A theoretical study shows that gravitational observations can leave material properties undetermined in two distinct ways: different states can share the same total source, and different spacetime histories can yield the same retained observations. The authors propose a property-level identifiability criterion to organise these inverse problems in general relativity, with applications to stars and black hole candidates.

The first type of ambiguity, which the authors call equal-source ambiguity, is illustrated with two ideal massless gases whose temperatures differ by a factor of (8/29)^(1/4) and whose entropies differ, even though their stress-energy tensors agree. The second type appears in a causal-fluid example starting from identical sources and the same geometric data but different constitutive responses: the curvature histories of the two models separate already at linear order in proper time, and their scale factors only at cubic order.

For black hole candidates, the framework distinguishes between stationary exterior parameters, perturbing fluxes and constituent history. It thus offers a test for whether the retained observational data identify a given material property, separating three notions that are often conflated: instantaneous source equality, history-wide source equality and observational projection.

The consequence is that the observed geometry does not by itself guarantee an inference of what the source is made of. The criterion helps delimit which properties are genuinely accessible from observations and which remain hidden behind degeneracy, a point of relevance for the interpretation of compact stars and black hole candidates.