Scope and status of the local-and-global-energy page: what is defined, what is conditional, and what is deferred

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

• What is defined. For a smooth symmetric covariant $2$-tensor field $T$ and a smooth observer congruence $n$, the page defines the measured energy density, energy-momentum current, energy flux and spatial stress (The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress), their physical reading and units (Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress), the acceleration and expansion of the congruence (Acceleration, expansion and the along-congruence derivative of an observer congruence), and, for a model with an everywhere timelike future-directed Killing field, the Killing current and the Killing charge of a finite three-chain (read as an energy in joules when the field is length-normalized under the framework's length-coordinate convention) (The timelike Killing current and the charge of a three-chain). The algebraic decomposition behind the measured fields is proved in The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness, and the divergence and finite-chain identities in Divergence of the measured energy-momentum current and the finite-chain balance identity and Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current.

• What is conditional. The local energy balance and conservation of the Killing charge (whose energy reading requires the stated normalization) are established by theorem Local energy balance of the measured current and conservation of the Killing charge in stationary models: they are conditional on the adopted field law, on the smooth hypotheses 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 conservation statement, on the existence of the timelike Killing field and on the vanishing of the side flux. Nothing on the page establishes that a Killing field exists for a given metric (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor), and the finite-chain flux statements are those of The metric volume form, current flux through 3-faces and finite-chain charges and Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.

• Smooth and finite branch. The local constructions and identities use smooth fields on nonempty open domains, which may be noncompact; the flux statements integrate over finite smooth singular chains. This restricts the integration objects, not the ambient manifold or matter domain. Distributional sources, interfaces with nonsmooth fields and rough regions need the additional mathematics deferred in Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations. Integration over unbounded slices and passage to asymptotic limits require additional convergence and boundary hypotheses not supplied by the finite-chain statements. The observer congruence is an idealization of a family of ideal-clock worldlines; the physical reading of measured quantities is the framework's reading of SI unit and coordinate conventions of the classical gravitational framework, and no numerical value is asserted by any item of the page.

• Deferred obligations. The absence of a universal global energy, the observer dependence of measured energy and the deferral of ADM-, Bondi- and Komar-type masses to a future boundary-asymptotic module are recorded in Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses. In particular, no asymptotic energy, quasilocal mass, radiative loss or initial-data energy is defined or used here. Later pages of this framework may consume the redshift-weighted Killing current and the measured-energy identities for stationary and spherical models and for linearized waves; each such use must verify the hypotheses of the items it consumes and carry the item's scope qualifications with it.

• Choice convention. The smooth-bundle structure used throughout the page assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) through Observer congruences, their observer curves and adapted charts (and The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress); no other choice principle is used, and no foundational item of this page introduces a choice principle of its own.

• Empirical status. The page contains no experiment, no measured input and no empirical premise. Its physics claims are conditional deductions from the adopted postulates and stated hypotheses; any later comparison with reported observations tests the combined model and apparatus assumptions and is evidence rather than proof.