# Physical Transverse-Traceless Decomposition

### From GRwiki

__Notation__: Let Newton's constant . The energy density is and the momentum density is .

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. Solve the following equations for and :

where

Indices on are lowered with the conformal metric. The physical metric and extrinsic curvature are

__Useful relations__:
The equations above are written compactly with the operators

Also define by where is the covariant derivative built from the physical metric. Let denote the physical curvature scalar. The following relations hold:

for any vector and any symmetric trace-free tensor .