# 214. Constitutive Origin of Background-Weighted Momentum Appendix 213 used the variable-background relation $$ \mathbf{g}=\frac{\mathbf{S}}{k^2} $$ as an adopted extension of the uniform-region momentum density. This appendix derives that relation exactly for the specific constitutive closure already used in the gravity chapters, once the resolved transport momentum is taken to be the constitutive momentum density $$ \mathbf{g}:=\mathbf{D}\times\mathbf{B}. $$ With that choice, the relation used in appendix 213 follows identically inside this constitutive class: $$ \varepsilon(\mathbf{r},t)=\varepsilon_0\,\alpha(\mathbf{r},t), \qquad \mu(\mathbf{r},t)=\mu_0\,\alpha(\mathbf{r},t), $$ with $$ \alpha=\frac{c}{k}. $$ It also derives the exact energy-exchange term for time-dependent background and the leading background force for radiative transport in the geometric-optics limit. What it does **not** derive is the full exact background force for arbitrary resolved field configurations. That remains open. ## 214.1 Symmetric Constitutive Closure Take a source-free linear isotropic medium with constitutive relations $$ \mathbf{D}=\varepsilon\,\mathbf{E}, \qquad \mathbf{B}=\mu\,\mathbf{H}, $$ and assume the symmetric constitutive scaling $$ \varepsilon=\varepsilon_0\,\alpha, \qquad \mu=\mu_0\,\alpha. $$ Then the local transport speed is $$ k=\frac{1}{\sqrt{\varepsilon\mu}} = \frac{1}{\sqrt{\varepsilon_0\mu_0}}\frac{1}{\alpha} = \frac{c}{\alpha}. $$ The local impedance is $$ Z=\sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{\mu_0}{\varepsilon_0}} = Z_0. $$ So this closure changes the propagation speed while keeping the impedance fixed. ## 214.2 Exact Momentum Density The Poynting vector is $$ \mathbf{S}=\mathbf{E}\times\mathbf{H}. $$ Take the resolved transport momentum density to be $$ \mathbf{g}:=\mathbf{D}\times\mathbf{B}. $$ Under the symmetric closure, $$ \mathbf{g} = (\varepsilon_0\alpha\mathbf{E})\times\mathbf{B}. $$ Since $$ \mathbf{H}=\frac{\mathbf{B}}{\mu} = \frac{\mathbf{B}}{\mu_0\alpha}, $$ we have $$ \mathbf{S} = \mathbf{E}\times\mathbf{H} = \frac{1}{\mu_0\alpha}\,\mathbf{E}\times\mathbf{B}. $$ Therefore $$ \mathbf{E}\times\mathbf{B} = \mu_0\alpha\,\mathbf{S}, $$ and hence $$ \mathbf{g} = \varepsilon_0\mu_0\alpha^2\,\mathbf{S} = \frac{\alpha^2}{c^2}\,\mathbf{S}. $$ But $$ \alpha=\frac{c}{k}, $$ so $$ \frac{\alpha^2}{c^2}=\frac{1}{k^2}. $$ Therefore $$ \boxed{ \mathbf{g}=\frac{\mathbf{S}}{k^2} }. $$ This is exact for the adopted symmetric constitutive closure. So appendix 213's background-weighted momentum density is not arbitrary inside this constitutive class. It is exactly the constitutive momentum density $\mathbf D\times\mathbf B$. ## 214.3 Exact Energy Density and Time-Dependent Exchange The electromagnetic energy density is $$ u := \frac{1}{2}\bigl(\mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}\bigr) = \frac{1}{2}\bigl(\varepsilon E^2+\mu H^2\bigr). $$ Maxwell's equations in the source-free medium are $$ \nabla\cdot\mathbf{D}=0, \qquad \nabla\cdot\mathbf{B}=0, $$ $$ \nabla\times\mathbf{E}=-\partial_t\mathbf{B}, \qquad \nabla\times\mathbf{H}=\partial_t\mathbf{D}. $$ Take the scalar product of the second curl equation with $\mathbf E$ and of the first with $\mathbf H$, then subtract: $$ \mathbf E\cdot(\nabla\times\mathbf H)-\mathbf H\cdot(\nabla\times\mathbf E) = \mathbf E\cdot\partial_t\mathbf D+\mathbf H\cdot\partial_t\mathbf B. $$ Using $$ \nabla\cdot(\mathbf E\times\mathbf H) = \mathbf H\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf H), $$ this becomes $$ \partial_t u+\nabla\cdot\mathbf S = -\frac{1}{2}\bigl(E^2\,\partial_t\varepsilon+H^2\,\partial_t\mu\bigr). $$ For the symmetric constitutive closure, $$ \partial_t\varepsilon=\varepsilon_0\,\partial_t\alpha, \qquad \partial_t\mu=\mu_0\,\partial_t\alpha. $$ Also, $$ u = \frac{\alpha}{2}\bigl(\varepsilon_0E^2+\mu_0H^2\bigr). $$ So the right-hand side becomes $$ -\frac{1}{2}\bigl(\varepsilon_0E^2+\mu_0H^2\bigr)\partial_t\alpha = -\frac{u}{\alpha}\,\partial_t\alpha. $$ Therefore $$ \partial_t u+\nabla\cdot\mathbf S = -u\,\partial_t\ln\alpha. $$ Since $\alpha=c/k$, $$ \partial_t\ln\alpha=-\partial_t\ln k, $$ and hence $$ \boxed{ \partial_t u+\nabla\cdot\mathbf S = u\,\partial_t\ln k }. $$ This is the exact energy balance for the time-dependent symmetric constitutive closure. Consequences: - if the background is static, $\partial_t k=0$, then energy continuity is source-free; - if the background varies in time, the resolved field exchanges energy with the constitutive background at the exact rate $u\,\partial_t\ln k$. So a time-dependent background already breaks the stronger claim that the resolved subsystem always has source-free energy continuity. ## 214.4 Radiative Packet Dynamics in Geometric Optics To derive the leading background force for transport itself, restrict to the geometric-optics regime of a narrow radiative packet. The local dispersion relation is $$ \omega(\mathbf{r},\mathbf{q},t)=k(\mathbf{r},t)\,|\mathbf{q}|. $$ Treat this as the packet Hamiltonian $$ H(\mathbf{r},\mathbf{p},t)=k(\mathbf{r},t)\,|\mathbf{p}|. $$ Hamilton's equations are $$ \dot{\mathbf{r}}=\nabla_{\mathbf p}H = k\,\frac{\mathbf p}{|\mathbf p|}, $$ $$ \dot{\mathbf p}=-\nabla_{\mathbf r}H = -|\mathbf p|\,\nabla k. $$ The packet energy is $$ U=H=k|\mathbf p|, $$ so $$ |\mathbf p|=\frac{U}{k}. $$ Therefore $$ \boxed{ \dot{\mathbf p} = -\frac{U}{k}\,\nabla k = -U\,\nabla\ln k }. $$ This is the leading force on a radiative packet due to spatial variation of the local transport speed. Likewise the energy changes as $$ \dot U=\partial_t H=|\mathbf p|\,\partial_t k=\frac{U}{k}\partial_t k, $$ so $$ \boxed{ \dot U=U\,\partial_t\ln k }. $$ This is exactly the packet version of the local energy-exchange law derived above. ## 214.5 Radiative Background Force Density For a narrow radiative packet with local energy density $u$ and negligible internal stress compared to the background gradient scale, the geometric-optics force density is $$ \boxed{ \mathbf{f}_{\mathrm{rad}} = -u\,\nabla\ln k }. $$ Equivalently, using $\mathbf g=\mathbf S/k^2$ and $|\mathbf S|=uk$ for pure radiation, $$ \mathbf{f}_{\mathrm{rad}} = -k\,|\mathbf g|\,\nabla\ln k. $$ This formula is not the full exact background force for arbitrary resolved field configurations. It is the leading transport-force density for radiative packets in the geometric-optics limit. So the situation is now clear: - exact constitutive result: $$ \mathbf g=\mathbf S/k^2, $$ - exact time-dependent energy exchange: $$ \partial_t u+\nabla\cdot\mathbf S=u\,\partial_t\ln k, $$ - leading radiative background force: $$ \mathbf f_{\mathrm{rad}}=-u\,\nabla\ln k. $$ ## 214.6 Relation to Appendices 212 and 213 Appendix 212 used the same symmetric constitutive closure to derive the static weak-field metric and the corresponding benchmark observables. Appendix 213 used $$ \mathbf g=\frac{\mathbf S}{k^2} $$ as the adopted variable-background momentum relation inside the hydrodynamic balance-law extension. The present appendix now clarifies the scope: - that momentum relation is exact inside the symmetric constitutive closure, - the time-dependent energy source term is also exact there, - but the full exact resolved background force density is still not known for arbitrary field configurations, - only its radiative geometric-optics form has been derived here. ## 214.7 Summary For the symmetric constitutive closure $$ \varepsilon=\varepsilon_0\,\frac{c}{k}, \qquad \mu=\mu_0\,\frac{c}{k}, $$ the exact field momentum density is $$ \mathbf g=\mathbf D\times\mathbf B=\frac{\mathbf S}{k^2}. $$ The exact energy balance is $$ \partial_t u+\nabla\cdot\mathbf S=u\,\partial_t\ln k. $$ So: - static background: source-free energy continuity for the resolved field, - time-dependent background: exact energy exchange with the constitutive background. For radiative packets in geometric optics, the background force is $$ \mathbf f_{\mathrm{rad}}=-u\,\nabla\ln k. $$ This provides the constitutive origin of the background-weighted momentum used in appendix 213 and identifies the next real open problem: > derive the full exact background force and stress exchange for general > variable-background field configurations, not only for radiative transport.