Minimal dynamical completion and memory

Minimal dynamical completion and memory

Linear completion: a quotient by the kernel of the brane response

Restrict first to the linearized, fixed-background problem with fixed brane forcing, for which the gauge-reduced bulk perturbation data form a vector space. Let $z_0$ denote the bulk gravitational initial data omitted by the readout (10), and define the brane-response operator $$\mathcal O_\Sigma z_0 =\text{the induced brane response generated by }z_0,\tag{86}$$ with brane initial data and applied brane sources held fixed. Put $N_\Sigma=\ker\mathcal O_\Sigma$.

This quotient discards exactly the bulk directions that never influence the brane, and retains no more. In the linear spectral representation it retains precisely the coupled KK amplitudes together with the allowed incoming homogeneous modes. Under a prescribed regular no-incoming state, the homogeneous amplitudes are fixed and the entire remaining response is the retarded convolution encoded by $G_R^{(\mathrm{KK})}$.

Schematically, in each relevant projected tensor sector, the exact linear brane equation takes the causal, generally nonlocal form $$\mathcal D_0 h_\Sigma(x) +\int d^4x'\;\mathcal K_R(x,x')\,h_\Sigma(x') =T_\Sigma(x)+\eta_{z_0}(x),\tag{88}$$ with $$\mathcal K_R =\bigl(G_R^{(0)}+G_R^{(\mathrm{KK})}\bigr)^{-1} -\bigl(G_R^{(0)}\bigr)^{-1}.\tag{89}$$ Tensor indices and normalizations are suppressed throughout (88)–(89): the symbols $\mathcal D_0$, $T_\Sigma$, and $\mathcal K_R$ denote the appropriately normalized operators and sources within the projected tensor sector under consideration. In equation (89), $G_R^{(0)}+G_R^{(\mathrm{KK})}$ is specifically the scalar coefficient multiplying $P^{(2)}$ in the full response operator (63); thus $\mathcal K_R$ is the spin-two-sector correction relative to the zero-mode operator. The $P^{(0)}$ contribution occurs only through the massless combination $G_R^{(0)}\bigl(P^{(2)}-\tfrac12P^{(0)}\bigr)$ and must be retained through the full projector-valued response (63). The combination $P^{(2)}-\tfrac12P^{(0)}$ is the complete massless-graviton exchange structure, not a scalar sector. Equation (88) is accordingly a schematic sector-by-sector statement, not a fully explicit tensor identity; the explicit tensor content is (62)–(63). The term $\eta_{z_0}$ is the homogeneous response of the unresolved bulk initial data. Two points deserve emphasis. First, $\mathcal K_R$ is computed by the KK spectral integral (56); it is not an arbitrary phenomenological memory function. Second, the same spectral density fixes both the branch cut of Section 6.2 and the long-distance $1/r^3$ residue of Section 6.5 , they are one prediction, not two. This structure agrees with the explicit integro-differential brane perturbation equation of Mukohyama [8].

Nonlinear completion

At the nonlinear level there is generally no vector-space kernel and hence no quotient by a subspace. Instead:

The tensor $E_{\mu\nu}$ is the local manifestation of the missing bulk response, not a substitute for it. Its divergence equation (42) constrains it but does not evolve it; supplying $E_{\mu\nu}$ instantaneously does not close the system, because its subsequent evolution requires the bulk problem. The nonlinear completion is therefore neither the linear quotient (87) nor, in general, a finite set of four-dimensional functions.