Diffeomorphism Gauge Freedom

Diffeomorphism gauge freedom is the redundancy obtained when all dynamical fields are moved together without changing physical boundary or reference data. It is not the statement that every diffeomorphism is unobservable. Transformations that act at a boundary, change asymptotic frames, or move fields relative to external apparatus can carry physical information and charges.

Required background. We use isometries and Lie derivatives, Einstein dynamics, and the Dirichlet gravitational boundary problem.

Helpful background. Preparation in special relativity, smooth tensor fields, and presymplectic currents clarifies gauge directions and boundary charges.

Infinitesimal gauge directions

For the active-pullback convention, a vector field $\xi^a$ generates

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

and corresponding Lie derivatives of all matter fields. Covariance implies that this variation maps solutions to nearby solutions. If $\xi^a$ has compact support in the interior, the transformation preserves all boundary data and represents a redundancy in the local description.

This redundancy explains why the ten metric components in four dimensions do not represent ten freely propagating physical modes. Four functions $\xi^a$ change the component description, while four differential identities relate the equations. A complete degree-of-freedom count additionally uses constraints and evolution; simply subtracting four twice is only a structural mnemonic.

Reall derives the infinitesimal metric variation and its relation to coordinate gauge Reall 2013, § 7.2, pp. 71–76, PDF.

Pure gauge in linearized gravity

On Minkowski background, a pure-gauge perturbation is

$$h_{\mu\nu}=2\partial_{(\mu}\xi_{\nu)}.$$

The linearized Riemann tensor is

$$R^{(1)}_{\rho\sigma\mu\nu} =\frac12\left( \partial_\mu\partial_\sigma h_{\rho\nu} +\partial_\nu\partial_\rho h_{\sigma\mu} -\partial_\nu\partial_\sigma h_{\rho\mu} -\partial_\mu\partial_\rho h_{\sigma\nu} \right).$$

Substituting $h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$ makes all third derivatives cancel pairwise because partial derivatives commute. Thus $R^{(1)}=0$. A nonzero component of $h_{\mu\nu}$ can be entirely descriptive; curvature distinguishes a local physical perturbation at this order.

The converse needs care. A perturbation with zero local linearized curvature may still encode global identifications or boundary changes. Gauge is determined by the allowed $\xi^a$ and the domain, not only by a local tensor calculation.

Boundary transformations and charges

Take a finite region with fixed induced metric. A vector field that vanishes appropriately at the boundary preserves the variational data. A vector field tangent to the boundary may act as an exact or asymptotic symmetry; one with a nonzero normal component can move the boundary itself. The presymplectic form tests whether the transformation is degenerate or has a nonzero Hamiltonian boundary generator.

At spatial or null infinity, diffeomorphisms preserving falloff conditions can include translations, rotations, boosts, or more general asymptotic transformations. They are not automatically quotiented out because their surface generators can measure energy, angular momentum, or other charges. The same local formula $\mathcal L_\xi g$ therefore represents gauge for compactly supported $\xi$ and a physical symmetry action for certain boundary-nonvanishing $\xi$.

Gauge fixing versus symmetry

A gauge condition, such as harmonic coordinates, selects representatives from gauge orbits to make equations tractable. It is not a physical restriction if every relevant orbit intersects the gauge slice with appropriate boundary conditions. Residual gauge transformations preserve the chosen condition and must still be handled.

A Killing field satisfies $\mathcal L_\xi g_{ab}=0$ and is an exact symmetry of one configuration, not merely a gauge change between distinct representatives. Its normalization and boundary behavior determine associated conserved quantities. Confusing a Killing symmetry with general diffeomorphism redundancy erases the difference between stabilizers of a solution and directions along its gauge orbit.

Carroll separates general covariance, gauge transformations, and isometries in a common geometric language Carroll 1997, Chapter 5, pp. 129–141, PDF.

Exercises

Why can a constant time translation at infinity be physically nontrivial even though it is generated by a diffeomorphism?

Solution

It does not vanish at the boundary and can preserve the asymptotic conditions while acting on the asymptotic time frame. In the presymplectic description its generator may have a nonzero surface term, identified with energy. Compactly supported diffeomorphisms are degenerate gauge directions under the same boundary conditions; the boundary action distinguishes the cases.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.