In their 1973 foundational paper 'Foundations for a Theory of Gravitation Theories', Thorne, Lee, and Lightman stated Schiff's conjecture — that any complete and self-consistent theory of gravity that obeys the weak equivalence principle (universality of free fall) must unavoidably also obey the Einstein equivalence principle (local Lorentz and position invariance of nongravitational physics) — and supported it only by a plausibility argument restricted to relativistic, Lagrangian-based theories, deferring a formal proof for a more limited class to a subsequent paper. That proof (Lightman and Lee) covers only electromagnetically interacting test bodies in static, spherically symmetric gravitational fields (the THεµ class), and later work establishes the implication only under further restrictions, strengthened premises, or added assumptions (Ni's rotation-state universality; Coley's WEP+UGR premise; Haugan's energy-conservation plausibility argument, which assumes energy conservation). Meanwhile explicit counterexamples to unrestricted readings exist: curvature-coupled and non-minimally coupled theories can satisfy universality of free fall while violating local position invariance, and Ni showed that trajectory universality alone does not imply the Einstein principle. Standard reviews through 2020 still describe the general conjecture as never proved in full generality, not rigorously provable as stated, and under active debate. The open problem is to settle the general conjecture under a precise, physically motivated characterization of 'complete and self-consistent' theories of gravity that goes beyond the restricted frameworks already proven.

Background

Two experimentally distinct principles anchor the foundations of gravitation theory. The weak equivalence principle (WEP), also called universality of free fall, states that the world line of an uncharged, shielded test body released at a given event with a given velocity is independent of the body's internal structure and composition; it is tested by Eötvös-type experiments, which compare the free fall of bodies of different materials on a torsion balance or in free-fall instruments. The Einstein equivalence principle (EEP) states that (i) WEP holds and (ii) the outcome of any local, nongravitational test experiment is independent of where and when in the universe it is performed (local position invariance, LPI) and of the velocity of the freely falling apparatus (local Lorentz invariance, LLI). A theory of gravity is called metric if its nongravitational physics couples only to a single spacetime metric, so that in a locally freely falling frame it reduces to special relativity; nonmetric theories let matter couple to additional gravitational fields or to prior geometric structure — fields fixed ab initio rather than determined by the theory's own equations, such as a flat background metric. Following Thorne, Lee, and Lightman (TLL), a gravitation theory is 'complete' if it makes definite predictions for every experiment current technology can perform, and it is 'self-consistent' if it harbors no internal contradiction.

Around 1960, Leonard Schiff conjectured that these principles are not independent: any complete, self-consistent theory of gravity that embodies WEP necessarily embodies the full EEP. If true, the Eötvös experiments alone would be the direct empirical foundation of the EEP and hence of the interpretation of gravity as a curved-spacetime, metric phenomenon. TLL's 1973 paper made this conjecture a centerpiece of a general framework for classifying gravitation theories. Their 'universal coupling theorem' delineates the overlap between Lagrangian-based and metric theories: a relativistic, Lagrangian-based theory is metric precisely when its matter sector couples to one and only one symmetric tensor gravitational field, under a definition of universal coupling that includes the condition that replacing that tensor by a Riemann-flat one throughout a laboratory leaves every local, nongravitational prediction unchanged. TLL then argued plausibly — but only for relativistic, Lagrangian-based theories — that WEP forces the matter Lagrangian into this universally coupled form, and they explicitly declined to claim a proof, deferring a formal proof for a more limited class of theories to a subsequent paper.

