Gravitational induction and the linearized single copy: Poincaré charges, the source map, and the breakdown of the gravito-electromagnetic analogy

Abstract Gravito-electromagnetism rewrites linearized gravity in Maxwell form, and gravitational induction extends it to bounded sources whose multipole moments evolve. The contraction A^μ=-h ̅^μν ξ_ν of Clark and Tucker that effects it is a linearized single copy in a restricted sense: it is the contraction with a fixed timelike vector on which Kerr–Schild and contraction-based double copies are built, but it lacks the structure of the double copy proper, and in Lorenz gauge its Maxwell equation is one row of the linearized Einstein equations. We characterize the source map T^μν↦T^μ0 completely. The characterization is kinematical, with no energy condition imposed: a conserved current confined to a fixed ball is the image of a localized symmetric conserved stress tensor if and only if its Poincaré charges are conserved. The mass dipole solves p^i (t)=K^i+tP^i with the boost and momentum charges as coefficients; the magnetic dipole, half the spin, is frozen. The cokernel is the space of iso(3)_ξ^*-valued functions of time, its instantaneous values the six rigid-motion obstructions of linear elasticity that drive our constructive converse; the kernel is the space of elastic self-stresses, and the exterior stress that torques a spinning-down dipole generates the torque obstruction class. Under the dominant energy condition the image second moment obeys |d^2/dt^2 ∫▒〖x^i x^j J^0 〗|≤2M. The two contraction maps in use differ by a term in the trace of the perturbation; for a pulsating source it gives the metric contraction an oscillating Gauss charge, a property of the auxiliary field of the harmonic representative rather than a gauge-invariant observable. Beyond linear order, the relaxed Einstein equations in harmonic coordinates allow the same bookkeeping with a flat-conserved but noncompact pseudotensor source, and the frozen charges become fluxes, fed at the standard gravitational-wave rates through the l=2 channel that the linear theory leaves open.