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REVIEW 2 major objections 5 minor 83 references

General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper derives gravity-corrected center-of-mass and relative coordinates for a two-particle atom and constructs a first-quantized atom-light Hamiltonian in Earth's Fermi-Walker frame in which the mass defect emerges naturally and…

desk verdict The Sec. III symmetry generators do not satisfy the Killing equations for the stated metric, so the c.m. coordinates and Hamiltonian built on them are unsupported as written; the framework and the Schwartz–Giulini comparison still merit serious referee attention. read the letter →

arxiv 2411.14307 v2 pith:CCO3YT2L submitted 2024-11-21 gr-qc quant-ph

Gregor Janson , Richard Lopp This is my paper · ORCID
classification gr-qcquant-ph
keywords generalrelativitycenter-of-masscoordinatescompositeparticlesFermi-Walkerframemassdefectatom-lightinteractionquantumatominterferometrySchwarzschildspacetime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to put general-relativistic effects in atom interferometry on a first-principles footing, rather than adding mass-defect and redshift terms to the Hamiltonian by hand. It constructs gravitationally corrected center-of-mass and relative coordinates for a two-particle atom such as hydrogen, starting from the ten symmetry generators of a static observer's non-rotating Fermi-Walker frame in Schwarzschild spacetime, expanded to first order in $\epsilon = R_S/R_E$, the ratio of the Schwarzschild radius to Earth's radius. In these coordinates the full quantum Hamiltonian of the atom coupled to light separates into center-of-mass and internal parts: the mass defect appears automatically as a state-dependent total mass (the internal energy divided by $c^2$ contributes to the mass), the internal energy levels are unaffected by gravity at this order, and the atom-light coupling acquires position-dependent gravitational prefactors. A residual coupling between internal and external motion remains but can be unitarily removed in quasi-one-dimensional geometries. The authors present the resulting Hamiltonian as the basis for describing general-relativistic effects in quantum sensors such as atom interferometers and clocks.

What carries the argument

The load-bearing objects are the ten gravitationally corrected Poincaré symmetry generators (16)–(19), obtained by inserting a quadratic ansatz into the Killing equations for the Fermi-Walker form of the Schwarzschild metric (11). These generators obey a deformed algebra (21) with position-dependent structure functions. The paper then enforces Eq. (22), the condition that the sum of single-particle generators must look like a single-particle generator, to determine the corrected center-of-mass coordinates (24), and uses the unitary-transformation method to obtain the relative coordinates (30). The final step is the Power-Zienau-Woolley transformation in the dipole approximation, which converts the two-particle light-matter Lagrangian into the Hamiltonian (48). Each step is the curved-spacetime analogue of the special-relativistic c.m. and relative coordinate construction of Refs. [35, 36, 55, 56, 34].

What would settle it

Measure the Rabi frequency of the same atomic transition for identical atoms held at two heights separated by $\Delta x$ in Earth's gravitational field: the Hamiltonian's atom-light coupling carries the prefactor $1 + \phi(R)/c^2$, so the predicted ratio of Rabi frequencies deviates from unity by $g\,\Delta x/c^2$ in the direction set by the local potential; a null result at that order, or a deviation of the opposite sign, would falsify the central claim.

Watch

Extended reading notes

Core claim

Starting from the Schwarzschild metric as seen by a static observer on Earth, the authors solve the Killing equations to first order in $\epsilon = R_S/R_E$ and obtain gravitationally corrected Poincaré symmetry generators in the Fermi-Walker frame, Eqs. (16)–(19). They then impose that the sum of the two particles' generators take the same functional form as a single-particle generator, Eq. (22), which yields the corrected center-of-mass position (24); the corrected relative coordinates (30) follow from the unitary-transformation method. Inserting these coordinates into the two-particle Lagrangian coupled to the electromagnetic field and applying the Power-Zienau-Woolley transformation gives the Hamiltonian (48). Its center-of-mass part contains the mass defect $M \to M + H_{\rm int}/c^2$ without being inserted ad hoc; the internal Hamiltonian (50) contains no gravitational correction terms at this order, so gravity does not shift internal energy levels or induce new transitions; and the atom-light Hamiltonian (51) has gravitational prefactors, so Rabi frequencies change with the atom's height. In contrast to the post-Newtonian Hamiltonian of Refs. [32, 33], the internal-gravity cross terms (54) and (55) vanish, while a new cross term (53) remains and is unitarily removable in quasi-one-dimensional settings.

