Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses

This note fixes what the framework does and does not provide under the name "energy". It proves nothing by itself and asserts no fact about nature.

• Observer dependence of measured energy. The measured energy density $\mu =T_{ab}{n}^a{n}^b$ and the SI measured energy flux $q=c\mathrm{\Phi }$ of Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress depend on the observer congruence $n$; only the tensor $T$ itself is observer independent. For a prescribed model different congruences generally assign different energies to the same region, and the framework contains no congruence-independent local energy density of matter.

• No local gravitational energy in the framework as built. The framework defines a stress-energy tensor only for the matter sector (Stress-energy tensor field of a model, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions): $T$ is the matter content of the field law, and the geometric side $G_{ab}+\mathrm{\Lambda }{g}_{ab}$ has identically vanishing divergence (The second Bianchi identity and the vanishing divergence of the Einstein tensor). The local balance of Local energy balance of the measured current and conservation of the Killing charge in stationary models is a balance of the matter current; no item of the framework defines a local gravitational energy density or flux that would complete it into a conservation law for a total energy. The framework's normal-coordinate material (Point-normal and exponential coordinate charts of the metric-selected connection, Scope of normal coordinates and of the local-inertial-frame reading) records that the Christoffel symbols can be made to vanish at a point in a point-normal chart; it asserts nothing about second metric derivatives or curvature. Separately, the curvature operator $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (a) is a tensor of type $(1,3)$ by Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a). Thus if $R_p\mathrm{\ne }0$, no chart can make all its components zero: zero components in a coordinate basis would imply, by multilinearity, that $R_p(u,v)w=0$ for all $u,v,w\in T_{p}M$. This tensorial observation does not establish a no-go theorem for gravitational energy. The normal-coordinate identities alone define no local gravitational energy density; the cited discussions of gravitational-energy localization provide context, not a no-go theorem proved or used here.

• A global energy needs a symmetry and control of the flux. The Killing charge $Q_{\xi }$ of The timelike Killing current and the charge of a three-chain exists only for a model that admits an everywhere timelike future-directed Killing field $\xi$; the theorem is conditional on this field and supplies no existence result for any specified metric (Killing fields of a Lorentzian metric). Even with such a $\xi$, the finite-chain statement of Local energy balance of the measured current and conservation of the Killing charge in stationary models gives $Q_{\xi }({\mathrm{\Sigma }}_2)-Q_{\xi }({\mathrm{\Sigma }}_1)=-Q_{\xi }(B)$: conservation requires the side flux to vanish. It is read as an energy in joules when $\xi$ is length-normalized under the framework's length-coordinate convention; for another flow-parameter normalization, its units carry that scale. For a slice of unbounded spatial extent one further needs the limiting flux and the integrals to converge; the flux theorem Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current is stated for finite smooth singular chains and supplies no such limit. Smooth noncompact manifolds and open matter domains are allowed; it is integration over unbounded slices and passage to asymptotic limits that need additional convergence hypotheses. The scope note Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations excludes distributional sources and low-regularity fields, not noncompact smooth domains.

• A Killing field need not be timelike everywhere. A Killing field that is timelike on one region may be spacelike or past-directed elsewhere, in which case the charge is not an energy there; the de Sitter example recorded in the cited Tong reference has no globally defined positive conserved energy of this kind. Nothing in the framework selects a preferred Killing field or normalization: the normalization is part of the model data, and changing $\xi$ changes $Q_{\xi }$ and can change its unit scale as recorded in The timelike Killing current and the charge of a three-chain clause (d).

• Asymptotic and quasilocal masses are deferred. The ADM energy and related Bondi-type quantities are defined by surface integrals at infinity for asymptotically flat spacetimes, and Komar-type integrals by fluxes at a chosen boundary; both require an asymptotic or boundary structure (fall-off conditions, a sphere at infinity, limit arguments) or a chosen quasilocal surface. None of this structure is defined on this page or on the pages it depends on. Such masses are deferred to a future boundary-asymptotic module; no asymptotic mass, quasilocal mass or radiative energy loss is defined, computed or used in this framework here.

• Empirical status. The note contains no experiment, no measured input and no empirical premise. It records the scope of the framework's energy statements and the obligations deferred to other pages; it does not assert that the framework's energy notions are the only possible ones, nor that the model describes nature.