Gauge Pathologies and Diagnostics

A coordinate description has failed when its chart, foliation, or gauge-fixed evolution ceases to represent a smooth spacetime solution reliably. The spacetime itself has failed only under a stronger geometric diagnosis, such as curvature blow-up, inextendibility, or incomplete causal curves under stated hypotheses. Separating the two requires several mutually independent checks: chart Jacobians, lapse and slice geometry, characteristic structure, constraint residuals, frame curvature, and continuation in a different gauge.

Required background. The Einstein Equation supplies the physical field equations, while Killing Fields and Isometry Groups supplies the distinction between an isometry, a gauge orbit, and coordinate-dependent components.

Helpful background. Hypersurface Normals and Tangential Projectors supplies foliation diagnostics, and PDEs and Green Functions for Gravitation supplies principal symbols, well-posedness, and continuation criteria.

Five different ways a description can fail

The phrase “the coordinates become singular” hides distinct mechanisms:

1. Chart failure. The map from coordinates to the manifold ceases to be one-to-one or its Jacobian loses rank. Components can diverge solely because the coordinate basis degenerates.
2. Foliation failure. The time-function gradient ceases to be timelike, the lapse reaches an inadmissible value, or normal generators form a caustic. The spacetime may still possess another smooth foliation.
3. Gauge steepening. A nonlinear lapse or shift equation develops discontinuous derivatives or a coordinate shock while curvature remains regular.
4. Formulation failure. The chosen reduced evolution loses strong hyperbolicity or its characteristic basis becomes ill conditioned. This is a failure of the PDE representation, not automatically of Einstein geometry.
5. Physical or geometric failure. Curvature measured in regular frames diverges, causal geodesics end without an admissible extension, or the solution cannot be continued in the specified regularity class.

A sixth possibility is neither gauge nor physical failure: an approximate solution can leave the Einstein constraint surface because of inconsistent data, boundary injection, or numerical error. Constraint growth diagnoses failure to solve the target equations, not a spacetime singularity.

Coordinate shocks and slice stretching

Consider a quasilinear scalar gauge mode written schematically as

$$\partial_t u+v(u)\partial_xu=0.$$

Along characteristics, $u$ is constant, but their speed depends on that constant. Differentiating gives

$$\frac{d}{dt}(\partial_xu) =-v'(u)(\partial_xu)^2$$

along a characteristic. If $v'(u)\partial_xu<0$ initially, the derivative can blow up in finite coordinate time even though $u$ remains bounded. Nonlinear slicing conditions can possess an analogous gauge characteristic steepening. The pathology lies in the coordinate field or its derivatives, not necessarily in $g_{\mu\nu}$ as a geometric tensor.

Slice stretching is different. If the lapse collapses in one region while the shift does not compensate, neighboring coordinate lines sample very different amounts of normal evolution. Spatial metric components and their coordinate gradients can grow. Proper distances on the slice may grow as well, but the extreme component behavior depends on how grid labels are transported. A new shift, slicing, or spatial chart can often redistribute it.

Gourgoulhon discusses foliation singularities, singularity avoidance, and spatial-coordinate choices in Gourgoulhon 2007, §§9.2–9.4, pp. 152–174, PDF.

Caustics and lapse collapse

Geodesic slicing follows a congruence of timelike normals. Let $J^i{}_A$ map initial labels $y^A$ to separation vectors on a later slice. A caustic occurs when

$$\det J=0.$$

Distinct coordinate lines then reach the same event and the label map is no longer invertible. The Raychaudhuri equation explains how negative expansion can drive this rank loss, but a caustic need not carry divergent curvature. In flat spacetime, inward geodesics normal to a curved initial surface can focus.

Lapse collapse is also ambiguous. In standard Schwarzschild coordinates,

$$\alpha=\sqrt{1-\frac{2GM}{r}}$$

vanishes at $r=2GM$. The Kretschmann scalar is

$$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =\frac{48G^2M^2}{r^6},$$

which is finite there. Horizon-penetrating coordinates continue the spacetime. The lapse zero diagnoses that particular time coordinate, not a curvature singularity.

At $r=0$ the same invariant diverges, so the horizon example cannot be used to claim that every lapse collapse is harmless. One must check a regular frame, invariants appropriate to the spacetime class, and extendibility.

Hyperbolicity and constraint diagnostics

For a first-order system

$$\partial_tU=A^i(U,x)\partial_iU+F(U,x),$$

freeze coefficients and form the principal symbol

$$P(s)=A^is_i$$

for every real unit covector $s_i$. A reliable continuum diagnosis asks:

• Are all characteristic speeds real?
• Is there a complete eigenbasis with a condition number bounded uniformly in $s_i$?
• Does a positive symmetrizer remain bounded?
• Do lapse, shift, or gauge parameters approach a value where these properties fail?

Real eigenvalues without a uniformly complete eigenbasis are insufficient. Near-degenerate characteristic variables can amplify high-frequency data even when metric components and curvature initially look regular. A change of formulation or gauge parameter can restore a well-posed representation of the same Einstein solution.

