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(gFg  ) , 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 18G RTRg 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 SdVRL 4 (2)g dx4 , where, defined by GR 16G  4 R          ,,    (1) dVgd 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  8G 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 ). 8G Multiplying Eq. (20) by4 , 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 8G    (22) tensor; multiplying the previous equation by4 , 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)

81G TgRR 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 TgRgR (30) 88G GG 16  Ricci scalar is proportional to the sum of Dirac and interaction Lagrangians as follow. 16G RiD  ()    m  (31) c4  Cosmological constant is a construction from electromagnetic field strength tensor and given by: 2G  F  F (32) c4 

Antisymmeric Ricci tensor is given by: 8G RjA (33)  c4  8G Antisymmetric Ricci tensor is the antisymmetric term of Eq. (13) multiplied by4 . 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

8G ,, and second set is,, . If we multiplied Eq. (13) by4 , we find c 8G F FRR (35) c4    8G RAFF j   (36)  c2    8G The symmetric Ricci tensor is the symmetric term of Eq. (13) multiplied by4 ; 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 88GG FF j A       (37) cc22    ,,

Substituting by Eq. (33) into Eq. (4), we find 8G jA  (38) c4   ,,

This tensor takes the form of curl of vector as follow: 8G 4 jAgg  log    log  (39) c Eq. (37) can be divided into two equations as follow. 8G FF   (40) c2   

8G jA   (41) c4  ,, 

Eq. (41) can be rewritten as: 8G jA log g  (42) c4   

8G 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. (46) into Eq. (43), we find 8G 11 log gjA g   gg (47)  c4  2 2

Equating the first term on the left hand side of the equation with the first term on the right hand side of the equation, we find 1 log g  g (48)  2

1 log g g (49)  2 

In Eq. (47) if we equate the second term by the second term, we find 81G jgA   g (50) c4  2  

Equating Eq. (50) with Eq. (39), we find 1 log gg  log   g  g (51)   2  

Eq. (40) can be rewritten in the form

8G    2 FF     log  g (52) c

 1 1   g   ggg       ggg      (53) 2 2

1   g ggg (54) 2

1111    g  ggg ggg  gggg   (55) 2222

Unification of Gravity and Electromagnetism 21

11     gg      g  ggg        g  2 2

1      g  ggggggggg          4

1    g  gg  g  g  gg   g  g  gg   g  4

       gggggggggggg          

1 g gg  g g  gg  g g  gg  g 4            

1         g  gggggggg   4

1  g gg  ggg   ggg    4      

1  g ggggg   4       

1  g ggggg     4   

1  gggg        4

1 ggg   (56) 4    

Using Eq. (49), we find 1 1   g  2log2log gg   g (57)             4 4 And substituting by this equation into Eq. (52), we have 8G 11 1 FF g  gggg  loglo g   g g (58) c2  4  2  2  

22 Unification of Gravity and Electromagnetism

Now, let’s construct the antisymmetric metric tensor; magnetic field in empty space is given by

 ik()11xx 14 x  ik ( 22 24 x)  ik()33x  34 x BBe 01 Be 02 Be 03 (59)   , kkkk(, , )are the wave frequency and wave vector. In general orthogonal 123 123  curvilinear coordinates a vector A defined as follow:    Ae 12hhh123 e e 3 (60)

Let’s suppose that (B01, B02, B03) is the unit vector then equate Eq. (59) with Eq. (60), we find ik((11x  14 x))ix kx 22 24 ik()33x  34 x h1 ee,h2 and he3  . Using these three coefficients to construct the antisymmetric metric tensor ( g ); this tensor is in the same form of electromagnetic field strength tensor F and with the same signs.

ikx()(11 14 x ikx 22 24 x )ikx()33 34 x 0 hhh1230 eee ikx() xikx() x ikx ( x ) hhh0 11 1433 34 2 2 2 4 132eee0 g  (61) ix()kx22 24 ikx()33 34 x ikxx ()11 14 hh230  h 1ee0 e  ikx()33 34 x i(kx22 2xikxx41114)( ) h321hh 0 eee 0

And now, we will return to the cosmological it is a continuous quantity. The second term of constant; the cosmological constant splits up into two cosmological constant can be written as:

 22 8G parts where F FEB22  22 (65)  c4

44GG22  EB (62) 1..2 B E 44  B (66) cc 2 2 A The first term of cosmological constant can be written as: Second term is proportional to density of vacuum magnetic energy; density of vacuum magnetic energy 8G   (63) equals to the absolute value of binding energy per 11c4 nucleon; second term is represented by a curve and it 1..BE is the image of the first term by reflection on the  E 2  (64) 1 2 A A-axis in A -plane; all expected values for the cosmological constant ( ) lie on the area between the First term is proportional to density of vacuum two curves. Symmetric gravitational field equation in electric energy; density of vacuum electric energy is empty space is B..E equivalent to binding energy per nucleon ; 1  RRgg   (67) A    2 BE..  m  c2 where 1 AA R R g (68) 2 mZmp (A  Z)mnN  M, A is atomic mass Equating Eq. (36) with Eq. (68), we find number, Z is atomic number, M N is a nucleus mass, mp is proton mass and m n is neutron mass [8]; 88GG  1 22j AFF  R  g (69) first term is a function of atomic mass number (A) and cc 2

