The gravitationally coupled Dirac equation (spin-connection formalism, with a Foldy–Wouthuysen transformation in the central-field case) yields a matching relation between inertial and gravitational mass, mG/mI=1m_G/m_I=1 to the computed order, valid for both particles and antiparticles at zero temperature. Separately, finite-temperature quantum field theory (Donoghue–Holstein–Robinett and successors) indicates that a particle in contact with a heat bath acquires temperature-dependent corrections to its gravitational mass, coupled through a term of the form hμντμνh_{\mu\nu}\tau^{\mu\nu} on the right-hand side of the Dirac equation, implying apparent violations of the Einstein equivalence principle that vanish at T=0T=0. Whether these two frameworks are mutually consistent — whether the thermal gravitational-mass shifts can be rederived within the covariant Dirac formalism and whether the hμντμνh_{\mu\nu}\tau^{\mu\nu} coupling maps onto the spin-connection coupling — was explicitly left open by Jentschura (2019) and remains unaddressed: the finite-temperature thread continues entirely within extended-gravity/effective-field-equation frameworks, while the curved-spacetime Dirac literature remains at zero temperature.

Background

The Einstein equivalence principle (EEP) comprises the universality of free fall (the weak equivalence principle), local Lorentz invariance, and local position invariance (LPI — the position- and time-independence of nongravitational experiments). For antimatter, the EEP is a genuinely open experimental question: the ALPHA Collaboration has only recently observed the free fall of antihydrogen (Anderson et al., Nature 621, 716 (2023)), finding it consistent with ordinary gravity at roughly the 20% level, while the AEgIS and GBAR experiments prepare similar measurements. On the theory side, a natural framework is the gravitationally coupled Dirac equation: the Dirac equation in curved spacetime, formulated with tetrads (local orthonormal frames) and the spin connection (the gravitational analogue of the gauge connection, which couples spin to spacetime curvature). In a weak, static central gravitational field, a Foldy–Wouthuysen transformation — a unitary block-diagonalization that separates the positive- and negative-energy (particle and antiparticle) sectors of the Dirac Hamiltonian — disentangles particles from antiparticles and yields a 'matching relation' between the inertial mass mIm_I and the gravitational mass mGm_G, satisfied identically for particles and antiparticles (Jentschura, Int. J. Mod. Phys. A 34, 1950180 (2019)). This zero-temperature analysis suggests that, at the level of the single-particle Dirac equation, antiparticles fall exactly like particles. A seemingly conflicting line of results goes back to finite-temperature field theory: Donoghue, Holstein, and Robinett (Phys. Rev. D 30, 2561 (1984); Phys. Rev. D 34, 1208 (1986)) found that the gravitational coupling of a particle in contact with a thermal bath of photons is modified — the renormalized energy-momentum tensor τμν\tau^{\mu\nu} at temperature TT induces a correction to the gravitational mass, so that mG≠mIm_G \neq m_I at T≠0T \neq 0 ('Aristotle was right: heavier objects fall faster'). Later work extended these finite-temperature equivalence-principle violations to scalar-tensor and other extended theories of gravity (Blasone et al., Eur. Phys. J. Plus 134, 169 (2019); Int. J. Geom. Methods Mod. Phys. 19, 2250055 (2022); Gasperini, arXiv:2101.00458; Battista et al., Phys. Rev. D 114 (2026)). In these approaches the thermal coupling enters as a term proportional to hμντμνh_{\mu\nu}\tau^{\mu\nu} (with hμνh_{\mu\nu} the metric perturbation) on the right-hand side of the Dirac equation, as in Eq. (B.7) of Jentschura (2019). All of these thermal effects vanish in the T→0T\to 0 limit in which the covariant spin-connection formalism is derived. Jentschura (2019) explicitly left open whether the two descriptions are equivalent: whether the finite-temperature gravitational-mass modifications can be rederived within the gravitationally coupled Dirac framework (for example, for a central gravitational field), and whether the hμντμνh_{\mu\nu}\tau^{\mu\nu} coupling maps onto the spin-connection coupling. Related zero-temperature background concerns LPI: high-precision spectroscopy and gg-factor calculations show quantum corrections to atomic transition frequencies that scale as Cn(M)∣Φ∣nC_n(M)|\Phi|^n in the gravitational potential Φ\Phi of a source of mass MM (Jentschura, Phys. Rev. A 98, 032508 (2018)), with coefficients Cn(M)C_n(M) so far estimated only at order of magnitude and only at zero temperature.

Problem Statement

Determine whether the finite-temperature indications of equivalence-principle violation — the temperature-dependent corrections to the gravitational mass of a particle in contact with a heat bath, derived from the renormalized energy-momentum tensor and coupled through a term of the form hμντμνh_{\mu\nu}\tau^{\mu\nu} on the right-hand side of the Dirac equation (Donoghue-Holstein-Robinett; Blasone et al.) — can be rederived and reconciled within the covariant, gravitationally coupled Dirac equation with spin connection, as formulated by Jentschura (2019), for example for a weak static central gravitational field via the Foldy-Wouthuysen transformation. Concretely: (1) construct (or prove impossible to construct) a thermal-state treatment of the Dirac field in a weak gravitational background — for example, the weak static central field mentioned above, though not restricted to it — that reproduces the finite-temperature shift of the gravitational-to-inertial mass ratio mG/mIm_G/m_I for both particles and antiparticles, recovering the zero-temperature matching relation mG/mI=1m_G/m_I=1 in the T→0T\to 0 limit; and (2) establish whether the hμντμνh_{\mu\nu}\tau^{\mu\nu} coupling of the finite-temperature field-theory approach is equivalent to, or distinct from, the spin-connection coupling of the covariant formalism, identifying any term present in one and absent in the other at a stated order in the weak-field and low-temperature expansion. An answer may take the form of a constructive reconciliation, a demonstration of inequivalence or no-go (for instance, because thermal effects are inherently many-body/medium effects that no single-particle Dirac framework can capture), or a precise delineation of the regimes in which the two formalisms agree; any such form resolves the task only if it answers both (1) and (2) within the stated order and regime.

References

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