Constraint residuals provide a different check. For the future normal, use $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$, $\rho=T_{\mu\nu}n^\mu n^\nu$, and $j_i=-\gamma_i{}^\mu T_{\mu\nu}n^\nu$:

$$\mathcal H =\,{}^{(3)}R+K^2-K_{ij}K^{ij}-16\pi G\rho-2\Lambda,$$ $$\mathcal M_i =D_jK^j{}_i-D_iK-8\pi G j_i.$$

If these become nonzero, the evolved fields no longer describe an Einstein solution with the stated matter. Yet small constraints do not establish that a large component is gauge, and zero constraints do not guarantee a regular coordinate chart or strong hyperbolicity.

Invariant and frame checks

Scalar polynomial curvature invariants are valuable but not universal. A diverging Kretschmann scalar in a regular orthonormal frame is strong evidence of curvature blow-up. A finite set of scalars can nevertheless vanish in nonflat radiative spacetimes or fail to distinguish inequivalent Lorentzian geometries.

A broader diagnostic set includes:

• Riemann components in a parallel-propagated orthonormal or null frame;
• eigenvalues and Cartan invariants of curvature and its derivatives;
• tidal distortion along causal geodesics;
• affine or proper-time completeness;
• existence of a smooth extension in a different chart;
• behavior of matter scalars and the stress tensor in regular frames.

The correct choice follows the claim. “The chart fails at a horizon” needs a regular extension and finite geometric fields. “The spacetime is singular” needs a definition such as inextendibility or geodesic incompleteness and the corresponding evidence. One scalar cannot certify every form of regularity.

A diagnostic decision table

Symptom	Immediate test	Discriminating follow-up
Metric component diverges	Coordinate Jacobian and determinant	Reexpress in an overlapping regular chart
Lapse approaches zero	Is $d t$ still timelike and is curvature finite?	Try a horizon-penetrating or alternate slicing
Spatial gradients steepen	Gauge characteristic crossing	Compare frame curvature and constraint residuals
Null or normal rays intersect	Jacobi-map determinant or expansion	Patch the congruence; test spacetime smoothness
Evolution becomes unstable	Principal-symbol eigenbasis and symmetrizer	Change formulation/gauge; perform an energy estimate
Constraints grow	$\mathcal H$, $\mathcal M_i$, and subsidiary fields	Locate initial, bulk, or boundary injection
Curvature frame components diverge	Parallel-propagated frame and invariants	Test causal completeness and extendibility

No row by itself proves the final diagnosis. The strength comes from agreement between independent geometric and formulation checks.

Two flat-spacetime counterexamples

Rindler coordinates on the right wedge are

$$T=\rho\sinh(a\eta), \qquad X=\rho\cosh(a\eta), \qquad \rho>0.$$

They give

$$ds^2=-a^2\rho^2d\eta^2+d\rho^2+dy^2+dz^2.$$

The lapse $a\rho$ vanishes at $\rho=0$, the chart Jacobian degenerates there, and all curvature components vanish. Minkowski coordinates extend smoothly across the Rindler horizon.

For a caustic, take ingoing radial null coordinates in flat spacetime. The area of round wavefronts is $4\pi r^2$ and the expansion is proportional to $-2/r$. At $r=0$ the generators meet and the angular Jacobi map loses rank. The vertex needs another local description, but Minkowski curvature remains zero. These examples independently show that lapse collapse and congruence focusing can both be pure coordinate failures.

Common pitfalls

Finite curvature scalars do not prove regularity. They may miss parallel-propagated blow-up or fail to characterize a degenerate Lorentzian geometry.

Constraint violation is not a physical singularity. It means the fields have left the Einstein solution set. The source may be continuum, boundary-related, or numerical.

A superluminal gauge speed is not automatically acausal. Gauge characteristics move coordinate information. Physical causality must be tested with gauge-invariant or constraint-compatible characteristic fields.

Exercise: diagnose the Rindler horizon

For the Rindler metric above:

1. compute the determinant and identify where the chart ceases to be nondegenerate;
2. show by the transformation to $(T,X)$ that the spacetime is smooth there; and
3. explain which observations distinguish this event from $r=0$ in Schwarzschild spacetime.

Solution

The determinant is

$$g=-a^2\rho^2,$$

so the coordinate metric becomes degenerate at $\rho=0$. The transformation maps the wedge to $X>\lvert T\rvert$; its inverse fails at the wedge boundary, but $(T,X,y,z)$ are regular Minkowski coordinates there. All Riemann components and scalar curvature invariants vanish.

For Schwarzschild $r=0$, the invariant $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=48G^2M^2/r^6$ diverges and no Schwarzschild-type coordinate relabeling removes that curvature blow-up. Thus the Jacobian and extension diagnose Rindler chart failure, while invariant and frame curvature distinguish the physical Schwarzschild singularity.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.