# A Note on the Gravitoelectromagnetic Analogy

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## Abstract

**:**

## 1. Introduction

## 2. Linear Gravitoelectromagnetic Form of Einstein Equations

## 3. Gravitoelectromagnetic Description of the Motion of Test Masses

## 4. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Note

1 | The spacetime signature is $(-1,1,1,1)$; Greek indices run from 0 to 3, while Latin indices from 1 to 3; boldface symbols, such as $\mathbf{x}$, refer to space vectors. |

## References

- Will, C.M. Theory and Experiment in Gravitational Physics; Cambridge University Press: Cambridge, UK, 2018. [Google Scholar]
- Ruggiero, M.L.; Tartaglia, A. Gravitomagnetic effects. Nuovo Cim.
**2002**, B117, 743–768. [Google Scholar] - Mashhoon, B. Gravitoelectromagnetism: A Brief review. In The Measurement of Gravitomagnetism: A Challenging Enterprise; Iorio, L., Ed.; Nova Science: New York, NY, USA, 2003. [Google Scholar]
- McDonald, K.T. Answer to Question #49. Why c for gravitational waves? Am. J. Phys.
**1997**, 65, 591–592. [Google Scholar] - Iorio, L.; Lichtenegger, H.I.M.; Ruggiero, M.L.; Corda, C. Phenomenology of the Lense-Thirring effect in the Solar System. Astrophys. Space Sci.
**2011**, 331, 351–395. [Google Scholar] [CrossRef][Green Version] - Cattaneo, C. General relativity: Relative standard mass, momentum, energy and gravitational field in a general system of reference. Il Nuovo Cimento (1955–1965)
**1958**, 10, 318–337. [Google Scholar] [CrossRef] - Costa, L.F.O.; Herdeiro, C.A. Gravitoelectromagnetic analogy based on tidal tensors. Phys. Rev. D
**2008**, 78, 024021. [Google Scholar] [CrossRef][Green Version] - Mashhoon, B.; McClune, J.C.; Quevedo, H. Gravitational superenergy tensor. Phys. Lett. A
**1997**, 231, 47–51. [Google Scholar] [CrossRef][Green Version] - Ramos, J.; Mashhoon, B. Helicity-rotation-gravity coupling for gravitational waves. Phys. Rev. D Part. Fields Gravit. Cosmol.
**2006**, 73, 084003. [Google Scholar] [CrossRef][Green Version] - Costa, L.F.O.; Natario, J. Gravito-electromagnetic analogies. Gen. Rel. Grav.
**2014**, 46, 1792. [Google Scholar] [CrossRef][Green Version] - Chicone, C.; Mashhoon, B. The generalized Jacobi equation. Class. Quantum Gravity
**2002**, 19, 4231. [Google Scholar] [CrossRef][Green Version] - Rizzi, G.; Ruggiero, M.L. The relativistic Sagnac effect: Two derivations. In Relativity in Rotating Frames; Rizzi, G., Ruggiero, M.L., Eds.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 179–220. [Google Scholar]
- Jantzen, R.T.; Carini, P.; Bini, D. The many faces of gravitoelectromagnetism. Ann. Phys.
**1992**, 215, 1–50. [Google Scholar] [CrossRef][Green Version] - Lynden-Bell, D.; Nouri-Zonoz, M. Classical monopoles: Newton, NUT space, gravomagnetic lensing, and atomic spectra. Rev. Mod. Phys.
**1998**, 70, 427. [Google Scholar] [CrossRef][Green Version] - Costa, L.F.O.; Natário, J. Frame-Dragging: Meaning, Myths, and Misconceptions. Universe
**2021**, 7, 388. [Google Scholar] [CrossRef] - Straumann, N. General Relativity, With Applications to Astrophysics; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Carroll, S.M. Lecture notes on general relativity. arXiv
**1997**, arXiv:gr-qc/9712019. [Google Scholar] - Mashhoon, B.; Gronwald, F.; Lichtenegger, H.I.M. Gravitomagnetism and the Clock Effect. In Gyros, Clocks, Interferometers...: Testing Relativistic Graviy in Space; Springer: Berlin/Heidelberg, Germany, 2001; pp. 83–108. [Google Scholar]
- Mashhoon, B. Gravitational couplings of intrinsic spin. Class. Quantum Gravity
