Gauge Freedom and Coordinate Conditions

Gauge freedom in general relativity is the freedom to change the map between geometric fields and coordinate labels without changing their relational physics. A useful coordinate condition must do more than simplify notation: it must select a regular representative, expose a suitable evolution system, preserve the Einstein constraints, and respect initial, boundary, and asymptotic data. This chapter builds those requirements from finite diffeomorphisms through lapse and shift choices to a practical diagnosis of gauge breakdown.

Helpful background. The Einstein Equation supplies covariant dynamics; Hypersurface Normals and Tangential Projectors supplies foliations; Killing Fields and Isometry Groups supplies Lie derivatives and exact symmetries; and PDEs and Green Functions for Gravitation supplies the initial-boundary and principal-symbol language.

From covariance to a coordinate system

A diffeomorphism $\Phi$ acts on every dynamical field by pullback:

$$(g,\psi)\longmapsto(\Phi^*g,\Phi^*\psi).$$

Covariance ensures that solutions map to solutions. It does not decide which maps are gauge redundancies after a domain, an initial surface, asymptotic frames, and boundary data have been specified. Nor does it choose a unique representative of a diffeomorphism orbit.

The chapter therefore separates four layers:

1. Gauge action: finite pullbacks and infinitesimal Lie derivatives.
2. Gauge condition: differential or geometric equations imposed on coordinate functions, lapse, or shift.
3. Gauge data: initial and boundary values that remove or retain the homogeneous residual transformations.
4. Gauge diagnostics: checks that the coordinate map, foliation, and reduced PDE system remain regular.

Keeping these layers distinct prevents several common errors. A chart can satisfy harmonic gauge and still possess residual waves. A lapse can avoid a region in coordinate time while physical curvature diverges there. A shift can move faster than light because it transports labels, not matter. A diffeomorphism at a boundary can carry a charge instead of representing a null direction.

The active–passive distinction and its relation to Killing symmetry are developed in Carroll 1997, §5, pp. 129–141, PDF, while the foliation gauge conditions used here are organized in Gourgoulhon 2007, Ch. 9, pp. 151–174, PDF.

Six questions, one dependency chain

Gauge Transformations in General Relativity fixes the active pullback convention, derives the Lie-derivative variation, and states when two Cauchy data sets are related by an admissible diffeomorphism. It also supplies the recurring flat-spacetime example: a time-dependent spatial relabeling changes metric components and shift while leaving curvature and intrinsic initial data unchanged.

Harmonic and Generalized Harmonic Coordinates then treats coordinate functions as scalar waves:

$$\Box_gx^\mu=-\Gamma^\mu=-H^\mu.$$

The condition removes the non-wave second-derivative terms from the Ricci tensor and defines the harmonic constraint $C^\mu=H^\mu-\Gamma^\mu$. The page distinguishes this principal-part reduction from the additional proof that $C^\mu$ propagates and that a boundary problem is well posed.

Time Slicing and Lapse Conditions asks which hypersurfaces represent successive times. It compares geodesic, harmonic, $1+\log$, maximal, and constant-mean-curvature slicings through

$$\partial_t\sqrt\gamma-\partial_i(\beta^i\sqrt\gamma) =-\alpha K\sqrt\gamma$$

and through the trace evolution equation for $K$. Hyperbolic lapse drivers control coordinate propagation; maximal and CMC conditions require elliptic boundary problems.

Spatial Gauge and Shift Conditions asks how points on neighboring slices are identified. Minimal distortion removes the longitudinal part of trace-free metric motion, while Gamma-freezing and Gamma drivers target conformal connection variables. The comparison makes the conformal-Killing kernel, reference connection, and parameter dependence explicit.

Residual Gauge Freedom and Boundary Data returns to the homogeneous equation. For changes between harmonic coordinate functions on a fixed geometry,

$$\Box_g^{\mathrm{sc}}\xi^\mu=0$$

acts on each coordinate function as a scalar. The page distinguishes this from the covariant vector-wave residual equation of de Donder perturbation gauge. Initial values, normal derivatives, incoming boundary fields, and asymptotic behavior decide which residual maps remain and whether they are redundancies or boundary symmetries.

Gauge Pathologies and Diagnostics closes the chain by separating chart failure, caustics, lapse collapse, gauge shocks, slice stretching, loss of hyperbolicity, constraint violation, and genuine geometric singularities.

