Curvature-enhanced inertia in curved spacetimes: an ADM-based formalism with multipole connections

Abstract We propose a covariant definition of an inertia tensor on spatial hypersurfaces in general relativity, constructed via integrals of geodesic distance functions using the exponential map. In the ADM 3 + 1 decomposition, we consider a spacelike slice (Σ, γ ij ) with induced metric γ ij , lapse N , shift N i , and a mass-energy density ρ ( x ) on Σ. At each point p ∈ Σ, we define a Riemannian inertia tensor I p ( u , v ) = ∫ Σ d γ ( p , x ) 2 γ p ( u , v ) − 〈 exp p − 1 ( x ) , u 〉 〈 exp p − 1 ( x ) , v 〉 ρ ( x ) d V γ ( x ) , where d γ ( p , x ) is the geodesic distance on (Σ, γ ) and exp p − 1 ( x ) is the inverse exponential map. This reduces to the Newtonian inertia tensor in the flat-space limit. An expansion in Riemann normal coordinates shows curvature corrections involving the spatial Riemann tensor. We apply this to two cases: (i) for closed or open FLRW slices, a spherical shell of matter has an effective moment of inertia scaled by ( χ 0 / sin χ