Displacement of an Extended Body by Metric Deformation: The Geometric Displacement Theorem

Titre : Displacement of an Extended Body by Metric Deformation: The Geometric Displacement Theorem Abstract : Let (M, ḡ) be a compact Riemannian manifold without boundary and Ω ⊂ M an extended body with smooth mass density ρ > 0, initially at rest. When the metric evolves along a smooth family $g(t)$ with g(0) = ḡ, the Karcher center of mass $p(t)$ of Ω is displaced — without any applied force and without initial velocity. We prove the Geometric Displacement Theorem: dḡ(p(T), p(0)) ≤ C(Ω, ρ, ḡ) · D_Ω where DΩ := ∫₀T ∫Ω (|ġ|g(t) + |∇g(t)ġ|g(t)) dVg(t) dt is the deformation functional restricted to Ω. A matching lower bound holds under a non-degeneracy condition, giving the equivalence c · DΩ ≤ dḡ(p(T), p(0)) ≤ C · DΩ. In particular, p(t) ≡ p(0) if and only if g is static on Ω — a Riemannian generalization of Newton's first law. Applications to the Ricci flow and to cosmological spacetimes via the ADM decomposition are given, including an exact closed-form expression for DΩ in FLRW geometry.