Load-bearing premise

The central premise is that ten symmetry generators that no longer obey the exact flat-spacetime rules can still be added together and repackaged in the same single-particle form, Eq. (22); if that repackaging fails, the atom's internal and external motion cannot be cleanly separated and the mass-defect picture breaks down.

Editorial extensions

If this is right

  • Internal atomic level spacings are unaffected by gravity to first order in $\epsilon$, and gravity induces no new transitions; only the rates of transitions that already exist are modified.
  • Atom-light coupling strengths acquire gravitational prefactors, so identical atoms at different heights in Earth's field experience different Rabi frequencies, giving a concrete observable for atom interferometry and clock experiments.
  • The mass defect enters the center-of-mass Hamiltonian as $M \to M + H_{\rm int}/c^2$ without being inserted by hand, giving a first-principles basis for equivalence-principle tests that rely on internal-state-dependent mass.
  • In quasi-one-dimensional geometries the residual internal-external cross term (53) can be removed by the unitary transformation (58), so the mass-defect picture is restored in laboratory-like settings.
  • Compared with the post-Newtonian Hamiltonian of Refs. [32, 33], the gravitationally corrected coordinates eliminate the internal-gravity cross terms (54) and (55), leaving the new cross term (53).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the decoupling is as clean as claimed, atom-interferometer phase calculations could be generated systematically from one Hamiltonian instead of by adding redshifts and mass-defect phases ad hoc; the difference should show up in detailed long-baseline phase predictions.
  • Editorial inference: the construction hinges on Eq. (22) holding for a deformed algebra; for long interrogation times where $c\tau/R_E$ is no longer small, the position-dependent structure functions in (21) may spoil decoupling, so the next-order short-time expansion is the natural stress test.
  • Editorial inference: the 'no internal level shift' statement is tied to the local observer's Fermi-Walker frame; recasting the same dynamics in Schwarzschild coordinates would move parts of the effect into redshift phases, so comparisons with coordinate-dependent experiments need to identify the frame explicitly.
  • Editorial inference: applying the same method to a Kerr background with a co-rotating observer would give semi-analytical, position-dependent corrections and could reveal rotational analogues of the predicted height-dependent Rabi frequency.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper aims to provide a first-principles, first-quantized description of a two-particle atom interacting with light in the Fermi-Walker frame of a static observer on Earth. It derives a Fermi-Walker form of the Schwarzschild metric, solves the Killing equations to first order in ε = R_S/R_E, and uses the resulting Poincaré-like generators to construct general-relativistic center-of-mass and relative coordinates. These coordinates are then used to derive a Hamiltonian containing the mass defect, gravitational corrections to the atom-light coupling, and a residual internal-external cross term. The stated physical conclusions are that the internal energy levels of the atom are unaffected by gravity to first order in ε and that the mass defect emerges naturally from the coordinate transformation, with only the transition rates acquiring gravitational corrections.

