Residual Gauge Freedom and Boundary Data

A differential coordinate condition constrains a gauge transformation but rarely determines it uniquely. Transformations that preserve the condition solve a homogeneous differential equation. Their freely specifiable initial values, normal derivatives, boundary values, and asymptotic behavior are the residual gauge data. A complete gauge choice therefore consists of the differential condition plus data that eliminate unwanted homogeneous solutions—or deliberately retain those that represent useful boundary symmetries.

Required background. The Einstein Equation supplies the covariant solution space, while Killing Fields and Isometry Groups supplies flows, Lie derivatives, and stabilizers.

Helpful background. Hypersurface Normals and Tangential Projectors supplies Cauchy and boundary decompositions, and PDEs and Green Functions for Gravitation supplies uniqueness, homogeneous solutions, and initial-boundary data.

The stabilizer of a gauge condition

Represent a coordinate condition by four functionals

$$\mathcal C^\mu[g]=0.$$

For an active infinitesimal diffeomorphism generated by $\xi^\mu$,

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

The transformed metric remains in the same gauge to first order when

$$\delta_\xi\mathcal C^\mu[g]=0.$$

This equation defines the infinitesimal stabilizer of the gauge condition. It should not be confused with the stabilizer of the metric. A Killing field satisfies $\mathcal L_\xi g=0$ and therefore leaves the entire configuration fixed. A generic residual generator changes the metric components while leaving only $\mathcal C^\mu$ fixed.

For an algebraic gauge condition, the stabilizer equation may be algebraic. For a differential condition it is another differential equation, so local solutions require data. The familiar statement “four coordinate conditions fix four gauge functions” ignores the kernel of this operator, just as an elliptic equation ignores its harmonic kernel if boundary conditions are not specified.

Residual transformations in harmonic gauge

Use the harmonic constraint convention

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

Let $H^\mu(x,g)$ be fixed as a functional prescription. For a metric variation $h_{\mu\nu}=\delta g_{\mu\nu}$ in a fixed chart, metric compatibility gives

$$\delta\Gamma^\mu =\nabla_\rho h^{\mu\rho}-\frac12\nabla^\mu h -h^{\rho\sigma}\Gamma^\mu{}_{\rho\sigma},$$

where $h=g^{\rho\sigma}h_{\rho\sigma}$ and the raised $h^{\rho\sigma}$ is the index-raised perturbation, so $\delta g^{\rho\sigma}=-h^{\rho\sigma}$. The last term comes from varying the inverse metric in the contraction. For $h_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$, the exact linear variation is

$$\delta_\xi C^\mu =\delta_\xi H^\mu+h^{\rho\sigma}\Gamma^\mu{}_{\rho\sigma} -\Box_g^{\mathrm{vec}}\xi^\mu-R^\mu{}_\nu\xi^\nu.$$

If $H^\mu$ depends on the metric, $\delta_\xi H^\mu$ must be evaluated by the chain rule even though the functional prescription is fixed. Omitting either this term or the inverse-metric term generally gives the wrong residual equation. The superscript $\mathrm{vec}$ means that both derivatives act covariantly on the vector index.

For ordinary harmonic coordinates, an easier exact argument keeps $g$ fixed and changes the coordinate functions. If $\Box_g x^\mu=0$, then $x'{}^\mu=x^\mu+\varepsilon\xi^\mu(x)$ also solves the coordinate condition precisely when

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

for each of the four functions, with an invertible coordinate map. This operator treats the label $\mu$ as a label, not as a covariant vector index. In a harmonic chart it equals $g^{\rho\sigma}\partial_\rho\partial_\sigma\xi^\mu$. The coordinate functions and their residual transformations are described in Carroll 1997, §4, pp. 123–124, PDF.

