Timelike and Null Geodesics in the Schwarzschild Space-time: Analytical Solutions

The theory of Schwarzschild geodesics is revisited. Using a theorem due to Weierstrass and Biermann, we derive concise formulas describing all timelike and null trajectories in terms of Weierstrass elliptic functions. The formulation given in this note uses an analogue of the so-called Mino time.

[1]  P. Mach,et al.  Equatorial Accretion on the Kerr Black Hole , 2023, Acta Physica Polonica B Proceedings Supplement.

[2]  A. Odrzywołek,et al.  Accretion of the relativistic Vlasov gas in the equatorial plane of the Kerr black hole , 2022, Physical Review D.

[3]  P. Mach,et al.  Revisiting timelike and null geodesics in the Schwarzschild spacetime: general expressions in terms of Weierstrass elliptic functions , 2022, Classical and Quantum Gravity.

[4]  A. Odrzywołek,et al.  Accretion of the Relativistic Vlasov Gas onto a Moving Schwarzschild Black Hole: Low-temperature Limit and Numerical Aspects , 2022, Acta Physica Polonica B Proceedings Supplement.

[5]  O. Sarbach,et al.  Kinetic Gas Disks Surrounding Schwarzschild Black Holes , 2022, Acta Physica Polonica B Proceedings Supplement.

[6]  O. Sarbach,et al.  Accretion of a Vlasov gas onto a black hole from a sphere of finite radius and the role of angular momentum , 2021, Physical Review D.

[7]  P. Mach,et al.  Accretion of Dark Matter onto a Moving Schwarzschild Black Hole: An Exact Solution. , 2021, Physical review letters.

[8]  P. Mach,et al.  Accretion of the relativistic Vlasov gas onto a moving Schwarzschild black hole: Exact solutions , 2020, Physical Review D.

[9]  P. Mach,et al.  Accretion of the Vlasov gas on Reissner-Nordström black holes , 2020, Physical Review D.

[10]  R. Sarpong,et al.  Bio-inspired synthesis of xishacorenes A, B, and C, and a new congener from fuscol† †Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sc02572c , 2019, Chemical science.

[11]  O. Sarbach,et al.  Phase space mixing in the equatorial plane of a Kerr black hole , 2018, Physical Review D.

[12]  O. Sarbach,et al.  Spherical steady-state accretion of a relativistic collisionless gas into a Schwarzschild black hole , 2017, 1701.07104.

[13]  L. Andersson,et al.  Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime , 2016, 1612.09304.

[14]  O. Sarbach,et al.  Accretion of a relativistic, collisionless kinetic gas into a Schwarzschild black hole , 2016, 1611.02389.

[15]  U. Kostic.  Analytical time-like geodesics in Schwarzschild space-time , 2012, 1201.5611.

[16]  G. Scharf.  Schwarzschild Geodesics in Terms of Elliptic Functions and the Related Red Shift , 2011, 1101.1207.

[17]  Y. Mino.  Perturbative approach to an orbital evolution around a supermassive black hole , 2003, gr-qc/0302075.

[18]  M. J. Reynolds.  An exact solution in non-linear oscillations , 1989 .

[19]  Subrahmanyan Chandrasekhar,et al.  The Mathematical Theory of Black Holes , 1983 .

[20]  Guilelmus Gustavus Adolphus Biermann.  Problemata quaedam mechanica functionum ellipticarum ope soluta , 1865 .

[21]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[22]  G Greenhill,et al.  The applications of elliptic functions , 2007 .

[23]  Paul F. Byrd,et al.  Handbook of elliptic integrals for engineers and scientists , 1971 .

[24]  Y. Hagihara.  Theory of the relativistic trajectories in a gravitational field of Schwarzsc hild , 1931 .