Significance. If the central derivation were sound, the paper would provide a useful bridge between relativistic two-body quantum mechanics and laboratory-frame atom interferometry. The program is well motivated: replacing ad hoc insertion of mass-defect and gravitational-potential terms by a systematic coordinate construction is valuable, and the Fermi-Walker setting is appropriate for describing experiments of finite spatial extent around an Earth-bound observer. The manuscript is also commendably explicit in its chain from metric to final Hamiltonian, and it identifies clearly which terms in previous work are removed by the generalized coordinates. However, the load-bearing generator derivation in Sec. III appears to be internally inconsistent, and the subsequent c.m. and relative coordinates inherit that problem. The significance of the paper is therefore conditional on a successful repair of Sec. III and a re-derivation of the coordinates and Hamiltonian that follow from it.

major comments (2)
  1. [Sec. III, Eq. (16) and Table I] The time-translation generator and its associated Killing vector do not satisfy the Killing equations for the metric (11). The FWC metric (11) is independent of the proper time τ, so ∂_τ is an exact Killing vector and any first-order-in-ε correction to the time-translation Killing vector must be independent of τ as well. The vector in Table I, however, has ξ^1 = ε c τ / R_E. Direct substitution into Eq. (12) gives a nonzero result even at the origin: with R_E = 1 and c = 1, the (0,1) component of the Killing equation for ξ^0 = 1 - 2ε x + ..., ξ^1 = ε τ evaluates to -ε to first order, not zero. This is not a matter of a deformed algebra or of an unjustified approximation; it is a failure of the equation that is claimed to define the generators. Since the c.m. coordinate (24), the relative coordinates (30), and the final Hamiltonian (48) are all constructed from these generators, the central result of the paper is unsupported by the presented derivation.
  2. [Sec. IV, Eq. (22)] The construction of the c.m. coordinates assumes that the sum of the ten single-particle generators can be expressed in the same functional form as a single-particle generator. This is the standard Poincaré-based argument of Osborn, Close, Liou, and Krajcik-Foldy, but the paper itself shows in Eq. (21) that the generators obey a deformed algebra with position-dependent structure functions. No argument is given that Eq. (22) remains valid in this deformed setting, particularly beyond the short-time regime cτ/R_E ≪ 1. Without such an argument, the generalized coordinates (24) and (30) cannot be claimed to decouple internal and external dynamics. This is a second load-bearing gap in the derivation, independent of the explicit Killing-equation failure noted above.
minor comments (5)
  1. [Sec. VI] The conclusion states that there is 'no other restriction on short times τ' in the results of Sec. II, but Sec. III explicitly restricts to 'a short time scale' when solving the Killing equations; these two statements should be reconciled and the domain of validity of the final Hamiltonian stated precisely.
  2. [Sec. III, Eq. (20)] The treatment of spin in the boost and rotation generators is left in a conditional state: spin is needed to avoid an overdetermined system, but the replacement (20) is declared to be beyond the scope of the paper. Since the final generators are used to define the c.m. coordinates, the reader cannot fully verify that the equations are consistent.
  3. [Sec. V, Eq. (58)] The claim that the residual cross-coupling Hamiltonian H_X can be removed by the unitary transformation (58) relies on the assumption P_y = P_z = 0 and a 'quasi-1D setting', but the size of the neglected transverse terms is not estimated. A quantitative statement about the regime in which this is justified would strengthen the physical interpretation.
  4. [Sec. V, Eqs. (42)-(48)] The Legendre transformation from the total Lagrangian (42) to the final Hamiltonian (48) is performed in a single step without intermediate algebra. Given the length of the calculation, the authors should either provide the main intermediate Hamiltonian in the c.m. and relative coordinates or state where the full calculation can be found.
  5. [Throughout] The equations contain several typographical ambiguities, such as the placement of parentheses in Eq. (17) and the use of 'h.c.' in Eq. (28). A careful proofreading pass would improve accessibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation is self-contained, with only a non-load-bearing self-citation.