By contrast, the background-covariant perturbation condition $F_\mu(h)=\nabla^\nu h_{\mu\nu}-\tfrac12\nabla_\mu h=0$ has the familiar de Donder residual equation $\Box_g^{\mathrm{vec}}\xi_\mu+R_\mu{}^\nu\xi_\nu=0$. It omits the contraction variation because it defines $F_\mu$ directly with a fixed background connection. Both residual operators are wave operators, but their lower-order terms are different. On a globally hyperbolic region without a timelike boundary, the relevant residual equation is determined by

$$\xi^\mu\big|_\Sigma=f^\mu, \qquad n^\nu\nabla_\nu\xi^\mu\big|_\Sigma=p^\mu.$$

For the scalar coordinate-function description the second datum is $n^\nu\partial_\nu\xi^\mu$ instead. This distinction does not change the number of data required.

A variable-coefficient example makes the distinction visible. For $ds^2=e^{2f(x)}(-dt^2+dx^2)+dy^2+dz^2$, all four coordinates are harmonic. The map $t'=t+\varepsilon x$ remains harmonic for every smooth $f$ and has unit Jacobian determinant. Its generator $\xi=x\partial_t$ satisfies the scalar residual equation, but

$$\left(\Box_g^{\mathrm{vec}}\xi+\operatorname{Ric}(\xi)\right)^t =2f'(x)e^{-2f(x)}.$$

This is generally nonzero. For $h=\mathcal L_\xi g$, the term $h^{\rho\sigma}\Gamma^t{}_{\rho\sigma}$ has exactly the same value, so the full $\delta\Gamma^t$ does vanish. A Minkowski calculation in Cartesian coordinates would miss this distinction because all the connection terms vanish there.

Setting both $f^\mu$ and $p^\mu$ to zero removes residual transformations connected to the identity in the domain of dependence. If either datum is nonzero, a nontrivial residual coordinate change generally remains.

For perturbations about Minkowski spacetime,

$$\Box_\eta\xi^\mu=0.$$

Plane-wave generators

$$\xi^\mu=\Re\!\left(A^\mu e^{ik_\rho x^\rho}\right), \qquad k_\rho k^\rho=0,$$

preserve the linearized harmonic condition. They can rearrange components of a metric perturbation without changing its linearized curvature. Harmonic gauge therefore does not by itself isolate the two physical gravitational-wave polarizations.

Carroll displays this residual wave equation in the weak-field theory in Carroll 1997, §6, Eqs. (6.36)–(6.44), pp. 149–150, PDF.

Initial data fix an identification

Gauge data on $\Sigma$ have a geometric meaning. The tangential part of $f^\mu$ changes spatial coordinates on the initial slice. Its normal part changes the first-order placement of the slice in the spacetime description. The derivative datum $p^\mu$ changes how that identification is continued away from $\Sigma$ and therefore affects lapse and shift.

Suppose two harmonic developments use the same geometric Cauchy data but different coordinate values or normal derivatives for the coordinate functions $x^\mu$. They can be isometric developments with different metric components. A uniqueness theorem for the reduced wave equations refers to the metric components after these gauge data have been fixed. Geometric uniqueness instead identifies solutions related by an admissible diffeomorphism.

Conversely, prescribing coordinate data cannot repair constraint-violating geometric initial data. Gauge, Hamiltonian, and momentum constraints are distinct. The coordinate data select a representative only after the geometric data lie on the Einstein constraint surface.

A timelike boundary adds incoming data

Let a finite domain have timelike boundary $\mathcal T$. A wave equation for $\xi^\mu$ has incoming characteristic fields at $\mathcal T$, so initial values alone no longer determine a residual transformation. Boundary conditions such as

$$\xi^\mu\big|_{\mathcal T}=0$$

remove maps that move boundary points in the chosen description. Other conditions can preserve the hypersurface while allowing tangential reparameterizations, or fix only a subset of gauge modes.

Initial and boundary gauge data must agree at the corner $\mathcal S=\Sigma\cap\mathcal T$. For Dirichlet data, the zeroth-order condition is simply

$$f^\mu\big|_{\mathcal S} =\xi^\mu\big|_{\mathcal T\cap\Sigma}.$$

Higher regularity requires compatibility of time derivatives obtained by differentiating the boundary condition and using the residual wave equation. Incompatible data can generate a corner singularity or place the solution outside the claimed differentiability class.

