Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions

Adopt as physical assumptions of the classical gravitational framework, in addition to Adopted spacetime model and classical domain of the gravitational framework, the following statements about the matter content of a model.

(i) Matter content is model input. A model of the framework specifies its nongravitational fields or matter and their explicit equations of motion and couplings. The geometric data $(M,g,N)$ of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and covariance of the model's equations under relabelling of events do not select those fields, equations or couplings. Here covariance means that, on overlapping smooth charts, the component transformations specified for the model's fields carry solutions of the equations in one chart to solutions of the same equations in the other. Different matter specifications over the same geometry are different models of the framework, and no dynamical selection principle is asserted.

(ii) Smooth matter and its stress-energy tensor. The matter of a model on its matter domain is represented by smooth fields on an open domain $U\subseteq M$, and the model's stress-energy tensor is the smooth symmetric covariant $2$-tensor field $T$ on $U$ of Stress-energy tensor field of a model, with the unit reading recorded there. The relation of $T$ to the matter fields is part of the model: a model derived from an action defines $T$ as the Hilbert stress-energy tensor of that action, while a model that prescribes $T$ directly is a separate source model.

(iii) Consistency with the field law. Where a model prescribes $T$ independently of matter fields, the prescription is admissible only if it meets the consistency conditions imposed by the gravitational field law adopted for the model, stated where that law is stated. If a model specifies a real-valued action functional on its admissible metric and matter configurations, the action, its dependence on those fields, its stated symmetries, and the matter equations obtained from it are part of the model. Any conservation identity used for $T$ when those equations hold must be established from that model's action, symmetries, equations, and regularity hypotheses; it is not assumed for an arbitrary action or arbitrary smooth symmetric $T$.

(iv) Energy conditions are optional additional assumptions. A model may adopt any of the pointwise inequalities of The pointwise energy conditions WEC, NEC, DEC and SEC as further assumptions on its stress-energy tensor. The general framework postulates impose none automatically. A particular matter model may imply one or more of them; any such implication must be established from that model's equations and hypotheses. Each model records which conditions, if any, it adopts, and a model that adopts none is not thereby excluded.

(v) Scope and exclusions. Idealizations such as unmodeled self-force, finite-size and tidal corrections, dispersion, dissipation, and quantum effects are outside the framework unless the model's explicit equations include them; matter concentrated in a distributional or lower-dimensional source, and matter configurations whose fields fail to be smooth on the model domain, are outside the stated smooth branch and require additional mathematics. The adoption of a matter model is not a claim that it describes nature.

• Adopted, not deduced. Clauses (i)-(v) are model assumptions. They constrain what counts as a model of the framework; they imply no theorem about matter and no measured property of any substance. Clause (iii) leaves any conservation equation to the item stating the field law; that item must define the derivative used and state the matter equations and regularity hypotheses under which the equation is asserted.

• Line of responsibility. The geometric objects, the inverse metric, the causal classification and the metric-selected connection class are supplied by the mathematical items of this page; the framework postulates adopt them physically. Empirical comparisons involving matter require a fully specified model and reported observations with apparatus, calibration, sampling and uncertainty qualifications, and would test the combined model rather than these assumptions individually.