Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations

This note fixes the scope of the page's field-law and matter content and the status of its claims; it proves nothing by itself.

• Two presentations, one field law. The framework adopts the classical field law in exactly one of two presentations for any given model: the action formulation Adopted classical gravitational action and variational field law or the direct-equation formulation Adopted Einstein field equation in the direct-equation presentation. The theorem Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling identifies the equation form of the two presentations, so they define one classical field law rather than two independent premises; a model never adopts both.

• Adopted versus derived. The field law, the matter sectors, the cosmological parameter and the Hilbert convention are adopted model input. The equation form of the action presentation, the conservation and trace identities, the parameter-variation identity and the rest-frame readings of the energy conditions are conditional deductions recorded in Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling and Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid. None of the adopted statements is derived from mathematics, symmetry or observation, and none of the deductions asserts that the model describes nature.

• Matter sectors. The matter content is one of the sectors of clause (iii) of Adopted classical gravitational action and variational field law: the scalar-field model with a supplied action and Hilbert tensor in the action presentation, or the dust and perfect-fluid models with prescribed stress-energy tensor and matter equations in the direct-equation presentation. The stress-energy tensor occupies the slot defined in Stress-energy tensor field of a model; its component reading in a $g$-orthonormal frame is in $\mathrm{J}{\mathrm{m}}^{-3}$, and the coupling satisfies $\kappa T\sim {\mathrm{m}}^{-2}$ with $\kappa =8\pi G_N\mathrm{/}{c}^4$ in the conventions of SI unit and coordinate conventions of the classical gravitational framework. The equation of state of a fluid sector is a further assumption of that model; the energy conditions of The pointwise energy conditions WEC, NEC, DEC and SEC are optional additional assumptions unless a particular matter model implies them; the field equation alone does not universally force them.

• Smooth oriented branch. The material assumes the smooth-branch restriction of Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions and, for the action and flux constructions, an orientation of the model's spacetime. Distributional or lower-dimensional sources, boundaries with junction conditions, low-regularity fields and quantum matter lie outside this page; they require further mathematics and, where physical, further adopted assumptions.

• Empirical status. The page contains no experiment, no measured input and no empirical premise. Predictions computed from a fully specified model — geometry, matter sector and field law — may later be compared with reported observations; such a comparison tests the combined model and its auxiliary assumptions, and agreement is evidence rather than a proof of any postulate.

• Deferred obligations. The broader observer-dependent energy-current discussion and the limits on global gravitational energy are treated on the local-and-global energy page; this page already proves the divergence and finite-chain flux balance for smooth currents, including Killing currents. The flat and Newtonian limits, the spherical vacuum, the homogeneous cosmologies and the linearized wave models are treated on their own pages. This page states no existence, uniqueness or propagation theorem for solutions of the field equation, and no initial-value or global causal claim.