Gauge Obstructions in Geometric Inverse Problems

An inverse problem cannot reconstruct distinctions that the measurement operator does not see. For geometric wave data, the basic invisible operation is a diffeomorphism that fixes the source and observation region. Null data may add conformal ambiguity, while coupled fields may admit their own internal gauge and source relabelings. A correct uniqueness theorem therefore identifies an equivalence class before it asks how to recover a representative.

Required background. Analysis for Mathematical Relativity supplies regularity and stability norms; PDEs and Green Functions for Gravitation supplies hyperbolic response operators; and Lorentzian metrics and signature supplies causal and conformal geometry.

Helpful background. The Einstein equation supplies diffeomorphism covariance, and Causality and Global Structure supplies chronological diamonds and accessibility.

The unavoidable diffeomorphism quotient

Let $(M,g)$ contain an observation region $V$, and consider a scalar source-to-solution map

$$L_{g,H}:f\longmapsto u\vert_V, \qquad \Box_g u+H(x,u)=f, \qquad \operatorname{supp}f\subset V.$$

Take the solution with vanishing past data, assume $H(x,0)=0$, and restrict to a small source class for which this retarded problem is well posed. These conditions select one response for each source; an equation without initial or retarded conditions would not define a single-valued measurement map.

If $F:M\to M$ is a diffeomorphism with $F\vert_V=\operatorname{id}$, define the pulled-back scalar nonlinearity by $(F^*H)(x,z)=H(F(x),z)$. Naturality of the wave operator then gives

$$\Box_{F^*g}(F^*u)+(F^*H)(x,F^*u) =F^*\!\left(\Box_gu+H(x,u)\right).$$

Sources and observations in $V$ are unchanged, hence $L_{F^*g,F^*H}=L_{g,H}$. If both coefficients are unknown, the coordinate-free target is therefore the orbit

$$[(g,H)]_V =\{(F^*g,F^*H):F\vert_V=\operatorname{id}\}.$$

Its metric component has the familiar ambiguity $[g]_V=\{F^*g:F\vert_V=\operatorname{id}\}$. If $H$ is instead fixed and known a priori, only diffeomorphisms satisfying $F^*H=H$ remain admissible. Thus covariance must be checked for the entire unknown coefficient tuple, not for the metric alone.

For boundary data, $F$ is required to fix the observed boundary portion. If the apparatus includes labeled emitters or tensor components, the pullback must act on those structures too. Fixing harmonic coordinates can select a representative for a proof or algorithm, but it does not turn that representative into an observable.

This is an independent check on every claimed reconstruction: pull back the entire forward problem and verify that its data are identical. A theorem claiming literal pointwise recovery in unspecified coordinates has failed this test.

Conformal ambiguity of light data

For a positive smooth function $c$, the metrics $g$ and $c^2g$ have the same unparametrized null geodesics and causal cones. Data recording only which light rays connect sources and receivers can therefore determine at most the conformal class

$$[g]_{\mathrm{conf}}=\{c^2g:c>0\},$$

again modulo diffeomorphism. Arrival parameters, amplitudes, a volume form, field equations, or timelike travel times can sometimes determine $c$, but that conclusion needs a separate argument.

Kurylev, Lassas, and Uhlmann make this boundary explicit. Given earliest light observation sets from source points in the theorem’s relatively compact region of a globally hyperbolic spacetime, they reconstruct its topological, differentiable, and conformal structure Kurylev, Lassas, and Uhlmann 2018, Theorem 1.2 and §§2, 5. For Ricci-flat metrics, Corollary 1.3 removes the conformal ambiguity when the apparatus is identified isometrically and every topological component of the source region meets it. Their active semilinear result instead obtains the conformal class in the causally accessible diamond Kurylev, Lassas, and Uhlmann 2018, Theorem 1.5 and §§3–5, not an arbitrary metric representative.

By contrast, the Lorentzian time-separation function uses proper time along maximizing causal curves. It changes under a nonconstant conformal rescaling. Under smoothness and timelike-convexity hypotheses, its restriction can determine the full metric jet in semigeodesic coordinates Lassas, Oksanen, and Yang 2016, Theorem 1 and §4. The coordinates are tied to the measurement hypersurface, which is how the gauge is fixed.

Field and source gauges

Einstein data bring further layers. A wave-coordinate reduction has gauge source functions; Maxwell and Yang–Mills fields have internal gauge; scalar fields can be relabeled if the coupling and measurement channels transform accordingly. The observation operator must specify which source components can be controlled and which solution components are read out.

In the Einstein–scalar active problem, sources cannot be assigned arbitrarily without violating stress-energy conservation. The theorem starts from a smooth globally hyperbolic background solution, assumes that background scalar gradients span the cotangent space throughout the source neighborhood, and uses adaptive sources compatible with the conservation law. Its data are small source–solution triples expressed in source-dependent Fermi coordinates. Under these hypotheses, Kurylev, Lassas, Oksanen, and Uhlmann recover the topological, differentiable, and conformal structure in the optimal chronological diamond. Their Ricci-flat full-metric conclusion is a separate corollary: every point of the target region must be connectible to the observation neighborhood by a path remaining in that region Kurylev, Lassas, Oksanen, and Uhlmann 2022, Theorem 1.1, Corollary 1.2, and §§1.2–1.3, 5.

The limitation is exact: this is not recovery outside the causally reachable and returnable region, not stability under noisy finite data, and not a claim that unconstrained physical matter sources can implement every microlocal probe used in the proof.

A hierarchy of conclusions

A precise inverse theorem should state, in order:

1. the manifold region and causal accessibility covered by the data;
2. the transformations leaving the observation operator invariant;
3. whether topology, smooth structure, conformal class, metric, or matter coefficient is determined;
4. whether determination is qualitative, constructive, or quantitatively stable in a named norm.

Uniqueness modulo gauge is not weaker drafting; it is the maximum physically meaningful conclusion. Stability and computation are additional theorems, not synonyms for uniqueness.

Check the invisible transformation

Let $F$ equal the identity on $V$ but move a point $q$ in the unknown region. An algorithm applied to $L_{g,H}$ reports coordinates for $q$. What must happen when the same algorithm is applied to $L_{F^*g,F^*H}$?

Solution

The two inputs are identical, so a deterministic algorithm must return the same output data. It cannot know whether the interior point represented by that output is $q$ in $(M,g)$ or $F^{-1}(q)$ in $(M,F^*g)$. Only the isometry class with the observation region marked is invariant. A chosen coordinate gauge may label a representative consistently, but the label is part of that choice rather than recovered physical information.

References

• Kurylev, Yaroslav, Matti Lassas, and Gunther Uhlmann. “Inverse Problems for Lorentzian Manifolds and Non-linear Hyperbolic Equations.” Inventiones Mathematicae 212 (2018): 781–857. DOI.
• Kurylev, Yaroslav, Matti Lassas, Lauri Oksanen, and Gunther Uhlmann. “Inverse Problem for Einstein–Scalar Field Equations.” Duke Mathematical Journal 171 (2022): 3215–3282. DOI.
• Lassas, Matti, Lauri Oksanen, and Yang Yang. “Determination of the Spacetime from Local Time Measurements.” Mathematische Annalen 365 (2016): 271–307. arXiv.