ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

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April, 2007 PROGRESS IN PHYSICS Volume 2 kind, because a non-stationary rotation of a bulky space body (planet, star, galaxy) — the generator of such a field — is a very rare phenomenon in the Universe. In this case the chr.inv.-Lorentz condition doesn’t change to the general formula (38) or for a vortical gravitational field of the “electric” kind, because the condition has no components of the field tensor Fαβ. The field invariants (34, 35) in the case are J1 = 4h im h kn ∗ ∗ ∂Aik ∂Amn , J2 = 0 . (74) ∂t ∂t The chr.inv.-Maxwell-like equations for a vortical gravitational field strictly of the “magnetic” kind are 1 c HikAik = −4πρ ∗ ∇k H ik − 1 c2 FkH ik = 4π c ji ⎫ ⎪⎬ Group I, ⎪⎭ ∗ ∇i H (75) ∗i = 0 ∗ ∗i ∂H ∂t + H∗i ⎫ ⎬ Group II, D = 0 ⎭ (76) which, after substituting for Hik and H∗i , are 1 c Aik ∗ ∂Aik = −2πρ ∂t ∗ 2 ik ∂ A ∂xk∂t +2 ∗ ∂ ∂xk � i∙ A∙nD kn −A k∙ ∙mD im� � + Δ j � 1 jk− Fk × c2 � ∗∂Aik × ∂t + 2 � A i∙ ∙nD kn − A k∙ ∙mD im�� = 2π c ji ⎫ ⎪⎬ ⎪⎭ ∗ 2 ∗i ∂ Ω ∂x (77) i∂t + ∗ ∂ ∂xi � � � ∗∂Ω∗i ∗i Ω D + ∂t + Ω∗i � D Δ j ji = 0 ∗ 2 ∗i ∂ Ω ∂t2 + � ∗ ∂ � � ∗∂Ω∗i ∗i Ω D + ∂t ∂t + Ω∗i ⎫ ⎪⎬ � D D = 0 ⎪⎭ (78) The chr.inv.-continuity equation for such a field, in a deforming inhomogeneous space, is ∗ 2 ∂ ∂xi∂x k � ∗∂A � ik − ∂t 1 c2 Aik ∗ ∂ 2 Aik 1 − ∂t2 c2 � ∗∂Aik ∂t +Aik � D × ∗ ∗ 2 ∂Aik ∂ × + 2 ∂t ∂xi∂x k � i∙ A∙nD kn −A k∙ ∙mD im� � ∗∂Aik + ∂t + + 2 � A i∙ ∙nD kn − A k∙ ∙mD im�� ∗ j ∂Δjk ∂xi 1 − c2 ∗ ∂Fk + ∂xi � + Δ j �� 1 jk − Fk Δ c2 l li − 1 � Fi c2 � = 0 . (79) If the space is homogeneous and free of deformation, the continuity equation becomes ∗ 2 ∂ ∂xi∂xk �∗∂A � ik − ∂t 1 c2 Aik ∗∂2Aik ∂t2 − − 1 c2 � ∗∂Aik ∂t + In such a case (a homogeneous space free of deformation) the chr.inv.-Maxwell-like equations (77, 78) become 1 c � ∗∂Fk 1 ∗∂Aik − FiFk = 0 . ∂xi c2 ∂t (80) Aik ∗∂Aik = −2πρ ∂t ∗ 2 ik ∂ A ∂xk ∗ ik 1 ∂A 2π − Fk = ∂t c2 ∂t c ji ⎫ ⎪⎬ Group I, ⎪⎭ ∗ 2 ∗i ∂ Ω ∂x (81) i = 0 ∂t ∗ 2 ∗i ∂ Ω ∂t2 ⎫ ⎪⎬ Group II. ⎪⎭ = 0 (82) The obtained field equations characterizing a vortical gravitational field of the “magnetic” kind specify the properties of such kinds of fields: 1. The field-inducing “charges” ρ are derived mainly from the non-stationary rotation of the space, while the field “currents” ji are derived mainly from the non-stationarity and its spatial inhomogeneity; 2. Such a field is permitted in a non-deforming homogeneous space, if the space rotates homogeneously at a constant acceleration; 3. Waves in such a field are standing waves of the acting gravitational inertial force. The waves are permitted only in a space which is inhomogeneous (Δi kn �= 0) and deforming (Dik �= 0); 4. The sources ρ and ji inducing such a field remain unchanged in a non-deforming homogeneous space where F i �= 0. 7 Conclusions According the foregoing results, we conclude that the main kind of vortical gravitational fields is “electric”, derived from a non-stationary gravitational inertial force and, in part, the space deformation. Such a field is a medium for traveling waves of the gravitational inertial force. Standing waves of a gravitational inertial force are permitted in a vortical gravitational field of the “magnetic” kind (spatial vortices of a gravitational inertial force or, that is the same, a nonstationary rotation of the space). Standing waves of the gravitational inertial force and their medium, a vortical gravitational field of the “magnetic” kind, are exotic, due to a nonstationary rotation of a bulky space body (the source of such a field) is a very rare phenomenon in the Universe. It is a matter of fact that gravitational attraction is an everyday reality, so the traveling waves of the gravitational inertial force transferring the attraction should be incontrovertible. I think that the satellite experiment, propounded in [6], would detect the travelling waves since the amplitudes of the lunar or the solar flow waves should be perceptible. Submitted on September 11, 2006 Accepted on November 15, 2006 Group I, Group II. D. Rabounski. The Theory of Vortical Gravitational Fields 9
Volume 2 PROGRESS IN PHYSICS April, 2007 References 1. Landau L. D. and Lifshitz E. M. The classical theory of fields. Butterworth–Heinemann, 2003, 428 pages (4th edition). 2. Zelmanov A. L. Chronometric invariants. Dissertation thesis, 1944. American Research Press, Rehoboth (NM), 2006. 3. Zelmanov A. L. Chronometric invariants and co-moving coordinates in the general relativity theory. Doklady Acad. Nauk USSR, 1956, v. 107(6), 815–818. 4. Rabounski D. Zelmanov’s anthropic principle and the infinite relativity principle. Progress in Physics, 2005, v. 1, 35–37. 5. Zelmanov A. L. Orthometric form of monad formalism and its relations to chronometric and kinemetric invariants. Doklady Acad. Nauk USSR, 1976, v. 227 (1), 78–81. 6. Rabounski D. A new method to measure the speed of gravitation. Progress in Physics, 2005, v. 1, 3–6; The speed of gravitation. Proc. of the Intern. Meeting PIRT-2005, Moscow, 2005, 106–111. 10 D. Rabounski. The Theory of Vortical Gravitational Fields
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