IndisputableMonolith.Relativity.GW.TensorDecomposition
TensorDecomposition decomposes a general spatial metric perturbation into its transverse-traceless tensor component plus scalar and vector sectors on an FRW background. Cosmologists and modified-gravity researchers cite it when isolating the two propagating gravitational-wave degrees of freedom. The module supplies the projection operator definition together with an algebraic uniqueness lemma for the TT part.
claimA spatial metric perturbation $h_{ij}$ on an FRW background decomposes as $h_{ij}=h_{ij}^{TT}+$ (scalar and vector sectors), where the transverse-traceless part $h_{ij}^{TT}$ is extracted by the projection operator satisfying $k^i h_{ij}^{TT}=0$ and $h^{TT i}_i=0$.
background
The module works in linearized gravity around a Friedmann-Robertson-Walker background imported from Cosmology.FRWMetric. It relies on the spatial geometry re-exports from the Geometry module aggregator for tensor operations on the spatial slices. Key objects are the general symmetric tensor perturbation, the decomposition map, the TT projection operator, and the uniqueness statement for the extracted tensor modes.
proof idea
The module first defines the projection operator via spatial derivatives and the Laplacian, then verifies the algebraic identities that isolate the transverse-traceless component. Uniqueness follows by direct computation showing that any two candidate TT tensors differ by a term annihilated by the projection.
why it matters in Recognition Science
This module supplies the tensor decomposition required by the parent Gravitational Waves module for constructing the quadratic action and propagation equations. It directly supports the ActionExpansion submodule by isolating the TT modes that carry the physical gravitational-wave degrees of freedom and for later imposition of GW170817 bounds.
scope and limits
- Does not derive the equations of motion for the perturbations.
- Does not include nonlinear or higher-order interaction terms.
- Does not treat vector or scalar mode dynamics.
- Does not incorporate matter sources or anisotropic stress.