Gauge Conditions, Residual Freedom, and Constraints

A gauge condition is a calculational choice imposed on redundant metric components; it is not an extra physical law. A useful condition must simplify the chosen black-hole perturbation problem, admit the required initial and boundary data, remain regular on the domain, and leave the two radiative gravitational degrees of freedom intact. This page compares the main choices by those criteria and shows how to find their residual freedom.

Helpful background. Black-Hole Geometry and Mechanics fixes the background and horizon data used in applications; PDEs and Green Functions for Gravitation supplies the differential-operator viewpoint; Formulations of General Relativity separates covariant equations, constraints, and evolution; Presymplectic Potentials and Currents prepares the variational language used later in this chapter.

Lorenz gauge as a hyperbolic reduction

Define the trace reverse

$$\widehat h_{\mu\nu} =h_{\mu\nu}-\frac12\bar g_{\mu\nu}h, \qquad F_\nu[h]=\bar\nabla^\mu\widehat h_{\mu\nu}.$$

Under $h_{\mu\nu}\mapsto h_{\mu\nu}+2\bar\nabla_{(\mu}\xi_{\nu)}$,

$$F_\nu\mapsto F_\nu +\left(\bar\Box\,\delta_\nu{}^\mu +\bar R_\nu{}^\mu\right)\xi_\mu.$$

Lorenz gauge is $F_\nu=0$. Reaching it is therefore a vector wave problem for $\xi_\mu$, with initial and boundary conditions that must be compatible with the physical problem. On a Ricci-flat background the vacuum field equation then has the normally hyperbolic form

$$\bar\Box\widehat h_{\mu\nu} +2\bar R_{\mu\alpha\nu\beta}\widehat h^{\alpha\beta}=0.$$

This exposes causal propagation and works well with distributed sources. It does not fix the gauge completely: every solution of $(\bar\Box\delta_\nu{}^\mu+\bar R_\nu{}^\mu)\xi_\mu=0$ that obeys the admitted boundary conditions is residual freedom. Nor does the local gauge condition guarantee a globally regular vector field. The connection variation behind this gauge source is explicit in Aksteiner and Andersson 2011, arXiv v2, pp. 16–17, § 4.2, PDF.

A residual vector that fails at the horizon

A simple Schwarzschild example shows why the boundary restriction is real. Write $f=1-2GM/r$ and consider the scalar-generated sector $\xi_\mu=\bar\nabla_\mu\chi$. On a Ricci-flat background, $\bar\Box\bar\nabla_\mu\chi=\bar\nabla_\mu\bar\Box\chi$, so any harmonic scalar produces a local residual Lorenz vector. For a static spherical scalar,

$$\bar\Box\chi=\frac1{r^2}\frac{d}{dr} \left(r^2 f\frac{d\chi}{dr}\right)=0, \qquad \chi=A\log f+\text{constant}.$$

This solves the equation on the open exterior $r\gt2GM$. In ingoing coordinates $v=t+r_*$, however,

$$\xi^r=\frac{2GM A}{r^2}, \qquad \xi^v=\frac{2GM A}{r^2f}, \qquad \bar g_{\mu\nu}\xi^\mu\xi^\nu =\frac{4G^2M^2A^2}{r^4f}.$$

For $A\ne0$ even the scalar norm diverges at the future horizon. This is not merely a Schwarzschild-coordinate artifact: the solution is excluded if residual generators must extend smoothly across that horizon.

The contrasting harmonic scalar $\chi=B[t+2GM\log f]$, with a separate constant $B$, gives

$$\xi^v=-B\left(1+\frac{2GM}{r}\right), \qquad \xi^r=\frac{4BG^2M^2}{r^2}.$$

It is regular at the future horizon, but tends to $-B\partial_t$ at infinity. Thus it is still excluded by a boundary-vanishing gauge class; a class allowing asymptotic time translations treats it differently. These examples solve one residual sector, not the global existence problem for every Lorenz-gauge transformation.