Unification of Gravity and Electromagnetism 23

Antisymmetric gravitational field equation in empty 6. Conclusion space by analogy to symmetric gravitational field General relativity is very successful theory; equation is differential geometry has been extended by new 1 g RRgg (70) tensors and operators. These tensors are  , g , 2   g , g ; the four dimensional gradient operators 1 became six operators, these operators are , , ,      R R g (71) 2   ,  ,   . Equating Eq. (71) with Eq. (33), we have This study introduced new relations in differential geometry and created new differential geometry 81G  analysis undertaken. 4 j ARg (72) c 2 References Eqs. (69) and (72) are two states of energy; [1] Sivaram, C., and Arun, K. 2012. “Primordial Rotation of gravitational object transits between them and changes the Universe, Hydrodynamics, Vortices and Angular its state from fermion to boson or vice versa; this Momenta of Celestial Objects.” The Open Astronomy transition followed by emitting or absorbing Journal 5: 7-11. gravitational field; if we added Eq. (69) into Eq. (72) [2] Maartens, R. 2000. “Cosmological Magnetic Fields.” Pramana Journal of Physics 55 (4): 575-83. we have [3] Kunze, K. E. 2009. “Cosmological Magnetic Fields.” Journal of Physics; Conference Series 189: 12-22. doi: 811G   10.1088/1742-6596/189/1/012022. 2 FFRgRg    (73) c 22 [4] Goenner, H. F. M. 2014. “On the History of Unified Field Theories: Part 2.” Living Rev. Relativity 17 (1): 5. If we equate Eq. (58) by Eq. (73), the first term of [5] Poplawski, N. J. 2007. “On the Nonsymmetric Purely Eq. (58) has not comparable one in Eq. (73) and Affine Gravity.” Modern Physics A 22 (36): 2701-20. equals to zero. [6] Ray, M. 1965. “Theory of Relativity: Special and 1 General.” Philosophy, Delhi, S. Chand.  [7] Greiner, W. Field Quantization. Berlin, Heidelberg, New g  0 (74) 4 York: Springer-Verlag. ISBN 3-540-59179-6. Equating second term of Eq. (58) by the first term [8] Parker, S. P. 1988. Nuclear and Particle Physics Source of Eq. (73), we find Book. ISBN10:0070455090. [9] Zbiral, G. 2012. “Does Gravitation Have an Influence on Electromagnetism?” Journal of Modern Physics 3: 11 ggR log (75) 1223-30. 22 [10] Williams, L. L. 2012. “Electromagnetic Control of Space Equating third term of Eq. (58) by second term of Time and Gravity:The Hard Problem of Interstellar Travel.” Astronomical Review 7 (2): 5. Eq. (73), we find [11] Shifflett. J. A. 2009. “A Modification of 11 Einstein-Schrodinger Theory That Contains  Einstein-Maxwell-Yang-Mills Theory.” Gen. Rel. Grav. ggR  log  (76) 22 41:1865-86. [12] Lal, A. K. 2011. “On Planetary Electromagnetism and Equating Eq. (75) by Eq. (76), we find Gravity.” International Journal of Astronomy & 11 Astrophysics 1 (2): 62-6. Doi:10.4236/ijaa.2011.12009.   [13] Sweetser, D. B. “Unifying Gravity and Electromagnetism gglog g   log g (77) 2 2 Using Analogies Based on Electromagnetism for Gravity.” 1340 Commonwealth Ave. Apt. 7, Allston, MA 02134.   (78) [14] Wiese, U. -J. 2009. Classical Field Theory. Institute for

