for orbiting binaries.

Conclusions:

• Conformal decompositions provide a framework for constructing physically realistic initial data.
• Quasi-equilibrium approximations are useful for studying binary systems emitting gravitational radiation.
• The methods discussed are crucial for numerical simulations in general relativity.