Gravitational field of a homogeneous cube. Classical and relativistic case

V. N. Makarov ; L. A. Shleiger ; A.A Karasev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 2, pp. 302-323.

Résumé

The problem of studying the gravitational field of cube-shaped bodies is of great interest to geophysics, astrophysics, mathematical physics, and other fields. The first part of the article presents a brief literary overview of various methods for calculating the gravitational field potential of a homogeneous cube within the framework of classical mechanics: obtaining an analytical solution; as a special case of the problem of finding the gravitational field of a polyhedron; by the finite element method; multipole decomposition. The method of calculating the gravitational field potential of a homogeneous cube using an analytical solution and multipole decomposition is analyzed in more detail. The second part of the article describes the relativistic case of the gravitational field of a homogeneous cube within the framework of post-Newtonian formalism in the first and second approximations. To solve the problem, a physical model was chosen that involved a balanced coordinate cube filled with an incompressible liquid with zero velocity and constant density. Relativistic corrections for the time and spatial coordinates are obtained. A precise analytical expression for these corrections in the region outside the cube, together with the components of the metric tensor, are obtained. A brief comparison of the results obtained for the relativistic case with the results of the classical Newtonian case is provided. The solution is obtained using numerical methods for the region inside the cube. The results obtained determine, with sufficient accuracy, the gravitational field parameters for a homogeneous cube considered in the framework of the relativistic approach.

Détail

Keywords: homogeneous cube, gravitational field, gravitational field potential, Newtonian mechanics, Post-Newtonian approximation

Bibliographie

[1] Haug E. G., Spavieri G., New exact solution to Einsteins field equation gives a new cosmological model, 2023 hal: hal-04286328 | DOI

[2] Stephani H., Kramer D., MacCallum M., et al., Exact solutions of Einstein's field equations, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge, 2009, xxix+701 pp. | DOI | Zbl

[3] Watanabe S., Hirabayashi M., Hirata N., et al., “Hayabusa2 arrives at the carbonaceous asteroid 162173 Ryugu — A spinning top-shaped rubble pile”, Science, 364:6437 (2019), 268–272 | DOI

[4] Domokos G., Sipos A. Á., Szabó Gy. M., Várkonyi P. L., “Formation of sharp edges and planar areas of asteroids by polyhedral abrasion”, Astrophys. J., 699:1 (2009), L13 | DOI

[5] Gibney E., “Freefall space cubes are test for gravitational wave spotter”, Nature, 527:7578 (2015), 284–285 | DOI

[6] Liu X., Baoyin H., Ma X., “Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube”, Astrophys. Space Sci., 2:333 (2011), 409–418 | DOI | Zbl

[7] Liu X., Baoyin H., Ma X., “Periodic orbits in the gravity field of a fixed homogeneous cube”, Astrophys. Space Sci., 2:334 (2011), 357–364 | DOI | Zbl

[8] Park R. S., Werner R. A., Bhaskaran S., “Estimating small-body gravity field from shape model and navigation data”, J. Guid. Control Dyn., 33:1 (2010), 212–221 | DOI

[9] Mufti I. R., “Iterative gravity modeling by using cubical blocks”, Geophys. Prospect., 23:1 (1975), 163–198 | DOI

[10] Fukushima T., “Mosaic tile model to compute gravitational field for infinitely thin non-axisymmetric objects and its application to preliminary analysis of gravitational field of M74”, Mon. Not. R. Astron. Soc., 459:4 (2016), 3825–3860 | DOI

[11] Sanny J., Smith D., “How spherical is a cube (Gravitationally)?”, Phys. Teacher, 53:2 (2015), 111–113 | DOI

[12] Lira J. A., “If the Earth were a cube, what would be the value of the acceleration of gravity at the center of each face?”, Phys. Educ., 53:6 (2018), 065013 | DOI

[13] Chappell J. M., Chappell M. J., Iqbal A., Abbott D., “The gravity field of a cube”, Physics International, 3:2 (2013), 50–57 | DOI

[14] Seidov Z. F., Skvirsky P. I., Gravitational potential and energy of homogeneous rectangular parallelepiped, 2000, arXiv: astro-ph/0002496 | DOI

[15] Bessel F. W., “Auszug aus einem Schreiben des Herrn Prof. Bessel an den Herausgeber”, Astron. Nachr., 2:32 (1823), 133–136 (In German)