That subsequent paper (Lightman and Lee) proved the conjecture within the THεµ formalism, a parametrization of all metric and many nonmetric theories describing charged point particles and electromagnetism in a static, spherically symmetric gravitational field of Newtonian potential U by four functions T(U), H(U), ε(U), and µ(U): every metric theory satisfies ε=μ=(H/T)1/2\varepsilon=\mu=(H/T)^{1/2} for all U, and Lightman and Lee showed that the rate of fall of a composite body of interacting charges is independent of its structure (WEP) if and only if this same metric condition holds, so that WEP implies EEP inside this class. Subsequent work pushed along two opposite directions. Proofs under further restrictions or strengthened hypotheses: Ni, in a general framework for electromagnetism in gravitational fields, found that trajectory universality plus universality of free-fall rotation states suffices to imply EEP; Haugan, via cyclic gedanken experiments, derived from energy conservation that any WEP violation entails a corresponding redshift (LPI) violation, and generalized the argument to LLI; Coley showed that WEP plus the universality of gravitational redshift (UGR) forces a broad class of theories into metric form. Counterexamples to unrestricted readings: Ohanian modified general relativity by adding a matter-curvature coupling and showed that all standard versions of WEP still hold while local nongravitational experiments become position-dependent; similar non-minimally coupled ('pseudogravitational') counterexamples were later constructed by Accioly and Wichoski. The modern standard references (Will's Living Reviews article, and the reviews of Di Casola, Liberati, and Sonego and of Tino and collaborators) summarize the state of the art as follows: the conjecture has never been proved in full generality, a rigorous proof is regarded as impossible without first pinning down what counts as a 'theory', special counterexamples are known, and yet the conjecture remains the working basis for interpreting Eötvös, redshift, and light-speed-isotropy experiments as a coherent experimental package establishing the EEP. The unresolved target is the general conjecture.

Problem Statement

Resolve Schiff's conjecture in its general form: does every complete and self-consistent theory of gravity that obeys the weak equivalence principle also, unavoidably, obey the Einstein equivalence principle?

(a) A proof. State precisely, as a set of hypotheses on theories of gravity, the conditions under which WEP implies local Lorentz invariance and local position invariance, and prove the theorem for the entire class of theories satisfying those hypotheses. The hypothesis class must credibly capture 'complete and self-consistent theories of gravity': in particular, curvature-coupled and other non-minimally coupled matter sectors — the basis of the known candidate counterexamples — must be addressed on principled grounds rather than excluded by assumption, and each hypothesis must be physically justified rather than chosen so as to make the theorem true. The hypothesis class must not be engineered to sidestep the disputed sectors: in particular, a class whose admission conditions rule out every curvature-coupled or non-minimally coupled matter sector by construction must independently show those excluded sectors to be contravened.

(b) A counterexample. Construct a complete, self-consistent theory of gravity, specified by its full action and field equations, in which WEP holds in the sense of the Background definition — the world line of an uncharged, shielded test body released at a given event with a given velocity is independent of the body's internal structure and composition — while local Lorentz invariance or local position invariance fails: some local, nongravitational test experiment has an outcome that depends on where, when, or at what velocity it is performed. The theory must be well-posed and make definite predictions (completeness), and the EEP violation must not be a disguised tidal or finite-size effect.

The already-proven restricted results — for electromagnetic systems in static, spherically symmetric fields within the THεµ and related frameworks, or for the strengthened WEP+UGR premise — do not answer this question, and the known special counterexamples do not settle it, because their admissibility as complete and self-consistent theories is precisely what the literature disputes; making that admissibility criterion exact is part of the problem. The conjecture is to be evaluated for all theories satisfying the stated hypotheses, not merely for theories inside a chosen parametrized test framework. In either form the admissibility criterion itself must be stated as a checkable property of the theory — decidable from its action, field equations, and predictions, and independently of whether it does or does not exhibit a WEP or EEP violation — and must be held fixed and applied uniformly to every candidate theory, so that a class is not defined to exclude WEP-violating sectors by construction and no theory is admitted only because it violates EEP. For form (b), the theory is specified by its full action and field equations, and the admissibility criteria are stated exactly and shown to be satisfied by the exhibited theory.

