Appendix C: WFR Stress-Energy Tensor (Full Derivation)

Appendix C: WFR Stress-Energy Tensor (Full Derivation)#

TLDR#

This appendix expands the full variational derivation of the WFR stress-energy tensor used in the geometry chapters.
Use it as reference when implementing WFR-consistency losses or when auditing the variational principles.

This appendix provides the full derivation of Theorem Theorem 8.

C.1 Setup#

\[ \mathcal{S}_{\mathrm{WFR}} = \frac12\int_0^T\int_{\mathcal{Z}} \rho\left(\|v\|_G^2+\lambda^2 r^2\right)\,d\mu_G\,ds,\]

with the continuity equation enforced separately:

\[ \partial_s\rho+\nabla\!\cdot(\rho v)=\rho r.\]

Define the Lagrangian density

\[ \mathcal{L}_{\mathrm{WFR}}:=\frac12\,\rho\left(\|v\|_G^2+\lambda^2 r^2\right).\]

We vary the metric \(G^{ij}\) while holding \((\rho, v, r)\) fixed as fields.

C.2 Metric variation#

Write the kinetic term using covariant components:

\[ \|v\|_G^2 = G_{ij} v^i v^j.\]

Under a variation of the inverse metric, \(\delta G^{ij}\), we have

\[ \delta G_{ij} = -G_{ia}G_{jb}\,\delta G^{ab},\]

so

\[\begin{split} \begin{aligned} \delta\|v\|_G^2 &= v^i v^j\,\delta G_{ij} \\ &= -v_i v_j\,\delta G^{ij}. \end{aligned}\end{split}\]

The volume form varies as

\[ \delta d\mu_G = -\frac12\,G_{ij}\,\delta G^{ij}\,d\mu_G.\]

Combine these:

\[ \delta\left(\sqrt{|G|}\,\mathcal{L}_{\mathrm{WFR}}\right) = \sqrt{|G|}\left[ -\frac12\,\rho v_i v_j\,\delta G^{ij} -\frac12\,\mathcal{L}_{\mathrm{WFR}}\,G_{ij}\,\delta G^{ij} \right].\]

Therefore,

\[ \delta\mathcal{S}_{\mathrm{WFR}} = -\frac12\int_0^T\int_{\mathcal{Z}} \left(\rho v_i v_j + \mathcal{L}_{\mathrm{WFR}} G_{ij}\right) \delta G^{ij}\,d\mu_G\,ds.\]

By definition,

\[ T^{\mathrm{WFR}}_{ij}:= -\frac{2}{\sqrt{|G|}}\frac{\delta(\sqrt{|G|}\,\mathcal{L}_{\mathrm{WFR}})}{\delta G^{ij}},\]

so we identify

\[ T^{\mathrm{WFR}}_{ij}=\rho v_i v_j + \mathcal{L}_{\mathrm{WFR}} G_{ij}.\]

C.3 Perfect-fluid form and pressure split#

Let

\[ P:=\mathcal{L}_{\mathrm{WFR}} =\frac12\,\rho\left(\|v\|_G^2+\lambda^2 r^2\right).\]

Then

\[ T^{\mathrm{WFR}}_{ij}=\rho v_i v_j + P G_{ij},\]

which is the perfect-fluid form in Riemannian signature. The reaction contribution is

\[ P_{\mathrm{react}}=\frac12\,\lambda^2\rho r^2,\]

and the transport contribution is \(P_{\mathrm{trans}}=\tfrac12\rho\|v\|_G^2\).

C.4 Relation to the metric law#

This auxiliary tensor is not substituted into the capacity-constrained metric law: that theorem uses the reward Risk Tensor from Definition 51. A combined source would require an explicit coupling and a unit conversion. The metric law remains the separate curvature–risk stationarity identity with source \(T^{\mathrm{risk}}_{ij}\). This completes the derivation of the auxiliary WFR tensor in Theorem Theorem 8.