[16] Monatliche Correspondenz zur Beförderung der Erd- und Himmelskunde / hrsg. vom Freyherrn von Zach, Band 24, November 1811, Becker, Gotha, 1811, 425–522 (In German) https://zs.thulb.uni-jena.de/receive/jportal_jpvolume_00203228

[17] Everest G., An Account of the Measurement of an Arc of the Meridian Between the Parallels of 18° 3' and 24° 7': Being a Continuation of the Grand Meridional Arc of India as Detailed by the Late Lieut.–Col. Lambton in the Volumes of the Asiatic Society of Calcutta, J.L. Cox, London, 1830, xii+337 pp. https://catalog.hathitrust.org/Record/012336511

[18] Haáz I. B., “Relations between the potential of the attraction of the mass contained in a finite rectangular prism and its first and second derivatives”, Geofizikai Közlemények, 2:7 (1953), 57–66 (In Hungarian) http://epa.niif.hu/02900/02941/00002/pdf/EPA02941_geofizikai_kozlemenyek_1953_02_057-066.pdf

[19] Mader K., Das Newtonsche Raumpotential prismatischer Körper u. seine Ableitungen bis zur dritten Ordnung, Sonderheft 11 der Österreichischen Zeitschrift für Vermessungswesen, Österr. Verein f. Vermessungswesen, Wien, 1951, 74 pp. (In German) https://www.ovg.at/static/vgi-sonderhefte/sonderheft1951_11_final_OCR.pdf

[20] Nagy D., “The gravitational attraction of a right rectangular prism”, Geophys., 31:2 (1966), 362–371 | DOI

[21] Botezatu R., Visarion M., Scurtu F., Cucu G., “Approximation of the gravitational attraction of geological bodies”, Geophys. Prospect., 19:2 (1971), 218–227 | DOI

[22] Mufti I. R., “Rapid determination of cube's gravity field”, Geophys. Prospect., 21:4 (1973), 724–735 | DOI

[23] Banerjee B., Das Gupta S. P., “Gravitational attraction of a rectangular parallelepiped”, Geophys., 42:5 (1977), 1053–1055 | DOI

[24] Waldvogel J., “The Newtonian potential of a homogeneous cube”, ZAMP, 27:6 (1976), 867–871 | DOI | Zbl

[25] Conway J. T., “Analytical solution from vector potentials for the gravitational field of a general polyhedron”, Celest. Mech. Dyn. Astron., 121 (2015), 17–38 | DOI | Zbl

[26] Barnett C. T., “Theoretical modeling of the magnetic and gravitational fields of an arbitrarily shaped three-dimensional body”, Geophys., 41:6 (1976), 1353–1364 | DOI

[27] Jessop C., Duncan M., Chau W. Y., “Multigrid methods for n-body gravitational systems”, J. Comput. Phys., 115:2 (1994), 339–351 | DOI | Zbl

[28] MacMillan W. D., Theoretical Mechanics. II: The Theory of Potential, McGraw-Hill, New York, London, 1930, xiii+469 pp. | Zbl

[29] Fock V. A., The Theory of Space, Time and Gravitation, Pergamon Press, New York, 1963, 411 pp. | Zbl

[30] O'Leary J., Barriot J. P., “Reconstructing the cruise-phase trajectory of deep-space probes in a general relativistic framework: An application to the Cassini gravitational wave experiment”, Astrodyn., 7:3 (2023), 301–314 | DOI

[31] Denisov V. I., Umnov A. N., “Parametrized post-Newtonian formalism for nonmetric theories of gravitation”, Theoret. Math. Phys., 96:1 (1993), 827–836 | DOI | MR | Zbl

[32] Zhu Y., Equations and Analytical Tools in Mathematical Physics. A Concise Introduction, Springer, Singapore, 2021, xii+252 pp. | DOI | Zbl

[33] Bagapsh A. O., “The Poisson integral and Green's function for one strongly elliptic system of equations in a circle and an ellipse”, Comput. Math. Math. Phys., 56:12 (2016), 2035–2042 | DOI | DOI | Zbl

[34] Kondrat'ev B. P., Teoriia potentsiala. Novye metody i zadachi s resheniiami [Theory of Potential. New Methods and Problems with Solutions], Mir, Moscow, 2007, 512 pp. (In Russian)