IndisputableMonolith.Relativity.GW.TensorDecomposition
The TensorDecomposition module decomposes a general spatial metric perturbation into its transverse-traceless tensor sector plus vector and scalar contributions for use in gravitational wave calculations. Researchers deriving quadratic actions or propagation equations in modified gravity cite it to isolate the TT component h_ij^TT. The module aggregates definitions and lemmas imported from the Geometry and FRWMetric modules without containing its own proofs.
claimDecomposition of a spatial metric perturbation $h_{ij}$ into transverse-traceless tensor part $h_{ij}^{TT}$, vector modes, and scalar modes, with projection operator satisfying $k^i P_{ij,kl}=0$, $P_{ij,kl} g^{kl}=0$, and uniqueness of the TT sector.
background
The module sits inside the Recognition Science treatment of relativity and imports the FRWMetric module for the background cosmology together with the Geometry module, which re-exports manifold and tensor structures. It introduces the splitting of perturbations around the FRW background into irreducible representations under spatial rotations. The central objects are the general tensor perturbation and the decomposition map that isolates the transverse-traceless part required for gravitational-wave propagation.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the TT decomposition required by the parent Gravitational Waves module, which states it provides tensor perturbation theory including TensorDecomposition for h_ij^TT, ActionExpansion for quadratic actions, and PropagationSpeed. It therefore supplies the geometric prerequisite for the GW170817 bounds on scalar coupling listed in the downstream module documentation.
scope and limits
- Does not derive the quadratic action for the tensor modes.
- Does not compute the propagation speed or dispersion relation.
- Does not incorporate constraints from specific observations such as GW170817.
- Does not treat nonlinear or higher-order metric corrections.