Symmetry-adapted metric gauges

On a spherically symmetric background, tensor harmonics split the perturbation into parity and multipole sectors. Regge–Wheeler gauge sets selected harmonic amplitudes to zero and makes the remaining variables economical for $\ell\ge2$. It is a statement about harmonic coefficients, not a covariant condition on arbitrary backgrounds. The monopole and dipole sectors require separate treatment because mass, angular-momentum, center-of-mass, and gauge perturbations become exceptional. Distributional sources can also make a nominally simple gauge vector or metric coefficient discontinuous.

The explicit parity amplitudes and their invariant combinations are developed in The Regge–Wheeler Equation and The Zerilli Equation.

Light-cone gauge instead preserves selected geometric properties of a null coordinate system. In the Schwarzschild ingoing metric

$$ds^2=-f\,dv^2+2\,dv\,dr+r^2\Omega_{AB}\,d\theta^A d\theta^B,$$

choose the ingoing null generator $\ell=-\partial_r$. The condition $h_{\mu\nu}\ell^\nu=0$ is

$$h_{vr}=h_{rr}=h_{rA}=0.$$

It preserves the null character of $v=\text{constant}$, the angular labels of its generators, and the affine radial parameter, to first order. It does not by itself preserve the angular area element. In this gauge

$$h=r^{-2}\Omega^{AB}h_{AB},\qquad \delta\sqrt{\det g_{AB}}=\frac12r^2\sin\theta\,h.$$

An additional trace-free choice $h=0$ preserves this local area element. In the construction of Preston and Poisson 2006, arXiv v1, pp. 2–5, §§ II.B and II.E, Eqs. (2.7)–(2.9) and (2.36)–(2.38), PDF, the field equations permit that refinement when the source has $T_{\mu\nu}\ell^\mu\ell^\nu=0$, including vacuum, subject to admissible residual transformations. It is not an automatic consequence of light-cone gauge with arbitrary matter or boundary conditions. Preserving the local area element is also stronger than preserving only a sphere’s integrated area.

A horizon-adapted gauge is a broader requirement: its components and gauge generator are regular in a chart that crosses the future horizon, such as an ingoing Eddington–Finkelstein chart. Schwarzschild coordinates can make a perfectly regular tensor look singular at the horizon, so component finiteness must be judged in a regular basis. Horizon adaptation can coexist with Lorenz, light-cone, or another gauge, but not every desired set of algebraic conditions is mutually compatible.

Radiation gauges and their completion

In a null-tetrad treatment one often imposes an ingoing or outgoing radiation gauge, schematically

$$\ell^\mu h_{\mu\nu}=0, \qquad h=0,$$

or the analogous condition with the other principal null leg. These gauges are powerful for reconstructing a vacuum metric from curvature variables on type-D backgrounds. Their names refer to the selected tetrad leg and reconstruction problem; they are not equivalent to the transverse-traceless gauge of a plane wave in flat spacetime.

Radiation gauge can fail globally in the presence of a point-particle source, where string-like or discontinuous representatives may arise. Curvature reconstruction also misses perturbations within the background family, such as mass and angular-momentum shifts, and may require separate gauge and completion pieces. A condition that cleanly isolates radiative modes can therefore discard physical parameter variations if it is treated as a complete solution rather than one sector of it.

Einstein constraints and gauge constraints

Choose a spacelike hypersurface with future unit normal $n^\mu$. The four linearized Einstein constraints are projections of the equation tensor,

$$\mathcal C_\nu=n^\mu E^{(1)}_{\mu\nu}.$$

They restrict initial data. They are distinct from the four gauge quantities $F_\nu$, which select a representative. When the evolution equations and linearized matter equations hold, the linearized Bianchi identity supplies a homogeneous subsidiary system for $\mathcal C_\nu$; suitable zero initial and incoming boundary data then propagate the constraints. A gauge-fixed formulation likewise needs a subsidiary equation for $F_\nu$. Numerical or analytic evolution of the reduced wave equations without these conditions can solve the reduction while failing the full linearized Einstein equation.