References

  1. K. S. Thorne, D. L. Lee, and A. P. Lightman, 'Foundations for a Theory of Gravitation Theories', Physical Review D 7, 3563-3578 (1973). DOI: 10.1103/PhysRevD.7.3563, https://doi.org/10.1103/PhysRevD.7.3563; open access: https://ntrs.nasa.gov/citations/19730008772.
  2. A. P. Lightman and D. L. Lee, 'Restricted proof that the weak equivalence principle implies the Einstein equivalence principle', Physical Review D 8, 364-376 (1973). DOI: 10.1103/PhysRevD.8.364, https://doi.org/10.1103/PhysRevD.8.364; open access: https://ntrs.nasa.gov/citations/19730012018.
  3. H. C. Ohanian, 'Comment on the Schiff conjecture', Physical Review D 10, 2041-2042 (1974). DOI: 10.1103/PhysRevD.10.2041, https://doi.org/10.1103/PhysRevD.10.2041.
  4. W.-T. Ni, 'Equivalence Principles and Electromagnetism', Physical Review Letters 38, 301-304 (1977). DOI: 10.1103/PhysRevLett.38.301, https://doi.org/10.1103/PhysRevLett.38.301.
  5. M. P. Haugan, 'Energy conservation and the principle of equivalence', Annals of Physics (N.Y.) 118, 156-186 (1979). DOI: 10.1016/0003-4916(79)90238-0, https://doi.org/10.1016/0003-4916(79)90238-0.
  6. A. Coley, 'Schiff's Conjecture on Gravitation', Physical Review Letters 49, 853-855 (1982). DOI: 10.1103/PhysRevLett.49.853, https://doi.org/10.1103/PhysRevLett.49.853.
  7. A. J. Accioly and U. F. Wichoski, 'Pseudogravitational nonminimally coupled theories', Classical and Quantum Gravity 7, L139-L141 (1990). DOI: 10.1088/0264-9381/7/7/004, https://doi.org/10.1088/0264-9381/7/7/004.
  8. C. M. Will, 'The Confrontation between General Relativity and Experiment', Living Reviews in Relativity 17, 4 (2014). DOI: 10.12942/lrr-2014-4, https://doi.org/10.12942/lrr-2014-4; arXiv:1403.7377, https://arxiv.org/abs/1403.7377.
  9. P. Wolf, L. Blanchet, C. J. Bordé, S. Reynaud, C. Salomon, and C. Cohen-Tannoudji, 'Does an atom interferometer test the gravitational redshift at the Compton frequency?', Classical and Quantum Gravity 28, 145017 (2011). DOI: 10.1088/0264-9381/28/14/145017, https://doi.org/10.1088/0264-9381/28/14/145017; arXiv:1012.1194, https://arxiv.org/abs/1012.1194.
  10. M. A. Hohensee, S. Chu, A. Peters, and H. Müller, 'Comment on: "Does an atom interferometer test the gravitational redshift at the Compton frequency?"', Classical and Quantum Gravity 29, 048001 (2012). DOI: 10.1088/0264-9381/29/4/048001, https://doi.org/10.1088/0264-9381/29/4/048001; arXiv:1112.6039, https://arxiv.org/abs/1112.6039.
  11. E. Di Casola, S. Liberati, and S. Sonego, 'Nonequivalence of equivalence principles', American Journal of Physics 83, 39-46 (2015). DOI: 10.1119/1.4895342, https://doi.org/10.1119/1.4895342; arXiv:1310.7426, https://arxiv.org/abs/1310.7426.
  12. G. M. Tino, L. Cacciapuoti, S. Capozziello, G. Lambiase, and F. Sorrentino, 'Precision Gravity Tests and the Einstein Equivalence Principle', Progress in Particle and Nuclear Physics 112, 103772 (2020). DOI: 10.1016/j.ppnp.2020.103772, https://doi.org/10.1016/j.ppnp.2020.103772; arXiv:2002.02907, https://arxiv.org/abs/2002.02907.
  13. L. I. Schiff, 'On Experimental Tests of the General Theory of Relativity', American Journal of Physics 28, 340-343 (1960). DOI: 10.1119/1.1935800, https://doi.org/10.1119/1.1935800.