full rationale

The paper's chain is: Fermi-Walker metric (11) from Schwarzschild; GR-corrected symmetry generators (16)-(19) obtained by solving the explicitly stated Killing equations (12) with the quadratic ansatz (15); c.m. and relative coordinates (24),(30) fixed by the algebraic consistency condition (22) taken from Osborn/Close/Liou; and finally the Hamiltonian (48) obtained by direct Legendre transformation of the Lagrangian (42). At no step is the target result inserted as an input: no parameter is fitted, R_GR is solved from the ten generator equations, the unitary generator u is fixed by Eq. (27) from R_GR, and H_int, H_cm, H_AL, H_L, H_X all come from the same Lagrangian. The mass defect appears because H_cm is expressed in terms of H_int after the coordinate substitution, matching the earlier flat-spacetime result [34]; the paper explicitly credits that prior work rather than presenting it as a new prediction. The only self-citation is Ref. [29] in a list of ad hoc treatments; it is not load-bearing. The possible failure of Table I to solve Eq. (12), if real, would be an internal correctness defect, not a circularity. Hence no circular step reduces the central claim to its inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation introduces no free parameters fitted to data. It relies on standard background results (Fermi-Walker transport, Killing equations, Osborn-Close and Liou coordinate methods) and on domain assumptions about the observer and the atom. The most delicate input is the perturbative ansatz for the Killing vectors, which produces a deformed algebra; the paper does not fully justify that this ansatz supports the c.m. coordinate construction.

assumptions (6)
  • standard math Fermi-Walker transport and the metric expansion to second order in spatial coordinates are valid for a static observer in Schwarzschild spacetime.
    Used in Sec. II, Eq. (4), to derive the metric (11). This is a standard result in differential geometry.
  • domain assumption Earth's rotation can be neglected, so a static Schwarzschild observer models an observer on the equator.
    Sec. II A justifies the Schwarzschild model for experiments shorter than the Earth's rotation period.
  • domain assumption The atom is a two-body system with vanishing total charge and negligible spin.
    Sec. IV restricts to hydrogenoid atoms and sets spins to zero after solving for coordinates (Sec. III).
  • ad hoc to paper The perturbative ansatz for the Killing vectors, truncated at second order in coordinates and first order in epsilon, provides a valid set of generators for defining c.m. and relative coordinates.
    Sec. III, Eq. (15), and Eq. (21) show the resulting algebra is deformed; the validity of using these generators to define composite particle coordinates is not fully established.
  • domain assumption The gravitational field is expanded only to first order in relative coordinates, neglecting tidal/geodesic deviation forces on the internal structure.
    Sec. V, around Eq. (57); the paper acknowledges Parker's geodesic deviation term is dropped.
  • domain assumption The lowest-order internal vector potential has no radiative part (A⊥(0)=0).
    Appendix A 1, Eq. (A12), standard for bound internal fields.

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Pith. "Pith review of General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles." pith.science (2026). https://pith.science/paper/CCO3YT2L

@misc{pith2026241114307,
  author       = {Pith},
  title        = {Pith review of: General Relativistic Center-of-Mass Coordinates for Composite Quantum Particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CCO3YT2L}},
  note         = {Machine review of arXiv:2411.14307}
}
read the original abstract

Recent proposals suggested quantum clock interferometry for tests of the Einstein equivalence principle. However, atom interferometric models often include relativistic effects only in an ad hoc fashion. Here, instead, we start from the multi-particle nature of quantum-delocalizable atoms in curved spacetime and generalize the special-relativistic center of mass (COM) and relative coordinates that have previously been studied for Minkowski spacetime to obtain the light-matter dynamics in curved spacetime. In particular, for a local Schwarzschild observer located at the surface of the Earth using Fermi-Walker coordinates, we find gravitational correction terms for the Poincar\'e symmetry generators and use them to derive general relativistic COM and relative coordinates. In these coordinates we obtain the Hamiltonian of a fully first-quantized two-particle atom interacting with the electromagnetic field in curved spacetime that naturally incorporates special and general relativistic effects.

Figures

Figures reproduced from arXiv: 2411.14307 by the authors.

