# Conformal Thin Sandwich Decomposition

### From GRwiki

#### Notation

Let Newton's constant . The matter energy density is , the momentum density is , and is the trace of the spatial stress.

#### Original Thin Sandwich Decomposition

Freely specify the following:

The variable is the trace-free part of the time derivative of the conformal metric . Also, is related to the scalar lapse function by , where is the determinant of the physical spatial metric .

Let and denote the covariant derivative and Ricci tensor built from the conformal metric. Indices on , and are raised with the inverse conformal metric. Define the operators

Solve the following equations for a scalar and a vector :

where

The physical metric and extrinsic curvature are

If the initial data , are evolved using for the initial scalar lapse and for the initial shift vector, then:

- the initial value of will be given by

#### Extended Thin Sandwich Decomposition

Freely specify the following:

Let and denote the covariant derivative and Ricci tensor built from the conformal metric. Indices on , and are raised with the inverse conformal metric. Define the operators

Solve the following equations for a scalar , vector , and weight -1 scalar density :

where

The physical metric and extrinsic curvature are

If the initial data , are evolved using for the initial scalar lapse and for the initial shift vector, then:

- the initial value of will coincide with the scalar field .
- the initial value of will be given by