The boundary also changes the meaning of “gauge.” A generator that vanishes appropriately on $\mathcal T$ normally preserves fixed boundary data and is a redundancy. A tangential generator that preserves the boundary conditions but acts nontrivially on boundary fields may be a boundary symmetry. A generator with a normal component can move the boundary and hence define a different domain unless boundary embedding is dynamical.

Kreiss, Reula, Sarbach, and Winicour show how gauge and constraint boundary conditions participate in the harmonic Einstein energy estimate in Kreiss et al. 2007, §§2–3, pp. 3–9, PDF.

Asymptotic data and large transformations

On a noncompact domain, falloff conditions replace a finite outer boundary. One must decide how fast $\xi^\mu$ and its derivatives decay and which nondecaying transformations preserve the asymptotic structure.

Behavior of $\xi^\mu$	Effect on the formulated problem
Compact support or sufficiently rapid decay	Interior gauge redundancy
Approach to an allowed asymptotic translation or rotation	Possible asymptotic symmetry with a charge
Change of the prescribed asymptotic frame or parity class	Different admissible data or a different problem
Growth that violates metric falloff	Not an allowed transformation

The differential gauge equation does not decide among these cases. It admits local homogeneous solutions; the phase space and its boundary conditions decide which solutions are quotient directions.

Large diffeomorphisms also need not be generated globally by a single vector field connected to the identity. Eliminating the infinitesimal kernel does not prove that a gauge slice intersects every orbit once. Global gauge copies, topology, fixed points, and chart breakdown remain separate questions.

A bounded one-dimensional model

The residual structure is already visible for a scalar wave on $0\le x\le L$:

$$(-\partial_t^2+\partial_x^2)\xi=0.$$

Initial conditions alone,

$$\xi(0,x)=f(x), \qquad \partial_t\xi(0,x)=p(x),$$

do not determine a solution on the bounded interval without boundary conditions. With homogeneous Dirichlet data,

$$\xi(t,0)=\xi(t,L)=0,$$

the residual solutions are standing waves

$$\xi_n(t,x) =\left(a_n\cos\frac{n\pi t}{L} +b_n\sin\frac{n\pi t}{L}\right) \sin\frac{n\pi x}{L}.$$

Zero initial data set every $a_n$ and $b_n$ to zero. Nonzero initial data retain residual transformations even though the boundary is fixed. If a nonzero boundary value is allowed, additional transformations enter from the boundary. The vector harmonic-gauge problem has the same logical structure, coupled to curvature and the gravitational boundary system.

Common pitfalls

A differential condition is not a unique solution. Its homogeneous equation, initial data, boundary data, and kernel must be controlled.

Residual gauge is not necessarily redundancy. A residual solution acting at infinity or a finite boundary can generate a physical symmetry and a surface charge.

Local removal does not prove global gauge fixing. Even if zero initial and boundary data eliminate infinitesimal residual waves, disconnected large maps or multiple orbit intersections may remain.

Exercise: removing linear residual gauge

On a globally hyperbolic Minkowski domain without timelike boundary, suppose $\xi^\mu$ obeys

$$\Box_\eta\xi^\mu=0, \qquad \xi^\mu\big|_{t=0}=0, \qquad \partial_t\xi^\mu\big|_{t=0}=0.$$

Use the wave energy to show that $\xi^\mu=0$ in the domain of dependence. Why does the same conclusion require boundary information on a finite spatial domain?

Solution

For each component, define

$$E(t)=\frac12\int_{\Sigma_t} \left[ (\partial_t\xi^\mu)^2+\lvert\nabla\xi^\mu\rvert^2 \right]d^3x.$$

With sufficient decay, integration by parts and the wave equation give $dE/dt=0$. The initial data imply $E(0)=0$, so all first derivatives vanish. The zero initial value then fixes the remaining constant to zero.

On a finite domain, the integration by parts produces a boundary flux. Incoming boundary values can make $E(t)$ nonzero even when the initial energy vanishes. A homogeneous maximally dissipative or Dirichlet boundary condition, plus corner compatibility, is needed to exclude that source.

References

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