Relativistic Newtonian dynamics

Relativistic Newtonian dynamics (RND) is an extension of Newtonian dynamics that overcomes its shortcomings by considering the influence of potential energy on space and time using some principles of Einstein's theories of special and general relativity. In its current form, it models the motion of objects with non-zero mass as well as massless particles under the attraction of a time independent conservative force in some inertial frame. Unlike general relativity, RND is not restricted solely to a gravitational potential and does not require the exigency of curving spacetime. Created in 2015, the centennial year of Einstein's general relativity, by Israeli scientists Yaakov Friedman[1] in collaboration with Joseph Steiner, RND predicts accurately both classical and modern tests of general relativity, such as, the perihelion precession of Mercury (planet)[2] which agree with the known observed perihelion precession,[3][4][5] the periastron advance of a binary star,[6] which is identical to the post-Keplerian equation[7][8] of the relativistic advance of the periastron in a binary, gravitational lensing [9] which is identical to Einstein's formula for weak gravitational lensing[3][10][5] using GR, and light travel (Shapiro delay) time delay [11] which agree with the known formula for the Shapiro time delay,[3][5] confirmed experimentally by several experiments[12]

Suppose that an object or a particle moves under a conservative, time independent force with a negative potential any space point , vanishing at infinity, in an (idealized) inertial frame . In order to express the influence of this potential energy at this point, introduce a normalized vector in the direction of the gradient of . Using an extension of the Equivalence principle, the influence due to on time intervals, space increments and velocities in the neighbourhood of in RND are quantified via the relativistic length contraction and time dilation due to the escape velocity of the escape orbit originating at . Specifically, the space increments in the direction of and the time intervals are altered by the Lorentz factor due to this influence, while space increments transverse to are not. This implies that the velocity components in the direction of are unaltered when transformed to , but those transverse to should be multiplied by resulting in an upper bound in , lower than the speed of light . This change in the direction of the velocity causes in turn a change in the classical trajectories.

For a central force this modification leads finally to the equations for the trajectory and time dependence

RND trajectory equation under a central force

RND time equation under a central force

in terms of integral of motion and integral of motion .

For motion in the gravitational field a spherically symmetric massive object of mass , the dimensionless gravitational potential is , where is its Schwarzschild radius. In this case the above equations yield the correct formulae for all the above-mentioned tests of general relativity.

References

  1. ↑ Friedman, Y. (2016). "Relativistic Newtonian dynamics under a central force". EPL. 116: 19001 arXiv:1705.04578. arXiv:1705.04578. Bibcode:2016EL....11619001F. doi:10.1209/0295-5075/116/19001.
  2. ↑ Friedman, Y.; Steiner, J. M. (2016). "Predicting Mercury's precession using simple relativistic Newtonian dynamics". EPL. 113: 39001 arXiv:1603.02560. arXiv:1603.02560. Bibcode:2016EL....11339001F. doi:10.1209/0295-5075/113/39001.
  3. 1 2 3 C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation (1973) Freeman and co.
  4. ↑ W. Rindler, Relativity, Special, General and Cosmological (2001) Oxford
  5. 1 2 3 S. Kopeikin, M. Efroimsky and G. Kaplan, Relativistic Celestial Mechanics of the Solar System (2011) Wiley-VCH, Berlin
  6. ↑ Friedman, Y.; Livshitz, S.; Steiner, J. M. (2016). "Predicting the relativistic periastron advance of a binary star without curving spacetime". EPL. 116: 59001 arXiv:1705.05705. arXiv:1705.05705. Bibcode:2016EL....11659001F. doi:10.1209/0295-5075/116/59001.
  7. ↑ Damour, T.; Deruelle, N. (1985). "General relativistic celestial mechanics of binary systems. I. The post-Newtonian motion". Ann. Inst. Henri Poincaré. 43: 107.
  8. ↑ Bagchi, M. (2013). "Periastron advance in neutron star–black hole binaries". Mon. Not. R. Astron. Soc. 428: 1201. arXiv:1210.0633. Bibcode:2013MNRAS.428.1201B. doi:10.1093/mnras/sts103.
  9. ↑ Friedman, Y.; Steiner, J. M. (2017). "Gravitational Deflection in Relativistic Newtonian Dynamics". EPL. 117: 59001 arXiv:1705.06967. arXiv:1705.06967. Bibcode:2017EL....11759001F. doi:10.1209/0295-5075/117/59001.
  10. ↑ Ö. Grön \and H. Sigbjörn, Einstein's General Theory of Relativity: With Modern Applications in Cosmology (2007) Springer
  11. ↑ Friedman, Y. (2017). "Relativistic Newtonian Dynamics for Objects and Particles". EPL. 117: 49003 arXiv:1705.06579. arXiv:1705.06579. Bibcode:2017EL....11749003F. doi:10.1209/0295-5075/117/49003.
  12. ↑ Will, C. M. (2014). "The Confrontation between General Relativity and Experiment". Living Rev. Relativ. 17: 1 .
  • New developments in RND can be found in
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