# Exact Solution for Relativistic Trajectories Using Modal Transseries

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Equation of the Orbit

## 3. Proposed Methodology

#### 3.1. Analysis of Convergence

**Theorem**

**1.**

## 4. Scattering and Bound Precessing Orbits

#### 4.1. Scattering Orbits

#### 4.2. Bounded Precessing Orbits

#### 4.3. Raabe–Duhamel’s Convergence Test

**Theorem**

**2.**

**Proof.**

## 5. Simulation Results and Applications

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Fields of Transseries and Modal Transseries

**Definition**

**A1**

**Lemma**

**A1.**

**Proof.**

**Definition**

**A2**

**Definition**

**A3**

## References

- Will, C.M. The confrontation between General Relativity and experiment. Living Rev. Relativ.
**2014**, 17, 4. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Weinberg, S. Cosmology, 1st ed.; Oxford University Press: Oxford, UK, 2008. [Google Scholar]
- Everitt, C.W.F.; DeBra, D.B.; Parkinson, B.W.; Turneaure, J.; Conklin, J.W.; Heifetz, M.I.; Keiser, G.M.; Silbergleit, A.; Holmes, T.J.; Kolodziejczak, J.; et al. Gravity Probe B: Final Results of a Space Experiment to Test General Relativity. Phys. Rev. Lett.
**2011**, 106, 221101–221105. [Google Scholar] [CrossRef] [PubMed] - Acedo, L. Modal series expansions for plane gravitational waves. Gravit. Cosmol.
**2016**, 22, 251–257. [Google Scholar] [CrossRef] [Green Version] - Baumgarte, T.W.; Shapiro, S.L. Numerical Relativity: Solving Einstein’s Equations on the Computer, 1st ed.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Galdi, G.P. An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady State Problems, 2nd ed.; Springer Science + Business Media: New York, NY, USA, 2011. [Google Scholar]
- Peskin, M.E.; Schroeder, D.V. An Introduction to Quantum Field Theory, 1st ed.; Addison-Wesley Publishing Company: Reading, MA, USA, 1995. [Google Scholar]
- Enns, R.H.; McGuire, G.C. Nonlinear Physics with Mathematica for Scientists and Engineers, 1st ed.; Birkhäuser: Boston, MA, USA, 2001. [Google Scholar]
- Hull, J.C. Options, Futures, and Other Derivatives, 7th ed.; Pearson Prentice Hall: Upper Saddle River, NJ, USA, 2009. [Google Scholar]
- FitzHugh, R. Mathematical models of excitation and propagation in nerve. In Biological Engineering; Schwan, H., Ed.; McGraw-Hill Book Co.: New York, NY, USA, 1969; Chapter 1; pp. 1–85. [Google Scholar]
- Nagumo, J.; Arimoto, S.; Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proc. IEEE
**1962**, 50, 2061–2070. [Google Scholar] [CrossRef] - Liao, S. Beyond Perturbation: Introduction to the Homotopy Analysis Method; Chapman & Hall/CRC: Boca Raton, FL, USA, 2004. [Google Scholar]
- Apostol, T.M. Calculus, Volume II, 2nd ed.; John Wiley & Sons: River Street Hoboken, NJ, USA, 1969. [Google Scholar]
- Abramowitz, M.; Stegun, I. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Books on Mathematics), 10th ed.; Dover Publications: Mineola, NY, USA, 1972. [Google Scholar]
- Dahn, B.; Göring, P. Notes on exponential logarithmic terms. Fundam. Math.
**1987**, 127, 45–50. [Google Scholar] [CrossRef] [Green Version] - Écalle, J. Six lectures on transseries, analysable functions and the constructive proof of Dulac’s conjecture. Bifurc. Period. Orbits Vector Fields
**1993**, 408, 75–184. [Google Scholar] - Edgar, G.A. Transseries for beginners. Real Anal. Exch.
**2010**, 35, 253–309. [Google Scholar] [CrossRef] - van der Hoeven, J. Transseries and Real Differential Algebra, 1st ed.; Springer: Berlin, Germany, 2006. [Google Scholar]
- Aschenbrenner, M.; van den Dries, L.; van der Hoeven, J. Asymptotic Differential Algebra and Model Theory of Transseries. arXiv
**2015**, arXiv:1509.02588v6. [Google Scholar] - Stewart, I.N. Galois Theory, 4th ed.; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2015. [Google Scholar]
- Acedo, L.; González-Parra, G.; Arenas, A.J. An exact global solution for the classical SIRS epidemic model. Nonlinear Anal. Real World Appl.
**2010**, 11, 1819–1825. [Google Scholar] [CrossRef] - Acedo, L.; González-Parra, G.; Arenas, A.J. Modal series solution for an epidemic model. Physica A
**2010**, 389, 1151–1157. [Google Scholar] [CrossRef] - González-Parra, G.; Acedo, L.; Arenas, A.J. Accuracy of analytical-numerical solutions of the Michaelis-Menten equation. Comput. Appl. Math.
**2011**, 30, 1–17. [Google Scholar] [CrossRef] [Green Version] - González-Parra, G.; Acedo, L.; Arenas, A.J. A novel approach to obtain analytical-numerical solutions of nonlinear Lorenz system. Numer. Algorithms
**2014**, 67, 93–107. [Google Scholar] [CrossRef] - Chandrasekhar, S. The Mathematical Theory of Black Holes, 1st ed.; Oxford University Press: Oxford, UK, 1983. [Google Scholar]
- Hobson, M.P.; Efstathiou, G.P.; Lasenby, A.N. General Relativity: An Introduction for Physicists, 1st ed.; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Bromwich, T.J.l. An Introduction to the Theory of Infinite Series, 1st ed.; MacMillan and Co., Limited: London, UK, 1908. [Google Scholar]
- Roseveare, N.T. Mercury’s Perihelion from Le Verrier to Einstein, 1st ed.; Oxford University Press: Oxford, UK, 1982. [Google Scholar]
- Danby, J.M.A. Fundamentals of Celestial Mechanics, 2nd ed.; Willmann-Bell, Inc.: Richmond, VA, USA, 1988. [Google Scholar]
- Williams, D.R. Mercury Fact Sheet. 2016. Available online: http://nssdc.gsfc.nasa.gov/planetary/factsheet/mercuryfact.html (accessed on 8 June 2016).
- Saca, J.M. An exact solution to the relativistic advance of perihelion: Correcting the Einstein approximation. Astrophys. Space Sci.
**2008**, 315, 365. [Google Scholar] [CrossRef] - Debnath, L. The Legacy of Leonhard Euler: A Tricentennial Tribute, 1st ed.; Imperial College Press: London, UK, 2010. [Google Scholar]
- Sundman, K. Mèmoire sur le problème des trois corps. Acta Math.
**1912**, 36, 105–179. [Google Scholar] [CrossRef] - Wang, Q. The global solution of the n-body problem. Celest. Mech.
**1991**, 50, 73–88. [Google Scholar] - Dahn, B. The limit behaviour of exponential terms. Fundam. Math.
**1984**, 124, 169–186. [Google Scholar] [CrossRef] - Geddes, K.; Gonnet, G. A New Algorithm for Computing Symbolic Limits Using Hierarchical Series. Lect. Notes Comput. Sci.
**1988**, 358, 490–495. [Google Scholar]