24 Unification of Gravity and Electromagnetism

Theoretical Physics—Bern University. [34] Carroll, S. “Does the Universe Need God?” Draft to [15] Elyasi, N., and Boroojerdian, N. 2011. “Affine Metrics appear in the Blackwell companion to science and and Algebroid Structures: Application to General Christianity. Relativity and Unification.” International Journal of [35] Iliopoulos, I. A., and Tomaras, T. N. 1985. “Gauge Theoretical Physics 51 (10): 3160-7. Invariance in Quantum Gravity.” SLAC-PUB-3768, [16] Tiwari, S. C. 2006. “Unified Field Theories and Einstein.” RU86/B/145. Physics. [36] Ellman, R. 2004. “The Origin and Its Meaning.” ISBN [17] Marquet, P. 2010. “Geodesics and Finslerian Equations in 978-1492100706 the EGR Theory.” The Abraham Zelmanov Journal 3: [37] Macken, J. A. “The Universe Is Only Space Time.” Santa 90-103. Rose California; Original Draft—February 2010; [18] Witten, L. 1959. “Geometry of Gravitation and Revision 7.1—May 2013. Electromagnetism.” Physical Review 115 (1): 206-14. [38] Rugh, S. E., and Zinkernagel, H. 2002. “The Quantum [19] Blau, M. 2008. Lecture Notes on General Relativity. Vacuum and the Cosmological Constant Problem.” Albert Einstein Center for Fundamental Physics, Institute Studies in History & Philosophy of Science Part B 33 (4): of Theoretical Physics, Bern University, CH-3012Bern, 663-705. Switzerland. [39] Staub, W. O. “Weyl’s Theory of the Combined [20] Greiner, W. 1980. Classical Electrodynamics. New York, Gravitational-electromagnetic Field.” PhD Thesis, Berlin, Heidelberg: Springer-Verlag. Pasadena, California. [21] Saha, G. B. 2006. Physics and Radiobiology of Nuclear [40] Poplawski, N. J. 2009. “Gravitation, Electromagnetism Medicine. New York: Springer. and the Cosmological Constant in Purely Affine Gravity.” [22] Carey, V. P. 1999. Statistical Thermodynamics and Foundations of Physics 39 (3): 307-30. Microscale Thermophysics. Cambridge: Cambridge [41] Goenner, H. F. M. 2004. “On the History of Unified Field University Press, ISBN10: 0521652774. Theories: Part 1.” Living Rev. Relativity 7 (2). [23] Kompaneyets, A. S. Theoretical Physics: Volume 2. [42] Ying, L. “Physical Null Conditions: Diameter of a Black ASIN: B004SAS17M. Hole Singularity.” DOI: [24] Liboff, R. L. Introductory Quantum Mechanics. 10.11648/j.ajmp.s.2015040101.18. ISBN-13:9788131704417. [43] Weinberg, S. 1989. “The Cosmological Constant [25] Gross, F. Relativistic Quantum Mechanics and Field Problem.” Reviews of Modern Physics 61: 1. Theory. ISBN-13:978-0471353867. [44] Clark, C. 2008. Quantum Electrodynamics. January 11, [26] Biswas, S. 2012. “Theory of Dynamic gravitational 2008 electromagnetism.” Adv. Studies theor. Phys. 6 (7): [45] Turro, N. J. 2009. Principles of Molecular 339-54. Photochemistry. University Science Books. [27] Boal, D. H. “Modern Physics from Quarks to Galaxies.” [46] Sorli, A. 2014. “Gravity as a Result Quantum Vacuum Physics Department, Simon Fraser University Energy Density.” In Proceedings of the IEEE [28] Sharif, M., and Bhatti, M. Z. U. H. 2012 “Gravitational International Conference on Electro/information binding energy in charged cylindrical symmetry.” Can. J. Technology, 100-2. Phys. 20 (90): 1-4. [47] Bondi, H., Pirani, F. A. E., and Robinson, I. [29] Tsagas, C, G. 2005. “Electromagnetic Fields in Curved “Gravitational Waves in General Relativity.” Space Time.” Class. Quantum Grav. 22: 393-407. DOI: Mathematical and Physical Sciences 251 (1267): 519-33. 10.1088/0264-9381/22/2/011. [48] Tanner, P. A., Chua, M., and Reid, M. F. 1993. “Energy [30] Clarkson, C. A., and Coley, A. A. 2001. “Magnetic Fields Transfer by Magnetic Dipole-magnetic Dipole and Cosmic Microwave Background.” 18: 1305. arxiv: Interaction.” Chemical Physics Letters 209 (5): 6. astr-ph/0102172v2. [49] Peiris, H. V. The Homogeneous Universe. Department of [31] Carroll, S. M. 1997. Lecture Notes on General Relativity. Physics and Astronomy, University College London, Arxiv: gr-qc/9712019v1. Gower Street, London, WCIE6BT, U.K. [32] Hung, G. Y. 2004. “The Motion and Structure of Matter [50] Saá, D. 2009. Gravitation and Electromagnetism Unified. under Universal Magnetism.” Journal of Theoretics 6 (6). Storrs. [33] Ale´cian, E., and Morsink, S. M. 2004. “The Effect of [51] De Witt, B. S. 1975. “Quantum Field Theory in Curved Neutron Star Gravitational Binding Energy on Space Time.” Physics Reports 19 (6): 295-357. Radiation-Driven Mass-Transfer Binaries.” The [52] Weinberg, S. “Why Quantum Mechanics Might Need an Astrophysical Journal 614: 914-21. Overhaul?” Science News.