Figure 1
Figure 1. Orthonormal tetrad 𝑒 𝛼 (𝜇) (𝜏) Fermi-Walker transported along the worldline 𝜎 𝛼(𝜏) of the observer. however, do not describe physical phenomena as seen by a realistic observer. Instead, we have to find a coordinate system attached to the worldline 𝜎 𝛼(𝜏) of the observer, parametrized by its proper time 𝜏. For this we choose the FWC 𝑥 𝛼. The goal of this section is to provide a short introduction to the Fermi￾Walker … view at source ↗

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Works this paper leans on

83 extracted references · 71 canonical work pages

  1. [1]

    (A8b) We can then solve the internal Maxwell’s equations order by order

    Internal part Firstly, let us consider the internal part of the electromagnetic fields and expand them in orders of𝑐−2, i.e., 𝜙el=𝜙(0) el +𝑐−2𝜙(2) el +O( 𝑐−4) (A8a) A⊥= A⊥( 0)+𝑐−2 A⊥( 2)+O( 𝑐−4). (A8b) We can then solve the internal Maxwell’s equations order by order. We are only interested in solutions up to the order𝑐−2, so that we are left with Δ𝜙(0) e...

  2. [2]

    (A16b) However, we now have a slightly different situation

    External part Analogously to the previous section we can expand the external electromagnetic fields in orders of𝑐−2, i.e., 𝜙el,ext=𝜙(0) el,ext+𝑐−2𝜙(2) el,ext+O( 𝑐−4) (A16a) 𝑨⊥= 𝑨⊥( 0)+𝑐−2 𝑨⊥( 2)+O( 𝑐−4). (A16b) However, we now have a slightly different situation. We assume that there are no external charges, i.e.,𝜌= 𝑗𝑎= 0. Therefore, the external Maxwell’...

  3. [3]

    Kasevich and S

    M. Kasevich and S. Chu, Phys. Rev. Lett. 67, 181 (1991)

    1991

  4. [4]

    Peters, K

    A. Peters, K. Y. Chung, and S. Chu, Metrologia 38, 25 (2001)

    work page 2001

  5. [5]

    Peters, K

    A. Peters, K. Y. Chung, and S. Chu, Nature 400, 849 (1999)

    work page 1999

  6. [6]

    Lan, P.-C

    S.-Y. Lan, P.-C. Kuan, B. Estey, P. Haslinger, and H. M¨ uller, Phys. Rev. Lett.108, 090402 (2012)

    work page 2012

  7. [7]

    G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, Nature 510, 518 (2014)

    work page 2014

  8. [8]

    Barrett, L

    B. Barrett, L. Antoni-Micollier, L. Chichet, B. Battelier, T. L´ev`eque, A. Landragin, and P. Bouyer, Nat. Commun. 7, 1 (2016)

    work page 2016

Show all 83 references
  1. [9]

    X. Wu, Z. Pagel, B. S. Malek, T. H. Nguyen, F. Zi, D. S. Scheirer, and H. M¨ uller, Sci. Adv.5, eaax0800 (2019)

    2019

  2. [10]

    Templier, P

    S. Templier, P. Cheiney, Q. d. de Castanet, B. Gouraud, H. Porte, F. Napolitano, P. Bouyer, B. Battelier, and B. Barrett, Sci. Adv. 8, eadd3854 (2022)

    2022

  3. [11]

    Stray, A

    B. Stray, A. Lamb, A. Kaushik, J. Vovrosh, A. Rodgers, J. Winch, F. Hayati, D. Boddice, A. Stabrawa, A. Niggebaum, M. Langlois, Y.-H. Lien, S. Lellouch, S. Roshanmanesh, K. Ridley, G. de Vil- liers, G. Brown, T. Cross, G. Tuckwell, A. Faramarzi, N. Metje, K. Bongs, and M. Holy...