**Figure 1.**Two examples of spiral orbits in the Schwarzschild spacetime for $\alpha =1/5$, $\beta =1$. The inner orbit corresponds to ${u}_{1}=-5$ and the outer one to ${u}_{1}=-10$.

**Figure 2.**A scattering orbit obtained for $\alpha =1/5$ and $\beta =1/100$ by summing 1000 terms in the series of Equation (20).

**Figure 3.**A bounded precessing orbit for $\alpha =0.05$ and $\beta =0.1$. Perihelion advance is $\simeq 1.82284$ sexagesimal degrees per revolution.

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**MDPI and ACS Style**

Acedo, L.; Arenas, A.J.; De La Espriella, N.
Exact Solution for Relativistic Trajectories Using Modal Transseries. *Symmetry* **2020**, *12*, 1505.
https://doi.org/10.3390/sym12091505

**AMA Style**

Acedo L, Arenas AJ, De La Espriella N.
Exact Solution for Relativistic Trajectories Using Modal Transseries. *Symmetry*. 2020; 12(9):1505.
https://doi.org/10.3390/sym12091505

**Chicago/Turabian Style**

Acedo, Luis, Abraham J. Arenas, and Nicolas De La Espriella.
2020. "Exact Solution for Relativistic Trajectories Using Modal Transseries" *Symmetry* 12, no. 9: 1505.
https://doi.org/10.3390/sym12091505