Vlasov–Maxwell–Einstein-type equations and their consequences. Applications to astrophysical problems

V. V. Vedenyapin   ; N. N. Fimin   ; V. M. Chechetkin

Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 2, pp. 258-279

Résumé

We consider a method for obtaining equations of the Hamiltonian dynamics for system of interacting massive charged particles using the general relativistic Einstein–Hilbert action. In the general relativistic case, Vlasov-type equations are derived in the nonrelativistic and weakly relativistic limits. Expressions are proposed for corrections to the Poisson equation, which can contribute to the effective action of dark matter and dark energy. In this case, an efficient approach to synchronizing the proper times of different particles of a many-particle system is proposed. Based on the obtained expressions for the action, we analyze the possibility of a composite structure of the cosmological term in the Einstein equations. Reduced Euler equations leading to the Milne–McCrea cosmological model are derived using a hydrodynamic substitution and are solved in the self-similar class.

Détail

Mots-clés : Lorentz action
Keywords: Lagrangian, Einstein–Hilbert action, Vlasov–Einstein equation.

Bibliographie

[1] V. Pauli, Teoriya otnositelnosti, Nauka, M., 1983 | MR | MR | Zbl

[2] V. A. Fok, Teoriya prostranstva, vremeni i tyagoteniya, Gostekhizdat, M., 1955 | MR | MR | Zbl

[3] S. Veinberg, Gravitatsiya i kosmologiya. Printsipy i prilozheniya obschei teorii otnositelnosti, Mir, M., 1975

[4] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. 2, Teoriya polya, Fizmatlit, M., 1988 | MR

[5] B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Sovremennaya geometriya. Metody i prilozheniya, Nauka, M., 1986 | MR | Zbl

[6] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Univ. Press, Oxford, 2009 | MR

[7] C. Cercignani, G. M. Kremer, The Relativistic Boltzmann Equation: Theory an Applications, Progress in Mathematical Physics, 22, Birkhäuser, Basel, 2002 | DOI | MR

[8] J. Ehlers, “General relativity and kinetic theory”, General Relativity and Cosmology, Proceedings of the International School of Physics “Enrico Fermi”. Course XLVII (Italian Physical Society, June 30 – July 12, 1969), ed. R. K. Sachs, Academic Press, New York, 1971, 1–70 | MR

[9] L. Andersson, M. Korzyński, Variational principle for the Einstein–Vlasov equations, arXiv: 1910.12152

[10] R. W. Lindquist, “Relativistic transport theory”, Ann. Phys., 37:3 (1966), 487–518 | DOI

[11] A. A. Vlasov, Statisticheskie funktsii raspredeleniya, Nauka, M., 1966 | MR

[12] N. A. Chernikov, “Kineticheskoe uravnenie dlya relyativistskogo gaza v proizvolnom gravitatsionnom pole”, Dokl. AN SSSR, 144:1 (1962), 89–92 | MR | Zbl

[13] Yu. L. Klimontovich, “Relyativistskie kineticheskie uravneniya dlya plazmy”, ZhETF, 37:3 (1959), 735–746 | Zbl

[14] Y. Choquet-Bruhat, Introduction to General Relativity, Black Holes, and Cosmology, Oxford Univ. Press, Oxford, 2015 | MR

[15] Yu. G. Ignatev, Relyativistskaya kineticheskaya teoriya neravnovesnykh protsessov, OOO “Foliant”', Kazan, 2010

[16] V. V. Vedenyapin, V. I. Parenkina, S. R. Svirschevskii, “O vyvode uravnenii elektrodinamiki i gravitatsii iz printsipa naimenshego deistviya”, Zh. vychisl. matem. i matem. fiz., 62:6 (2022), 1016–1029 | DOI | DOI | MR

[17] V. V. Vedenyapin, V. I. Parenkina, A. G. Petrov, Chzhan Khaochen, “Uravnenie Vlasova–\allowbreakEinshteina i tochki Lagranzha”, Preprinty IPM im. M. V. Keldysha, 23, 2022 | DOI

[18] V. V. Vedenyapin, N. N. Fimin, V. M. Chechetkin, “K voprosu o vyvode uravneniya Vlasova–\allowbreakMaksvella–\allowbreakEinshteina i ego svyaz s kosmologicheskim lyambda-chlenom”, Vestn. Mosk. gos. obl. un-ta. Ser. Fiz.-matem., 2019, no. 2, 24–48 | DOI

[19] V. V. Vedenyapin, “Uravnenie Vlasova–Maksvella–Einshteina”, Preprinty IPM im. M. V. Keldysha, 188, 2018 | DOI

[20] V. V. Vedenyapin, M. A. Negmatov, “O vyvode i klassifikatsii uravnenii tipa uravneniya Vlasova i magnitnoi gidrodinamiki. Tozhdestvo Lagranzha i forma Godunova”, TMF, 170:3 (2012), 468–480 | DOI | DOI | MR

[21] V. V. Vedenyapin, M. A. Negmatov, N. N. Fimin, “Uravneniya tipa Vlasova i Liuvillya, ikh mikroskopicheskie, energeticheskie i gidrodinamicheskie sledstviya”, Izv. RAN. Ser. matem., 81:3 (2017), 45–82 | DOI | DOI | MR

[22] V. V. Vedenyapin, M. A. Negmatov, “O vyvode i klassifikatsii uravnenii tipa Vlasova i magnitnoi gidrodinamiki. Tozhdestvo Lagranzha, forma Godunova i kriticheskaya massa”, Trudy Shestoi Mezhdunarodnoi konferentsii po differentsialnym i funktsionalno-differentsialnym uravneniyam (Moskva, 14–21 avgusta, 2011). Chast 3, SMFN, 47, RUDN, M., 2013, 5–17 | DOI

[23] V. V. Vedenyapin, M. A. Negmatov, “O topologii statsionarnykh reshenii gidrodinamicheskikh i vikhrevykh sledstvii uravneniya Vlasova i metod Gamiltona–Yakobi”, Dokl. RAN, 449:5 (2013), 521–526 | DOI | DOI

[24] V. V. Vedenyapin, M. Yu. Voronina, A. A. Russkov, “O vyvode uravnenii elektrodinamiki i gravitatsii iz printsipa naimenshego deistviya”, Dokl. RAN. Ser. Fizika, tekhn. nauki, 495:1 (2020), 9–13 | DOI | DOI

[25] G. Rein, A. D. Rendall, “Smooth static solutions of the spherically symmetric Vlasov–Einstein system”, Ann. Inst. H. Poincaré Phys. Théor., 59:4 (1993), 383–397 | MR

[26] T. Okabe, P. J. Morrison, J. E. Friedrichsen, L. C. Shepley, “Hamiltonian dynamics of spatially-homogeneous Vlasov–Einstein systems”, Phys. Rev. D, 84:2 (2011), 024001, 11 pp. | DOI

[27] H. E. Kandrup, P. J. Morrison, “Hamiltonian structure of the Vlasov–Einstein system and the problem of stability for spherical relativistic star clusters”, Ann. Phys., 225:1 (1993), 114–166 | DOI | MR

[28] V. V. Vedenyapin, N. N. Fimin, V. M. Chechetkin, “The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equations system”, Eur. Phys. J. Plus, 136:6 (2021), 670, 11 pp. | DOI

[29] V. Vedenyapin, N. Fimin, V. Chechetkin, “The system of Vlasov–Maxwell–Einstein-type equations and its nonrelativistic and weak relativistic limits”, Internat. J. Modern Phys. D, 29:1 (2020), 2050006, 23 pp. | DOI | MR