Diffeomorphisms, Symmetry, and Conservation

Diffeomorphisms connect coordinate descriptions, gauge redundancy, exact symmetries, differential identities, and boundary charges, but those concepts are not interchangeable. This chapter separates passive relabeling from active pullback, identifies which active transformations are gauge only after boundary data are fixed, derives the off-shell Noether identity, and builds conserved and observable quantities from symmetries and relations.

Helpful background. Special relativity and smooth maps and tensors support this overview.

Enter the chapter

The coordinate and measurement questions can be studied before the dynamical arguments. The later Noether and gravitational-charge discussions build on Einstein’s equation, the gravitational boundary problem, and presymplectic potentials. Killing fields supply the exact symmetries used to construct matter currents.

A coordinate chart assigns numbers to events; changing those numbers leaves the geometric fields untouched. An active diffeomorphism instead pulls all fields to new values at the same manifold labels. Covariant equations map solutions to solutions under that pullback. The active map becomes gauge only if every dynamical field is transformed and the chosen boundary, asymptotic, and reference data are preserved.

The infinitesimal metric variation is

$$\delta_\xi g_{ab}=\mathcal L_\xi g_{ab}=2\nabla_{(a}\xi_{b)}.$$

For compactly supported $\xi^a$, invariance of the action turns this transformation into an off-shell identity among Euler–Lagrange expressions. For a boundary-nonvanishing $\xi^a$, the same integration by parts can leave a surface generator. The support and boundary behavior—not the local Lie-derivative formula alone—distinguish redundancy from physical symmetry.

Reall develops the active geometric framework and its connection to field variations Reall 2013, Chapter 7, pp. 67–76, PDF.

Six questions in sequence

1. Active and Passive Diffeomorphisms compares one field in two charts with two actively related field configurations and tracks complete relational measurements.
2. Diffeomorphism Gauge Freedom tests a pure-gauge linear perturbation and distinguishes compact support from boundary symmetries.
3. Diffeomorphism Noether Identity and Bianchi Consistency varies the full metric–matter action and derives the off-shell identity before recovering on-shell stress conservation.
4. Killing Fields and Conserved Matter Currents proves $\nabla_a(T^{ab}\xi_b)=0$ and states the side-flux assumptions needed for a slice-independent charge.
5. Observables and Coordinate-Invariant Questions shows why scalar character alone does not anchor an event and why tetrad readings and relational quantities can be physical.
6. Why Gravitational Energy Is Not Generally Local separates matter stress, coordinate-dependent pseudotensors, quasi-local surfaces, and asymptotic charges.

Each page uses the same test: name what is held fixed, what is transformed, and what happens at the boundary.

Identity, symmetry, and conservation

Diffeomorphism invariance of a scalar-matter action gives schematically

$$\nabla_a\mathcal E^{ab}-\mathcal E_\phi\nabla^b\phi=0.$$

This is true off shell. Pure gravity reduces it to $\nabla_aG^{ab}=0$; matter equations reduce the matter identity to $\nabla_aT^{ab}=0$. An exact Killing field then converts the conserved tensor into a current $J^a=T^{ab}\xi_b$. A global charge follows only after the side-boundary flux and normalization of $\xi^a$ are controlled.

Thus an identity is not a conserved charge. The identity comes from local gauge redundancy; the charge comes from a physical or asymptotic symmetry plus boundary conditions.

Observables and gravitational energy

Coordinate invariance asks whether a complete expression is unchanged under passive relabeling of the same geometric events. Gauge invariance asks whether it is constant on allowed active diffeomorphism orbits. A curvature scalar has a coordinate-independent value at a specified event, but fixed coordinate numbers alone do not identify the same physical event across actively related field configurations. Curvature at an event selected by material clocks and rods can satisfy both requirements. Tetrad components are apparatus dependent but remain coordinate independent when the apparatus is included.

Gravitational energy exhibits the same boundary sensitivity. Connection-quadratic densities can be set to zero at an event in normal coordinates and are not universal tensors. Curvature remains, as do quasi-local and asymptotic surface charges. Matter energy is locally tensorial; gravitational energy is generally encoded through gauge-fixed, quasi-local, or boundary structures whose assumptions must be stated.

Review the chapter

A vector field $\xi^a$ generates a metric variation $\mathcal L_\xi g_{ab}$. In one case $\xi^a$ has compact support; in another it approaches a normalized time translation at infinity. Why can the first be gauge while the second generates energy?

Solution

The compactly supported transformation preserves all boundary data. Its action variation has no surface contribution, and it is a degenerate direction of the physical description. The asymptotic translation acts nontrivially on the boundary frame. Integration by parts leaves a surface generator whose value can be the total energy. The same local Lie derivative has different physical status because its boundary behavior differs.

Onward connection

Congruences, Focusing, and Energy Conditions now uses a congruence rather than a symmetry: Raychaudhuri evolution converts curvature contractions into finite-parameter focusing under explicit kinematic and energy hypotheses. The distinction between local identity and global conclusion remains essential there.

References

• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.

Further reading

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.