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Unification of Gravity and Electromagnetism

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Unification of Gravity and Electromagnetism Journal of Physical Science and Application 7 (3) (2017) 15-24 D doi: 10.17265/2159-5348/2017.03.002 DAVID PUBLISHING Unification of Gravity and Electromagnetism Mohammed A. El-Lakany Physics Department, Faculty of Science, Cairo University, Giza 12613, Egypt Abstract: Gravity and electromagnetism are two sides of the same coin, which is the clue of this unification. Gravity and electromagnetism are represented by two mathematical structures, symmetric and antisymmetric respectively. Einstein gravitational field equation is the symmetric mathematical structure. Electrodynamics Lagrangian is three parts, for electromagnetic field, Dirac field and interaction term. The definition of canonical energy momentum tensor was used for each term in Electrodynamics Lagrangian to construct the antisymmetric mathematical structure; symmetric and antisymmetric gravitational field equations are two sides of the same Lagrangian. Key words: Gravity, electromagnetism, general theory of relativity, quantum field theory, nuclear and particle physics, astrophysics and cosmology. 0.5 1. Gravity and Electromagnetism Are Two =10ergs.s ~ 9 cc . e 4 3 Sides of the Same Coin 3 re For proton 9 and also for the Gravitational objects have spin and angular p ~10 ergs.s cc momentum; spin and angular momentum of solar system we have 9 ; for a solsys ~gs10 er .s cc gravitational objects are related to basic quantum 10100 properties of elementary particles. The angular galaxy =10ergs.s ~ 9 cc , spin density gal 4 R 3 momentum for the sun is given by 3 gal 120 250; for solar system 10 9 JRsun=M sun sun 10 ergs.s for universe =10ergs.s~ cc [1]. Not univ 4 R 3 it is JR=M 252 10 ergs.s . In the 3 H solsys solsys solsys only this, but also magnetic fields seem to be case of a galaxy the angular momentum is given by everywhere that we can look in the universe [2]. 2 where 13 Jgal = M gal R gal Magnetic fields are observed to be of the order of 10 45 2 47 2 18 3 Mgal = 10 g; R gal = 10 cm ; 2 10 HZ G in neutron stars, 10 G in solar type stars. Magnetic and the value of angular momentum is fields of order a few μG also have been detected in 74 . Similarly for cluster of galaxies, radio galaxies [3]. Magnetic fields are associated with Jgal 10 ergs.s the angular momentum is given by all gravitational objects; gravitational objects are 2110 in Hubble scale magnetic dipoles; electromagnetism not only tied to JRClust=M Clust Clust 10 charged particles, but the planets, stars, galaxies and and for the universe 120 . Spin density (σ = Juniv 10 clusters. spin/volume) is the same for a wide range; for an electron, the spin density is given by 2. Symmetric and Antisymmetric Mathematical Structures Unification of gravity and electromagnetism has Corresponding author: Mohammed A. El-Lakany, M.Sc., been pursued by many scientists, like Weyl, research fields: astrophysics and cosmology, general relativity and gravity, electromagnetism. Eddington, Einstein, Infeld, Born and Schrodinger. 16 Unification of Gravity and Electromagnetism Weyl initiated this unification; Eddington considered RR (3) connection as the central concept then decomposed its ,, Ricci tensor to symmetric Ricci tensor ( R ) which RR (4) represents gravity and antisymmetric Ricci tensor ,, ( R ) represents electromagnetism. Symmetric and antisymmetric Ricci tensors give us Infeld and Born followed the path of Eddington the opportunity to have symmetric and antisymmetric then derived the Lagrangian gravitational field equations. L=GR det(gFg ) , they considered 4. General Theory of Relativity the asymmetric metric g()gF, its General relativity is the modern theory of gravity; symmetric term g represents gravity and General theory of relativity relates gravitational field antisymmetric term ( F ) represents to the curvature of space time. Symmetric stress electromagnetism, g is the determinant of the energy tensor T is the source of gravitational field in g μν symmetric metric tensor [4]. Schrodinger general theory of relativity. generalized Eddington Lagrangian to a new form In the presence of permanent gravitational field, the containing the cosmological constant () [5]; symmetric gravitational field equation is despite the failure of these previous attempts, they in its entirety refer to something cannot be neglected that 18G RTRg g 4 (5) gravity and electromagnetism should be represented 2 c by two mathematical structures. R is the Ricci scalar and G is the gravitational 3. Curvature Tensor constant. Einstein-Hilbert action for gravity is given by Riemann tensor in terms of Christoffel’s symbols is c4 SdVRL 4 (2)g dx4 , where, defined by GR 16G 4 R ,, (1) dVgd x is invariant volume element and Riemann Christoffel tensor is