A comparison of gauge controls

Control	Mathematical target	Data still required	Principal failure to test
Harmonic source $H^\mu$	Coordinate wave equations and metric wave principal part	Coordinate initial data and incoming gauge boundary data	Residual waves or subsidiary-constraint injection
Lapse $\alpha$	Normal separation and slicing geometry	Initial lapse; hyperbolic or elliptic boundary data	Caustics, lapse collapse, gauge steepening
Shift $\beta^i$	Tangential coordinate transport	Initial shift/driver and spatial boundary frame	Slice stretching, kernel, ill-conditioned gauge modes
Asymptotic restrictions	Allowed large transformations	Falloff, parity, and reference frame	Mistaking a charged symmetry for pure gauge
Diagnostic suite	Regularity of chart, foliation, and PDE representation	Independent geometric and formulation checks	Confusing coordinate failure with spacetime failure

The objects in the first three rows are coordinate controls, not observables. Their characteristic speeds and damping parameters describe how a representation evolves. Physical modes are identified only after constraints and gauge directions have been separated.

Convention locks

The active finite action is $\Phi^*g$. Its infinitesimal form is therefore

$$\delta_\xi g_{\mu\nu} =\mathcal L_\xi g_{\mu\nu} =2\nabla_{(\mu}\xi_{\nu)}.$$

A passive coordinate displacement produces an opposite component variation at fixed numerical argument. The sign is not a physical ambiguity; it records which map has been chosen.

For spacelike slices, this volume uses

$$t^\mu=\alpha n^\mu+\beta^\mu, \qquad n^\mu n_\mu=-1, \qquad K_{ij}=-\frac12\mathcal L_n\gamma_{ij}.$$

Hence

$$(\partial_t-\mathcal L_\beta)\gamma_{ij}=-2\alpha K_{ij}.$$

All lapse and shift formulas in this chapter follow this sign. A source using $K_{ij}=+\tfrac12\mathcal L_n\gamma_{ij}$ must be translated as a whole.

For generalized harmonic coordinates the chapter uses

$$\Box_gx^\mu=-H^\mu, \qquad C^\mu=H^\mu-\Gamma^\mu.$$

Later generalized-harmonic evolution pages retain this definition.

Diagnostic workflow

When a gauge-fixed evolution develops large components or gradients, proceed in this order:

1. Check the coordinate Jacobian, metric determinant, lapse, and causal character of the slice.
2. Compute Hamiltonian, momentum, and gauge-constraint residuals.
3. Inspect the physical, gauge, and constraint principal symbols for a complete uniformly conditioned characteristic basis.
4. Evaluate curvature in a regular frame and use invariants suited to the spacetime class.
5. Attempt an overlapping regular chart or a different foliation.
6. Only then make a claim about curvature singularity, incompleteness, or inextendibility, with the relevant definition and hypotheses.

Rindler lapse collapse and flat null caustics show why this order matters: both are coordinate failures in a curvature-free spacetime. Schwarzschild $r=0$ shows the opposite case, where a curvature invariant diverges and a coordinate change does not remove the geometric failure.

Review the chapter

In flat spacetime, introduce $T=t$ and $X=x+F(t)$.

1. Which part of the ADM data changes?
2. When is the new chart harmonic?
3. Which extra data must be fixed before the residual harmonic transformation is unique?
4. If an outer boundary remains at fixed $x$, when is the transformation still gauge within the original boundary-value problem?

Solution

The spatial metric and extrinsic curvature of the constant-time slices remain $\gamma_{xx}=1$ and $K_{xx}=0$, while the shift becomes $\beta^x=\dot F$; lapse remains one. Since

$$\Box_\eta x=\ddot F,$$

the chart is harmonic exactly when $\ddot F=0$. Even then, the homogeneous scalar wave equation for the coordinate functions admits residual transformations until the generator and its normal derivative are fixed initially and its incoming boundary behavior is specified.

A boundary at $x=x_0$ follows $X=x_0+F(t)$ in the inertial chart. It is the same worldtube as $X=x_0$ only if $F=0$; a nonzero constant $F$ requires a corresponding shift of the boundary label. The transformation is gauge within the original problem only if the boundary and all its prescribed data are pulled back consistently or if the allowed boundary symmetry explicitly includes that motion.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.
• Kreiss, Heinz-Otto, Oscar Reula, Olivier Sarbach, and Jeffrey Winicour. “Well-Posed Initial-Boundary Value Problem for the Harmonic Einstein Equations Using Energy Estimates.” Classical and Quantum Gravity 24 (2007): 5973–5984. DOI.