Curvature-Enhanced Inertia in Curved Spacetimes: An ADM-Based Formalism with Multipole Connections

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 (Sigma, gamma_ij) with induced metric gamma_ij, lapse N, shift N^i, and a mass-energy density rho(x) on Sigma. At each point p in Sigma, we define a Riemannian inertia tensor I_p(u,v) as an integral over Sigma involving geodesic distances and the 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 (chi_0 / sin(chi_0))^2 > 1 or (chi_0 / sinh(chi_0))^2 < 1, confirming that positive curvature increases and negative curvature decreases inertia; and (ii) for a slowly rotating relativistic star (Hartle--Thorne approximation), we recover the known result: I = (8 pi / 3) integral from 0 to R of rho(r) exp(-(nu + lambda)/2) [omega_bar(r)/Omega] r^4 dr. In the Newtonian limit, this becomes I_Newt = (8 pi / 3) integral from 0 to R of rho(r) r^4 dr. We show that I_p reproduces these corrections and encodes post-Newtonian contributions consistent with Thorne's and Dixon's multipole formalisms. We also discuss the relation to Geroch--Hansen moments and propose an extension to dynamical slices involving extrinsic curvature. This work thus provides a unified, geometric account of inertia and multipole structure in general relativity, bridging Newtonian intuition and relativistic corrections.