    2022

  4. [12]

    Parker, C

    R. Parker, C. Yu, W. Zhong, B. Estey, and H. M¨ uller, Science 360, 191 (2018)

    2018

  5. [13]

    Morel, Z

    L. Morel, Z. Yao, P. Clad´e, and S. Guellati-Kh´elifa, Nature 588, 61 (2020)

    2020

  6. [14]

    Roura, Phys

    A. Roura, Phys. Rev. X 10, 021014 (2020)

    2020

  7. [15]

    Ufrecht, F

    C. Ufrecht, F. Di Pumpo, A. Friedrich, A. Roura, C. Schubert, D. Schlippert, E. M. Rasel, W. P. Schleich, and E. Giese, Phys. Rev. Res.2, 043240 (2020)

    2020

  8. [16]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Rev. D78, 122002 (2008)

    2008

  9. [17]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Lett. B 678, 37 (2009)

    2009

  10. [18]

    P. W. Graham, J. M. Hogan, M. A. Kasevich, and S. Rajendran, Phys. Rev. D94, 104022 (2016)

    2016

  11. [19]

    A. A. Geraci and A. Derevianko, Phys. Rev. Lett.117, 261301 (2016)

    2016

  12. [20]

    Arvanitaki, P

    A. Arvanitaki, P. W. Graham, J. M. Hogan, S. Rajendran, and K. Van Tilburg, Phys. Rev. D97, 075020 (2018)

    2018

  13. [21]

    Badurina, V

    L. Badurina, V. Gibson, C. McCabe, and J. Mitchell, Phys. Rev. D 107, 055002 (2023)

    2023

  14. [22]

    M. Abe, P. Adamson, M. Borcean, D. Bortoletto, K. Bridges, S. P. Carman, S. Chattopadhyay, J. Coleman, N. M. Curfman, K. DeRose, T. Deshpande, S. Dimopoulos, C. J. Foot, J. C. Frisch, B. E. Garber, S. Geer, V. Gibson, J. Glick, P. W. Graham, S. R. Hahn, R. Harnik, L. Hawkins, ...

    2021

  15. [23]

    Bertoldi, K

    A. Bertoldi, K. Bongs, P. Bouyer, O. Buchmueller, B. Canuel, L.-I. Caramete, M. L. Chiofalo, J. Coleman, A. De Roeck, J. Ellis, P. W. Graham, M. G. Haehnelt, A. Hees, J. Hogan, W. von Klitzing, M. Krutzik, M. Lewicki, C. McCabe, A. Peters, E. M. Rasel, A. Roura, D. Sabulsky, S...

    2021

  16. [24]

    Badurina, D

    L. Badurina, D. Blas, and C. McCabe, Phys. Rev. D105, 023006 (2022)

    2022

  17. [25]

    M. Zych, F. Costa, I. Pikovski, and ˇC. Brukner, Nat. Commun. 2, 505 (2011)

    2011

  18. [26]

    Zych, Quantum Systems under Gravitational Time Dilation (2017)

    M. Zych, Quantum Systems under Gravitational Time Dilation (2017). 13

    2017

  19. [27]

    Loriani, A

    S. Loriani, A. Friedrich, C. Ufrecht, F. Di Pumpo, S. Kleinert, S. Abend, N. Gaaloul, C. Meiners, C. Schubert, D. Tell,´E. Wodey, M. Zych, W. Ertmer, A. Roura, D. Schlippert, W. P. Schleich, E. M. Rasel, and E. Giese, Sci. Adv. 5, eaax8966 (2019)

    2019

  20. [28]

    Di Pumpo, C

    F. Di Pumpo, C. Ufrecht, A. Friedrich, E. Giese, W. P. Schleich, and W. G. Unruh, PRX Quantum 2, 040333 (2021)

    2021

  21. [29]

    Di Pumpo, A

    F. Di Pumpo, A. Friedrich, A. Geyer, C. Ufrecht, and E. Giese, Phys. Rev. D105, 084065 (2022)

    2022

  22. [30]

    Di Pumpo, A

    F. Di Pumpo, A. Friedrich, C. Ufrecht, and E. Giese, Phys. Rev. D 107, 064007 (2023)

    2023

  23. [31]

    Janson, A

    G. Janson, A. Friedrich, and R. Lopp, A VS Quantum Science 6, 024403 (2024), https://pubs.aip.org/avs/aqs/article- pdf/doi/10.1116/5.0178230/19932745/024403 1 5.0178230.pdf