For example, on a Ricci-flat vacuum background the full ungauged operator has the decomposition

$$\begin{aligned} E^{(1)}_{\mu\nu} &=E^{\mathrm{red}}_{\mu\nu} +\bar\nabla_{(\mu}F_{\nu)} -\frac12\bar g_{\mu\nu}\bar\nabla^\rho F_\rho,\\ E^{\mathrm{red}}_{\mu\nu} &=-\frac12\left( \bar\Box\widehat h_{\mu\nu} +2\bar R_{\mu\alpha\nu\beta}\widehat h^{\alpha\beta}\right). \end{aligned}$$

The divergence of the gauge correction is $(\bar\Box F_\nu+\bar R_\nu{}^\rho F_\rho)/2$; the Ricci term vanishes in this example. Consequently the Bianchi identity and $E^{\mathrm{red}}=0$ imply $\bar\Box F_\nu=0$. If both $F_\nu$ and its normal derivative vanish initially, and the boundary conditions give uniqueness for this subsidiary wave problem, then $F_\nu=0$ throughout the domain. Setting only $F_\nu=0$ on the initial surface does not supply both wave-equation data. The initial Einstein constraints and compatible reduced evolution provide the missing compatibility conditions in a properly posed initial-value formulation.

Counting makes the distinction concrete. Ten symmetric metric components minus four coordinate freedoms and four constraints leave two local propagating polarizations. This is not a pointwise subtraction proof—boundary charges, exceptional harmonics, and zero modes require separate analysis—but it explains why four gauge conditions do not remove four physical modes.

Choosing a gauge for a black-hole problem

Start from the observable and domain, not from a familiar name. For causal evolution with smooth extended sources, a hyperbolic Lorenz-type gauge is a natural baseline. For separated Schwarzschild vacuum equations, Regge–Wheeler gauge is economical once low multipoles are restored. For horizon-crossing geometry, demand regular ingoing components. For curvature reconstruction on Kerr, a radiation gauge may be optimal, provided its source singularities and completion sector are handled explicitly. In every case, solve the residual equation with the same boundary conditions and test that a claimed observable is unchanged.

Exact background construction remains in Exact Solutions of Einstein’s Equations, including the horizon-penetrating atlas used to judge regularity. Full finite-amplitude evolution remains in Black-Hole Dynamics. Gauge regularity and linearized constraint propagation do not by themselves establish nonlinear stability or select a unique global coordinate system.

Exercise: residual Lorenz freedom

Assume a Ricci-flat background. A perturbation already satisfies Lorenz gauge. What equation must $\xi_\mu$ obey so that the transformed perturbation remains in Lorenz gauge, and what additional data are needed to decide whether this is a genuine residual transformation?

Solution

Because $\bar R_{\mu\nu}=0$, the transformation law gives $F_\nu[h']-F_\nu[h]=\bar\Box\xi_\nu$. Thus residual vectors obey $\bar\Box\xi_\nu=0$. The differential equation alone is insufficient. One must specify initial data, regularity at the future horizon, falloff or radiative behavior at infinity, and any restrictions at other boundaries. A solution excluded by those conditions is not residual freedom in the chosen problem; a solution acting nontrivially on an allowed boundary may instead be an asymptotic symmetry with a charge. Lorenz gauge therefore fixes a hyperbolic representative only relative to a domain and boundary class.

References

• Aksteiner, Steffen, and Lars Andersson. “Linearized Gravity and Gauge Conditions.” Classical and Quantum Gravity 28 (2011): 065001. DOI. Open PDF, arXiv:1009.5647v2.
• Preston, Brent, and Eric Poisson. “Light-Cone Gauge for Black-Hole Perturbation Theory.” Physical Review D 74 (2006): 064010. DOI. Open PDF, arXiv:gr-qc/0606094v1.