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Hodge dual gauge symmetry in minimal Einstein-aether theory

Kamal Hajian*

  • *Contact author: kamal.hajian@uni-oldenburg.de
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Phys. Rev. D 111, 124028 – Published 18 June, 2025

DOI: https://doi.org/10.1103/hpdv-yd6q

Abstract

Einstein-aether gravity is a theory that breaks the local Lorentz symmetry by introducing a preferred direction via a vector field, which is considered to play the role of an aether. The theory is identified by four coupling constants between the aether and gravity. Minimal Einstein-aether is the special case in which only one of the couplings is non-zero. We show that the aether vector field in its minimal version is Hodge dual to a gauge field. The gauge symmetry in the dual description has been known for decades and has been used to implement a cosmological constant into the Lagrangian. As a result, solutions to the well-established gauge theory can be transferred into the minimal Einstein-aether theory straightforwardly. On the other hand, some of the proposed solutions to the minimal Einstein-aether theory could be discarded as pure gauges of the vanishing aether. We prove as a theorem that this holds true for all divergenceless aether fields.

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References (68)

  1. T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64, 024028 (2001).
  2. T. Jacobson, Einstein-aether gravity: A status report, Proc. Sci. QG-PH2007 (2007) 020. [arXiv:0801.1547].
  3. T. Jacobson, Undoing the twist: The Hořava limit of Einstein-aether theory, Phys. Rev. D 89, 081501 (2014).
  4. D. Mattingly, Modern tests of Lorentz invariance, Living Rev. Relativity 8, 5 (2005).
  5. S. Liberati, Tests of Lorentz invariance: A 2013 update, Classical Quantum Gravity 30, 133001 (2013).
  6. T. Jacobson, Einstein-aether gravity: Theory and observational constraints, arXiv:0711.3822.
  7. J. Oost, S. Mukohyama, and A. Wang, Constraints on Einstein-aether theory after GW170817, Phys. Rev. D 97, 124023 (2018).
  8. B. Z. Foster and T. Jacobson, Post-Newtonian parameters and constraints on Einstein-aether theory, Phys. Rev. D 73, 064015 (2006).
  9. T. Gupta, M. Herrero-Valea, D. Blas, E. Barausse, N. Cornish, K. Yagi, and N. Yunes, New binary pulsar constraints on Einstein-æther theory after GW170817, Classical Quantum Gravity 38, 195003 (2021).
  10. J. W. Elliott, G. D. Moore, and H. Stoica, Constraining the new aether: Gravitational Cerenkov radiation, J. High Energy Phys. 08 (2005) 066.
  11. A. Adam, P. Figueras, T. Jacobson, and T. Wiseman, Rotating black holes in Einstein-aether theory, Classical Quantum Gravity 39, 125001 (2022).
  12. C. Ding, A. Wang, and X. Wang, Charged Einstein-aether black holes and Smarr formula, Phys. Rev. D 92, 084055 (2015).
  13. C. Gao and Y. G. Shen, Static spherically symmetric solution of the Einstein-aether theory, Phys. Rev. D 88, 103508 (2013).
  14. E. Barausse, T. Jacobson, and T. P. Sotiriou, Black holes in Einstein-aether and Horava-Lifshitz gravity, Phys. Rev. D 83, 124043 (2011).
  15. T. Tamaki and U. Miyamoto, Generic features of Einstein-aether black holes, Phys. Rev. D 77, 024026 (2008).
  16. D. Garfinkle, C. Eling, and T. Jacobson, Numerical simulations of gravitational collapse in Einstein-aether theory, Phys. Rev. D 76, 024003 (2007).
  17. C. Eling and T. Jacobson, Black holes in Einstein-aether theory, Classical Quantum Gravity 23, 5643 (2006); 27, 049802(E) (2010).
  18. C. Zhang, X. Zhao, K. Lin, S. Zhang, W. Zhao, and A. Wang, Spherically symmetric static black holes in Einstein-aether theory, Phys. Rev. D 102, 064043 (2020).
  19. M. Gürses and Ç. Şentürk, Gödel-type metrics in Einstein-aether theory II: Nonflat background in arbitrary dimensions, Gen. Relativ. Gravit. 48, 63 (2016).
  20. W. Donnelly and T. Jacobson, Stability of the aether, Phys. Rev. D 82, 081501 (2010).
  21. S. M. Carroll, T. R. Dulaney, M. I. Gresham, and H. Tam, Instabilities in the aether, Phys. Rev. D 79, 065011 (2009).
  22. S. Tsujikawa, C. Zhang, X. Zhao, and A. Wang, Odd-parity stability of black holes in Einstein-aether gravity, Phys. Rev. D 104, 064024 (2021).
  23. T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Growth of structure in theories with a dynamical preferred frame, Phys. Rev. D 77, 084010 (2008).
  24. S. M. Carroll and E. A. Lim, Lorentz-violating vector fields slow the universe down, Phys. Rev. D 70, 123525 (2004).
  25. J. Zuntz, T. G. Zlosnik, F. Bourliot, P. G. Ferreira, and G. D. Starkman, Vector field models of modified gravity and the dark sector, Phys. Rev. D 81, 104015 (2010).
  26. T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Modifying gravity with the aether: An alternative to dark matter, Phys. Rev. D 75, 044017 (2007).
  27. T. Y. Alpin and A. B. Balakin, The Einstein–Maxwell-aether-axion theory: Dynamo-optical anomaly in the electromagnetic response, Int. J. Mod. Phys. D 25, 1650048 (2016).
  28. A. B. Balakin and J. P. S. Lemos, Einstein-aether theory with a Maxwell field: General formalism, Ann. Phys. (Amsterdam) 350, 454 (2014).
  29. C. Bonvin, R. Durrer, P. G. Ferreira, G. Starkman, and T. G. Zlosnik, Generalized Einstein-aether theories and the solar system, Phys. Rev. D 77, 024037 (2008).