    2024 doi

  24. [32]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, Phys. Rev. D78, 042003 (2008)

    2008

  25. [33]

    Werner, P

    M. Werner, P. K. Schwartz, J.-N. Kirsten-Siemß, N. Gaaloul, D. Giulini, and K. Hammerer, Phys. Rev. D109, 022008 (2024)

    2024

  26. [34]

    P. K. Schwartz and D. Giulini, Phys. Rev. A100, 052116 (2019)

    2019

  27. [35]

    P. K. Schwartz,Post-Newtonian Description of Quantum Systems in Gravitational Fields, Ph.D. thesis, Gottfried Willhelm Leibniz Universit¨at Hannover (2020)

    2020

  28. [36]

    Sonnleitner and S

    M. Sonnleitner and S. M. Barnett, Phys. Rev. A 98, 042106 (2018)

    2018

  29. [37]

    Osborn, Phys

    H. Osborn, Phys. Rev. 176, 1514 (1968)

    1968

  30. [38]

    F. E. Close and H. Osborn, Phys. Rev. D 2, 2127 (1970)

    1970

  31. [39]

    A. I. Nesterov, Class. Quantum Gravity 16, 465 (1999)

    1999

  32. [40]

    F. K. Manasse and C. W. Misner, J. Math. Phys. 4, 735 (2004)

    2004

  33. [41]

    Klein and P

    D. Klein and P. Collas, Class. Quantum Gravity 25, 145019 (2008)

    2008

  34. [42]

    Klein and P

    D. Klein and P. Collas, Class. Quantum Gravity 26, 045018 (2009)

    2009

  35. [43]

    Poisson, Living Rev

    E. Poisson, Living Rev. Relativ. 7, 6 (2004)

    2004

  36. [44]

    Klein and W.-S

    D. Klein and W.-S. Yang, Math. Phys. Anal. Geom.15, 61–83 (2012)

    2012

  37. [45]

    Mart´ın-Mart´ınez, T

    E. Mart´ın-Mart´ınez, T. R. Perche, and B. de S. L. Torres, Phys. Rev. D 101, 045017 (2020)

    2020

  38. [46]

    T. R. Perche and J. Neuser, Class. Quantum Gravity38, 175002 (2021)

    2021

  39. [47]

    T. R. Perche, Phys. Rev. D 106, 025018 (2022)

    2022

  40. [48]

    Kajari, M

    E. Kajari, M. Buser, C. Feiler, and W. P. Schleich, Proceedings of the International School of Physics ”Enrico Fermi”, Course CLXVIII 168, 45–148 (2009)

    2009

  41. [49]

    Kajari, Perspectives on Relativistic Sagnac Interferometry , Ph.D

    E. Kajari, Perspectives on Relativistic Sagnac Interferometry , Ph.D. thesis, Universit¨at Ulm (2011)

    2011

  42. [50]

    Llosa, Class

    J. Llosa, Class. Quantum Gravity 34, 205003 (2017)

    2017

  43. [51]

    L. H. Thomas, Phys. Rev. 85, 868 (1952)

    1952

  44. [52]

    P. A. M. Dirac, Rev. Mod. Phys. 21, 392 (1949)

    1949

  45. [53]

    A. J. Macfarlane, J. Math. Phys. 4, 490 (1963)

    1963

  46. [54]

    Bakamjian and L

    B. Bakamjian and L. H. Thomas, Phys. Rev. 92, 1300 (1953)

    1953

  47. [55]

    L. L. Foldy, Phys. Rev. 122, 275 (1961)

    1961

  48. [56]

    V. J. Mart ´ınez-Lahuerta, S. Eilers, T. E. Mehlst ¨aubler, P. O. Schmidt, and K. Hammerer, Phys. Rev. A106, 032803 (2022)

    2022

  49. [57]

    M. K. Liou, Phys. Rev. D 9, 1091 (1974)

    1974

  50. [58]

