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Lagrange equations of motion of particles and photons in the Schwarzschild field


Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation

The equations of motion of a particle in the gravitational field of a black hole are considered in a formulation which uses generalized coordinates, velocities and accelerations and is convenient for finding the integrals of motion. The equations are rewritten in terms of the physical velocities and accelerations measured in the Schwarzschild frame by a stationary observer using proper local length and time standards. The attractive force due to the field and the centripetal acceleration of a particle are proportional to the particle's kinetic energy m/√1−v2, consistent with the fact that the particle's kinetic energy and the photon's energy ħω in the field increase by the same amount from their out-of-the-field values. The attraction exerted on particles and photons by the gravitational field source is proportional to their kinetic energies. The particle trajectory in the ultrarelativistic limit v → 1 coincides with the photon trajectory.

Fulltext is available at IOP
Keywords: gravitational field, Schwarzschild's geometry, mass and energy in gravitation
PACS: 03.30.+p
DOI: 10.3367/UFNe.0185.201511h.1229
URL: https://ufn.ru/en/articles/2015/11/g/
Citation: Ritus V I "Lagrange equations of motion of particles and photons in the Schwarzschild field" Phys. Usp. 58 1118–1123 (2015)
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Received: 2nd, July 2015, 22nd, September 2015

Оригинал: Ритус В И «Лагранжевы уравнения движения частиц и фотонов в шварцшильдовском поле» УФН 185 1229–1234 (2015); DOI: 10.3367/UFNr.0185.201511h.1229

References (11) ↓ Cited by (2) Similar articles (20)

  1. Landau L D, Lifshits E M Teoriya Polya (M.: Nauka, 1988); Landau L D, Lifshitz E M The Classical Theory Of Fields (Oxford: Butterworth-Heinemann, 2000)
  2. Weinberg S Gravitation And Cosmology: Principles And Applications Of The General Theory Of Relativity (New York: Wiley, 1972); Vainberg S Gravitatsiya i Kosmologiya: Printsipy i Prilozheniya Obshchei Teorii Otnositel’nosti (M.: Mir, 1975)
  3. Fok V A Izv. AN SSSR Otd. Matem. Estestv. Nauk (4 - 5) 551 (1937); Fock V A Phys. Z. Sowjetunion 12 404 (1937)
  4. Schwinger J Phys. Rev. 82 664 (1951)
  5. Misner C W, Thorne K S, Wheeler J A Gravitation (San Francisco: W.H. Freeman, 1973); Mizner Ch, Torn K, Uiler Dzh Gravitatsiya (M.: Mir, 1977)
  6. Lightman A P, Press W H, Price R N, Teukolsky S A Problem Book In Relativity And Gravitation (Princeton, NJ: Princeton Univ. Press, 1975); Laitman A, Press V, Prais R, Tyukol’ski S Sbornik Zadach Po Teorii Otnositel’nosti i Gravitatsii (M.: Mir, 1979)
  7. Novikov I D, Frolov V P Fizika Chernykh Dyr (M.: Nauka, 1986); Frolov V P, Novikov I D Black Hole Physics. Basic Concepts And New Developments (Dordrecht: Kluwer Acad. Publ., 1998)
  8. Pauli W Relativitätstheorie (Leipzig: Teubner, 1921); Pauli W Theory Of Relativity (New York: Pergamon Press, 1958); Pauli V Teoriya Otnositel’nosti (M.: Nauka, 1983)
  9. Okun’ L B Usp. Fiz. Nauk 158 511 (1989); Okun’ L B Sov. Phys. Usp. 32 629 (1989)
  10. Einstein A Sitzungsber. Königl. Preuß. Akad. Wissenschaft. Berlin 47 (2) 831 (1915)
  11. Weyl H Raum, Zeit, Materie. Vorlesungen über Allgemeine Relativitätstheorie (Berlin: Verlag von Julius Springer, 1923); Veil’ G Prostranstvo, Vremya, Materiya. Lektsii Po Obshchei Teorii Otnositel’nosti (M.: Yanus, 1996)

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