# Hamiltonian (ADM)

### From GRwiki

## Contents |

#### Notation:

where is Newton's constant. is the spatial metric, is the scalar lapse function, is the shift vector, and is the spatial covariant derivative. is the conjugate momentum, related to the extrinsic curvature by

#### Hamiltonian

Time evolution is defined via Poisson brackets with the Hamiltonian

The Hamitonian and momentum constraints are

The fundamental Poisson brackets relations are where is the three--dimensional Dirac delta function.

#### Equations of motion:

In terms of the time derivative operator , the ADM (Hamiltonian) equations are

where is the spatial Einstein tensor. The constraint evolution system is:

#### Other relations:

The time derivative of the extrinsic curvature is

where and are the spatial Ricci tensor and spatial curvature scalar. The zero density constraints defined by York (see the gdot-Kdot system) are and . They are related to the ADM constraints by and . The York form of the constraints evolved with the Hamiltonian (ADM equations) are