    R. A. Krajcik and L. L. Foldy, Phys. Rev. D 10, 1777 (1974)

    1974

  51. [59]

    Wigner, Ann

    E. Wigner, Ann. Math. 40, 149 (1939)

    1939

  52. [60]

    Bargmann and E

    V. Bargmann and E. P. Wigner, Proc. Natl. Acad. Sci. U.S.A.34, 211 (1948)

    1948

  53. [61]

    Braathen and L

    H. Braathen and L. Foldy, Nucl. Phys. B 13, 511 (1969)

    1969

  54. [62]

    W. G. Unruh, Phys. Rev. D 14, 870 (1976)

    1976

  55. [63]

    Singh and N

    D. Singh and N. Mobed, Class. Quantum Gravity 24, 2453 (2007)

    2007

  56. [64]

    Peres, P

    A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. 88, 230402 (2002)

    2002

  57. [65]

    Terashima and M

    H. Terashima and M. Ueda, Phys. Rev. A 69, 032113 (2004)

    2004

  58. [66]

    Singh and N

    D. Singh and N. Mobed, Phys. Rev. D 79, 024026 (2009)

    2009

  59. [67]

    Singh and N

    D. Singh and N. Mobed, Ann. Phys. 522, 555 (2010)

    2010

  60. [68]

    P. M. Alsing, G. J. S. J. au2, and P. Kilian, Spin-induced non- geodesic motion, gyroscopic precession, wigner rotation and epr correlations of massive spin 1/2 particles in a gravitational field (2009), arXiv:0902.1396 [quant-ph]

    2009 arXiv

  61. [69]

    M. L. W. Basso and J. Maziero, Phys. Rev. A103, 032210 (2021)

    2021

  62. [70]

    Levy-Nahas, J

    M. Levy-Nahas, J. Math. Phys. 8, 1211 (1967)

    1967

  63. [71]

    Lukierski, A

    J. Lukierski, A. Nowicki, and H. Ruegg, Phys. Lett. B 293, 344 (1992)

    1992

  64. [72]

    Bacry, J

    H. Bacry, J. Phys. A 26, 5413 (1993)

    1993

  65. [73]

    Ma´slanka, J

    P. Ma´slanka, J. Math. Phys. 34, 6025 (1993)

    1993

  66. [74]

    Fernandez, Phys

    J. Fernandez, Phys. Lett. B 368, 53 (1996)

    1996

  67. [75]

    Zhang and X

    L. Zhang and X. Xue, The deformation of poincar´e subgroups concerning very special relativity (2012), arXiv:1204.6425 [math- ph]

    2012 arXiv

  68. [76]

    Pfeifer and J

    C. Pfeifer and J. Relancio, Eur. Phys. J. C 82 (2022)

    2022

  69. [77]

    Andrews, G

    D. Andrews, G. Jones, A. Salam, and R. G. Woolley, J. Chem. Phys. 148 (2018)

    2018

  70. [78]

    R. G. Woolley, Phys. Rev. Res.2, 013206 (2020)

    2020

  71. [79]

    I. I. Rabi, Phys. Rev. 51, 652 (1937)

    1937

  72. [80]

    Parker, Phys

    L. Parker, Phys. Rev. D 22, 1922 (1980)

    1980

  73. [81]

    Delva, N

    P. Delva, N. Puchades, E. Sch¨onemann, F. Dilssner, C. Courde, S. Bertone, F. Gonzalez, A. Hees, C. Le Poncin-Lafitte, F. Mey- nadier, R. Prieto-Cerdeira, B. Sohet, J. Ventura-Traveset, and P. Wolf, Phys. Rev. Lett.121, 231101 (2018)

    2018

  74. [82]

    Takamoto, I

    M. Takamoto, I. Ushijima, N. Ohmae, T. Yahagi, K. Kokado, H. Shinkai, and H. Katori, Nat. Photonics 14, 411 (2020)

    2020

  75. [83]

    Bothwell, C

    T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron, and J. Ye, Nature602, 420 (2022)

    2022

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