**2000**, 17, 2399–2409. [Google Scholar] [CrossRef][Green Version] - Padmanabhan, T. Gravitation: Foundations and Frontiers; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Bakopoulos, A.; Kanti, P. From GEM to electromagnetism. Gen. Relativ. Gravit.
**2014**, 46, 1742. [Google Scholar] [CrossRef][Green Version] - Harris, E.G. Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields. Am. J. Phys.
**1991**, 59, 421–425. [Google Scholar] [CrossRef] - Clark, S.J.; Tucker, R.W. Gauge symmetry and gravito-electromagnetism. Class. Quantum Gravity
**2000**, 17, 4125. [Google Scholar] [CrossRef][Green Version] - Pascual-Sanchez, J.F. The Harmonic gauge condition in the gravitomagnetic equations. Nuovo Cim. B
**2000**, 115, 725–732. [Google Scholar] - Bertschinger, E. Cosmological dynamics: Course 1. In Les Houches Summer School on Cosmology and Large Scale Structure (Session 60); Cornell University: Ithaca, NY, USA, 1993; pp. 273–348. [Google Scholar]
- Damour, T.; Soffel, M.; Xu, C. General-relativistic celestial mechanics. I. Method and definition of reference systems. Phys. Rev. D
**1991**, 43, 3273. [Google Scholar] [CrossRef] - Carroll, S.M. Spacetime and Geometry; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Bini, D.; Cherubini, C.; Chicone, C.; Mashhoon, B. Gravitational induction. Class. Quantum Gravity
**2008**, 25, 225014. [Google Scholar] [CrossRef] - Thorne, K.S.; Hartle, J.B. Laws of motion and precession for black holes and other bodies. Phys. Rev. D
**1985**, 31, 1815. [Google Scholar] [CrossRef][Green Version] - Flanagan, E.E.; Hughes, S.A. The basics of gravitational wave theory. New J. Phys.
**2005**, 7, 204. [Google Scholar] [CrossRef] - Ruggiero, M.L. Gravitational waves physics using Fermi coordinates: A new teaching perspective. Am. J. Phys.
**2021**, 89, 639. [Google Scholar] [CrossRef] - Ruggiero, M.L.; Ortolan, A. Gravito-electromagnetic approach for the space-time of a plane gravitational wave. J. Phys. Commun.
**2020**, 4, 055013. [Google Scholar] [CrossRef] - Bini, D.; Geralico, A.; Ortolan, A. Deviation and precession effects in the field of a weak gravitational wave. Phys. Rev. D
**2017**, 95, 104044. [Google Scholar] [CrossRef][Green Version] - Ruggiero, M.L.; Ortolan, A. Gravitomagnetic resonance in the field of a gravitational wave. Phys. Rev. D
**2020**, 102, 101501. [Google Scholar] [CrossRef]

**Figure 1.**The behavior of the particle coordinate parallel to the wave propagation direction; we set $\alpha =\frac{{V}_{0}^{2}}{c}{A}^{+}$.

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Ruggiero, M.L. A Note on the Gravitoelectromagnetic Analogy. *Universe* **2021**, *7*, 451.
https://doi.org/10.3390/universe7110451

**AMA Style**

Ruggiero ML. A Note on the Gravitoelectromagnetic Analogy. *Universe*. 2021; 7(11):451.
https://doi.org/10.3390/universe7110451

**Chicago/Turabian Style**

Ruggiero, Matteo Luca. 2021. "A Note on the Gravitoelectromagnetic Analogy" *Universe* 7, no. 11: 451.
https://doi.org/10.3390/universe7110451