of rank four, gravity Lagrangian is defined by contravariant in δ and covariant in μ, ν, and σ, and 2Λ (6) also 0 (2) Gravity Lagrangian is a combination of Ricci scalar and cosmological constant. Is the necessary condition for the validity of the special theory of Relativity and for the absence of 5. Electrodynamics permanent gravitational field or the necessary and Electrodynamics Lagrangian is given by sufficient condition that the space time is flat [6]. Lowering the last index in the Riemann Christoffel – (7) tensor with the symmetric metric tensor, the lowered where, F is the electromagnetic field strength tensor RRg is symmetric under tensor, D is the gauge contravariant derivative, interchanging of the first and last pair of indices and is matter field, † is their adjoint, 0 i 1 antisymmetric in μ, ε and in ν, σ. Symmetric and and is the four Dirac matrices with (ν = 0, 1 … 3). antisymmetric Ricci tensors can be written as follow The electromagnetic field strength tensor ( F ) is Unification of Gravity and Electromagnetism 17 given by Employing the Maxwell equation, we obtain 0 EEE 123 FFjj A FF A (13) EBB1320 F The antisymmetric stress energy tensor for EB230 B 1 electromagnetic field can be written in the form: EBB321 0 1 and TjAgFFem. (14) 4 0 EEE123 8G EBB1320 If we multiplied this equation by , we find F 4 c EB230 B 1 .. G .. g (15) EBB3210 and they lowered index counterpart. The second term in electrodynamics Lagrangian for The first term of Electrodynamics Lagrangian for Dirac field is given by the electromagnetic field is given by D – (16) .. (8) The canonical energy momentum tensor is defined Canonical energy momentum tensor for by electromagnetic field Lagrangian is .. .. g .. (9) g ()FF 17 Using the identity 4F , we find (A) θ γ g 1 em. FF g FF (10) 18 4 The canonical energy momentum tensor that has Eq. (10) is not antisymmetric due to the asymmetric been presented in this equation is not antisymmetric tensor ()FF [7]; for this, suppose the due to the symmetric term ( γ ). For this, the asymmetric tensor is the sum of symmetric, antisymmetric stress energy tensor can be written as antisymmetric tensor can be written as follow. the canonical energy momentum tensor minus this symmetric term as follow: FF FF (11) θ γ (19) The divergence tensor is arbitrary antisymmetric tensor in their first two indices ( ), it is g constructed from electromagnetic field strength tensor (20) ( F ) and electromagnetic vector potential ( A ). 8G Multiplying Eq. (20) by4 , we have Eq. (11) in terms of this definition can be rewritten c as g (21) FF() FA FF () FA (12) Third term is the interaction Lagrangian and given 18 Unification of Gravity and Electromagnetism by 8G (22) tensor; multiplying the previous equation by4 , c The canonical energy momentum tensor is given by g (23) we find g (25) And antisymmetric stress energy tensor is If we added Eqs. (15) and (21) to Eq. (25), we have g (24) Antisymmetric stress energy tensor for interaction Lagrangian is the same canonical energy momentum . g .. (26) . g g – (27) If gauge contravariant derivative ( ) is used in the previous equation, we find . g g – (28) 81G TgRR g (29) c4 2 Antisymmetric gravitational field equation is gauge invariant and antisymmetric stress energy tensor can be written in the form. cc44 c 4 TgRgR (30) 88G GG 16 Ricci scalar is proportional to the sum of Dirac and interaction Lagrangians as follow. 16G RiD () m (31) c4 Cosmological constant is a construction from electromagnetic field strength tensor and given by: 2G F F (32) c4 Antisymmeric Ricci tensor is given by: 8G RjA (33) c4 8G Antisymmetric Ricci tensor is the antisymmetric term of Eq. (13) multiplied by4 . Substituting Eqs. (31) c and (32) into Eq. (6), we have – .. (34) Gravity Lagrangian equal to electrodynamics Lagrangian, but in terms of the secondset of indices. Electrodynamics Lagrangian and its parts can be written in terms of one of two sets of indices, first set is Unification of Gravity and Electromagnetism 19 8G ,, and second set is,, . If we multiplied Eq. (13) by4 , we find c 8G F FRR (35) c4 8G RAFF j (36) c2 8G The symmetric Ricci tensor is the symmetric term of Eq. (13) multiplied by4 ; it has two parts, first term c is a construction of current density and electromagnetic vector potential, and second term is the gravitational field tensor. If we substituted by Eq. (36) into Eq. (3), we find 88GG FF j A (37) cc22 ,, Substituting by Eq. (33) into Eq. (4), we find 8G jA (38) c4 ,, This tensor takes the form of curl of vector as follow: 8G 4 jAgg log log (39) c Eq. (37) can be divided into two equations as follow. 8G FF (40) c2 8G jA (41) c4 ,, Eq. (41) can be rewritten as: 8G jA log g (42) c4 8G log gjA (43) c4 11 g ggg ggg (44) 22 gggggg 0 (45) 20 Unification of Gravity and Electromagnetism 11 11 gg gg g gg (46) 22 22 Substitute by Eq.
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