  30. M. Gürses and Ç. Şentürk, A modified gravity theory: Null aether, Commun. Theor. Phys. 71, 312 (2019).
  31. A. Chatrabhuti, P. Patcharamaneepakorn, and P. Wongjun, Æ ther field, Casimir energy and stabilization of the extra dimension, J. High Energy Phys. 08 (2009) 019.
  32. S. M. Carroll and H. Tam, Aether compactification, Phys. Rev. D 78, 044047 (2008).
  33. L. Ackerman, S. M. Carroll, and M. B. Wise, Imprints of a primordial preferred direction on the microwave background, Phys. Rev. D 75, 083502 (2007); 80, 069901(E) (2009).
  34. A. Einstein, Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin) 142 (1917).
  35. S. Perlmutter et al. (Supernova Cosmology Project Collaboration), Measurements of Ω and Λ from 42 high redshift supernovae, Astrophys. J. 517, 565 (1999).
  36. A. G. Riess et al. (Supernova Search Team), Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998).
  37. J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999); Adv. Theor. Math. Phys. 2, 231 (1998).
  38. J. D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: An example from three-dimensional gravity, Commun. Math. Phys. 104, 207 (1986).
  39. A. Aurilia, H. Nicolai, and P. K. Townsend, Hidden constants: The theta parameter of QCD and the cosmological constant of N=8 supergravity, Nucl. Phys. B176, 509 (1980).
  40. M. J. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. 94B, 179 (1980).
  41. M. Henneaux and C. Teitelboim, The cosmological constant as a canonical variable, Phys. Lett. 143B, 415 (1984).
  42. M. Henneaux and C. Teitelboim, Asymptotically anti-de Sitter spaces, Commun. Math. Phys. 98, 391 (1985).
  43. C. Teitelboim, The cosmological constant as a thermodynamic black hole parameter, Phys. Lett. 158B, 293 (1985).
  44. M. Henneaux and C. Teitelboim, The cosmological constant and general covariance, Phys. Lett. B 222, 195 (1989).
  45. D. Chernyavsky and K. Hajian, Cosmological constant is a conserved charge, Classical Quantum Gravity 35, 125012 (2018).
  46. K. Hajian, H. Özşahin, and B. Tekin, First law of black hole thermodynamics and Smarr formula with a cosmological constant, Phys. Rev. D 104, 044024 (2021).
  47. K. Hajian and B. Tekin, Coupling constants as conserved charges in black hole thermodynamics, Phys. Rev. Lett. 132, 191401 (2024).
  48. B. Withers, Einstein-aether as a quantum effective field theory, Classical Quantum Gravity 26, 225009 (2009).
  49. M. Gürses, Ç. Şentürk, and B. Tekin, Minimal Einstein-aether theory, Eur. Phys. J. C 84, 945 (2024).
  50. E. Franzin, S. Liberati, and J. Mazza, Kerr black hole in Einstein–æther gravity, Phys. Rev. D 109, 084028 (2024).
  51. J. Lee and R. M. Wald, Local symmetries and constraints, J. Math. Phys. (N.Y.), 31, 725 (1990).
  52. R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993).
  53. V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994).
  54. R. M. Wald and A. Zoupas, A general definition of “conserved quantities” in general relativity and other theories of gravity, Phys. Rev. D 61, 084027 (2000).
  55. A. Ashtekar, L. Bombelli, and R. Koul, Phase space formulation of general relativity without a 3+1 splitting, Lect. Notes Phys. 278, 356 (1987).
  56. A. Ashtekar, L. Bombelli, and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Mechanics, Analysis and Geometry: 200 Years after Lagrange, edited by M. Francaviglia (1990), pp. 417–450.
  57. C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three Hundred Years of Gravitation, edited by S. W. Hawking and W. Israel (1987), pp. 676–684.
  58. K. Hajian and M. M. Sheikh-Jabbari, Solution phase space and conserved charges: A general formulation for charges associated with exact symmetries, Phys. Rev. D 93, 044074 (2016).
  59. K. Hajian, On thermodynamics and phase space of near horizon extremal geometries, Ph.D. thesis, 2015, arXiv:1508.03494.
  60. M. J. Duncan and L. G. Jensen, Four forms and the vanishing of the cosmological constant, Nucl. Phys. B336, 100 (1990).
  61. Zh. Wu, The cosmological constant is probably zero, and a proof is possibly right, Phys. Lett. B 659, 891 (2008).
  62. Roy P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11, 237 (1963).
  63. E. T. Newman and A. I. Janis, Note on the Kerr spinning particle metric, J. Math. Phys. (N.Y.) 6, 915 (1965).
  64. E T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence, Metric of a rotating, charged mass, J. Math. Phys. (N.Y.) 6, 918 (1965).
  65. B. Carter, Hamilton-Jacobi and Schrodinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10, 280 (1968).
  66. B. Carter, The commutation property of a stationary, axisymmetric system, Commun. Math. Phys. 17, 233 (1970).
  67. B. Carter, in Les Astre Occlus, Proceedings of 1972 Les Houches Summer School (2nd ed.), edited by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973).
  68. K. Hajian, Conserved charges and first law of thermodynamics for Kerr–de Sitter black holes, Gen. Relativ. Gravit. 48, 114 (2016).

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Phys. Rev. D 111, 124028– Published 18 June, 2025
  • Received 18 April 2025
  • Accepted 8 June 2025
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