vacuum_stress_energy
plain-language theorem explainer
The vacuum_stress_energy supplies the zero tensor for empty spacetime in the Recognition Science gravity module. Researchers deriving vacuum Einstein solutions or confirming conservation without sources would cite this base case. The construction is a direct structure instantiation that sets all components to zero and confirms symmetry by reflexivity.
Claim. The vacuum stress-energy tensor is the zero tensor defined by $T_{mu nu} = 0$ for all indices, satisfying the symmetry condition $T_{mu nu} = T_{nu mu}$ identically.
background
The StressEnergy structure in this module represents the matter stress-energy tensor as an abstract symmetric map $T : Idx → Idx → ℝ$, obtained from the variational definition $T_{mu nu} = -(2/sqrt(-g)) delta S_matter / delta g^{mu nu}$. The module proves conservation nabla^mu T_{mu nu} = 0 from the contracted Bianchi identity together with the sourced Einstein field equations, thereby establishing Axiom 3 on matter coupling.
proof idea
This is a direct definition that populates the StressEnergy structure by assigning the constant-zero function to the T field and discharging the symmetry obligation by reflexivity.
why it matters
It supplies the zero-source instance required by the downstream theorem vacuum_is_special_case, which recovers the vacuum Einstein equations, and by StressEnergyCert, which certifies conservation for the RS coupling kappa = 8 phi^5. The construction closes the conservation chain for the empty case, consistent with the framework's T8 derivation of D = 3 and the phi-ladder constants.
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