Hyperbolic Octonion Formulation of Gravitational Field Equations
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Hyperbolic Octonion Formulation of Gravitational Field Equations Süleyman Demir
Received: 21 March 2012 / Accepted: 11 August 2012 / Published online: 30 August 2012 © Springer Science+Business Media, LLC 2012
Abstract In this paper, the Maxwell-Proca type field equations of linear gravity are formulated in terms of hyperbolic octonions (split octonions). A hyperbolic octonionic gravitational wave equation with massive gravitons and gravitomagnetic monopoles is proposed. The real gravitoelectromagnetic field equations are recovered and written in compact form from an octonionic potential. In the absence of charges, this reduces to the Klein-Gordon equation of motion for the massive graviton. The analogy between massive gravitational theory and electromagnetism is shown in terms of the present formulation. Keywords Octonion · Gravitational field equations · Proca-Maxwell equations · Monopole
1 Introduction The term gravitoelectromagnetism (GEM) is used to state the close formal analogies between gravitation and electromagnetism. Although Maxwell himself [1] has noticed this possibility, the theoretical analogy between the Maxwell’s equations and the gravitational fields has first been suggested by Heaviside [2] in 1893. Either the sources are static (in which case the electric charge is only source to the electric field but not to the electromagnetic field) or they are moving. The origin of gravitational field is the mass of the body, whereas the source for electromagnetic field is the electric charge of the particle. Similarly, a moving matter (mass current) generates gravitomagnetic field according to Einstein’s theory of General Relativity just as a moving charge (electric current) produces a magnetic field according to Ampère’s law [3]. The similarity between Newton’s law in gravity and Coulomb’s law in electricity leads to introduction of some formulations almost identical to the Maxwell equations in electro˜ and the gravitomagnetic field H ˜ are defined in magnetic theory. The gravitoelectric field E close analogy with classical electrodynamics, S. Demir () Department of Physics, Science Faculty, Anadolu University, 26470, Eski¸sehir, Turkey e-mail: [email protected]
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Int J Theor Phys (2013) 52:105–116
˜ ˜ = −∇ϕe − ∂ A , E ∂t ˜ = ∇×A˜ e , H e
(1a) (1b)
where the ϕe and A˜ e terms represent the scalar potential and vector potential of GEM, respectively. In this case, the following Maxwell-type field equations for linear gravitation can be written(assuming with G = c = 1), ˜ = −e , ∇.E
(2a)
˜ = 0, ∇.H
(2b)
˜ ˜ = − ∂H , ∇ ×E ∂t ˜ = −J˜ + ∇ ×H e
(2c) ˜ ∂E , ∂t
(2d)
where e and J˜ are the gravitoelectric mass density and gravitoelectric mass current density, respectively [4]. Furthermore, the Lorenz gauge condition in GEM is defined by e
∂ϕe =0 ∇.A˜ e + ∂t
(3)
which states the law of mass conservation. The possibility of a particle that carries magnetic charge was first introduced by Dirac [5] in 1931. Although there is no experimental evidence for their existence, up to the present time
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