Abstract
The motion of a particle in a static, spherically symmetric gravitational field is investigated in Euclidean space. The gravitational effects are described as due to a scalar field: To every point in space there is assigned a refractive index deciding the velocity of light in that point. The motion of light in the vacuum is described by the equation of classical optics. An equation of motion for material test particles is then derived by employing the usual Lagrangian formalism. The motion of the planets around the sun is explained, in particular the perihelion motion of Mercury. The present theory fully explains the four “classical” tests of general relativity in a mathematically far simpler way, and it can be equivalent to the Schwarzschild solution. It is also found that the effect of gravitation depends on the velocity of the particle, becoming repulsive for radial velocities larger thanc/\(\sqrt 3 \) (c is the velocity of light). This seemingly odd result can also be obtained from the equations of general relativity, as was shown by Cavalleri and Spinelli.
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References and Footnotes
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In order to avoid any misunderstandings, it should be said that the literature on gravitation in flat space is rather vast; cf., e.g., N. Rosen,Phys. Rev. 57, 147 (1940); W. E. Thirring,Ann. Phys. Leipzig 16, 96 (1961); W. Petry,Gen. Relativ. Gravit. 13, 865 (1981); E. R. Bagge,Kerntechnik 39, 223 (1981);40, 47 (1982); and especially the review article by G. Cavalleri and G. Spinelli,Nuovo Cimento 3, (8), 1–92 (1980). The approaches adopted by the cited authors are however rather different from the one adopted in the present paper.
A. Maréchal, “Optique géométrique générale,” inHandbuch der Physik, Vol. 34, S. Flügge, ed. (Springer, Berlin, 1956), p. 44; M. Born and E. Wolf,Principles of Optics, 6th edn. (Pergamon, New York, 1980), § 3.2.
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An equivalent one can be found in, e.g., R. Adler, M. Bazin, and M. Schiffer,Introduction to General Relativity (McGraw-Hill, New York, 1965), § 6.1.
G. Cavalleri and G. Spinelli,Lett. Nuovo Cimento 6, 5 (1973).
G. Cavalleri and G. Spinelli [13] call the velocityω used in our paperunrenormalized. They also define thesemi-renormalized velocity 556-1 =ω/Φ(ρ), measured by “ideal” clocks but “affected” material meter rods, and therenormalized velocity 556-2 =ω/Φ(ρ)Ω(ρ), which is measured by “affected” meter rodsand clocks. Whereas for the unrenormalized and the semi-renormalized coordinates the sign of the acceleration of a particle will depend on its velocity, the acceleration of a particle expressed in the renormalized coordinates will always be in the direction of the origin. One could also define theself-measured renormalized velocity 556-3 =ω/Φ(ρ)Ω(ρ, w/c), where the clock used to measure the velocity is moving with the particle. H. E. Ives,J. Opt. Soc. Am. 29, 294 (1939);39, 757 (1949);Proc. Am. Phil. Soc. 25, 125 (1951); T. Sjödin,Indian J. Theor. Phys. 29, 89 (1981). Presupposing, as is customary, that Ω(w/c) = (1 -ω 2/c 2)1/2, we obtain 556-4 =ωγ/Φ(ρ)Ω(ρ). The advantage of this velocity is that its measurement does not presuppose that clocks in different locations have been synchronized. Assuming the validity of the relation (5.21) and that limc(r) = lim Ω(r) = 0 for some r0 ≥ 0, we easily obtainr→r 0+r→r 0+ (cf. Ref. 14) that limω 2 = lim 556-52 = 0, lim 556-62 =c 02 , andr→r 0+r→r 0+r→r 0+ lim 556-72 = ∞.r→r 0+
Let us however note that the fact that the function representing the refractive index is not fixed in the present theory may also be taken as a strength. If, e.g., the sun would have an appreciable quadropole moment—as R. H. Dicke and H. M. Goldenberg claim inPhys. Rev. Lett. 18, 313 (1967)—general relativity would be falsified, but the present theory could probably be saved through a clever choice of the functionρ.
B. Kursunoglu,Phys. Rev. D 9, 2723 (1974); “A non-technical history of the generalized theory of gravitation dedicated to the Albert Einstein Centennial,” inOn the Path of Albert Einstein, A. Perlmutter and L. Scott, eds. (Plenum, New York and London, 1979).
Cf. eg. N. Rosen, “Bimetric theory of gravitation,” inTopics in Theoretical and Experimental Gravitation Physics, V. De Sabbata and J. Weber, eds. (Plenum, New York, 1977). Rosen writes “...a singularity must be regarded as representing a breakdown of the physical law described by the equations.”
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Sjödin, T. A scalar Euclidean theory of gravitation: motion in a static spherically symmetric field. Found Phys Lett 3, 543–556 (1990). https://doi.org/10.1007/BF00666023
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DOI: https://doi.org/10.1007/BF00666023