General Field Equation for Electromagnetism and Gravitation

Transcription

1 International Journal of Modern Phyic and Application 07; 4(5: ISSN: General Field Equation for Electromagnetim and Gravitation Sadegh Mouavi Department of Phyic Amirkabir Univerity of Technology Tehran Iran Citation Sadegh Mouavi. General Field Equation for Electromagnetim and Gravitation. International Journal of Modern Phyic and Application. Vol. 4 No pp Keyword Gravitation Electromagnetim Electrogravity Received: March 3 07 Accepted: July 8 07 Publihed: September 8 07 Abtract Eintein howed that the effect of gravitational field on a pace-time i explained mathematically uing Ricci tenor. Alo it i clear that the effect of electromagnetic filed on a pace-time i explained with electromagnetic tenor which atifie Maxwell' equation. In real phyic world both electromagnetic and gravitational field exit in a pace-time imultaneouly. So the pace-time hould be conidered imultaneouly uing two econd rank tenor. In thi manucript a new approach for writing a general field equation for both gravitation and electromagnetim in a four dimenional pacetime i propoed. A a reult a relationhip between electromagnetim and gravitation i obtained.. Introduction Since 94 that Albert Eintein propoed a new and complete definition of gravitation there have been everal attempt to unify electromagnetim and gravitation included Kaluza theory with the fifth dimenion and ome attempt in the domain of Quantum mechanic. But all of them were unucceful in unifying electromagnetim with gravitation. For example ome year after the uggetion of Kaluza theory at 939 Eintein refued to accept hi theory in a letter mentioning that hi reearch i frutrated: A no arbitrary contant occur in the equation the theory would lead to electromagnetic and gravitation field of the ame order of magnitude. Therefore one would be unable to explain the empirical fact that the electrotatic force between two particle i o much tronger than the gravitational force. Thi mean that a conitent theory of matter could not be baed on thee equation []. Eintein himelf tried to do thi unification a he aid: It would be a great tep forward to unify in a ingle picture the gravitational and electromagnetic field then there would be a worthy completion of the epoch of theoretical phyic []. Eintein wa trying to unify thee two field in a theory baed on field and not particle a it ha been mentioned in hi paper in 935: A complete field theory know only field and not the concept of particle and motion [3]. Actually Eintein wanted the field to be aborbed in geometry and to formulate electromagnetim in geometry like gravitation. But electromagnetim ha not been aborbed in geometry in none of the previou theorie. Eintein howed that the effect of gravitational field on a pace time i explained with a ymmetric rank tenor namely Ricci tenor. In addition it i known that the effect of electromagnetic filed on a pace-time i explained with a rank antiymmetric tenor which atifie Maxwell' equation.

2 45 Sadegh Mouavi: General Field Equation for Electromagnetim and Gravitation In real phyic world both electromagnetic and gravitational field exit in a pace-time imultaneouly. Therefore the pace-time hould be conidered imultaneouly by two econd rank tenor which one of them i ymmetric and the other one i antiymmetric. But what i the relationhip between thee tenor and how can they be introduced in a ingle equation? In the following a new way i propoed for writing a general field equation for gravitation and electromagnetim in a four dimenional pace-time while conidering the above mentioned tenor.. Mathematical Background The cla of totally antiymmetric tenor i an important cla of tenor of type (0. Thi cla contain covariant tenor with antiymmetric property in every pair of their argument i.e. T( X... X... X... X = T( X... X... X... X ( µ µ for all pair of indice µ and and for all X [4]. By applying the alternating operator A to a general tenor T of type (0 thi kind of tenor can be formed. Applying operator A to T give the linear combination defined by AT( X... X = gn(... T( X... X (!... where in thi ummation (... are an even or an odd permutation of ( integer number and baed on that gn(... = ± and equation ( i to be valid for every ( X... X. When T i totally antiymmetric applying the A operator to it imply reproduce T. However when n (the dimenion of the vector pace applying the A operator to T reduce T to zero; imply put there i no totally antiymmetric tenor of type (0 for n. Antiymmetric tenor of type (0 are called -form. Since they mut vanih when any two of their argument coincide it follow that the dimenion of the vector pace for n! -form can be obtained uing:.!( n! * T p Thi pace i denoted by Λ. By applying the A operator to the bai element of the * following tenor product a bai for Λ can be obtained: T p A( e... e (3 The reulting bai element can be written a the exterior or the wedge product of the e a the following: e e... e (... (4 By extending the ummation only over trictly decending equence a general -form can be written a: Ω = Ω (... (5... e e... e Conidering that interchanging a pair of indice i equal to interchanging the correponding element in the wedge product it can be deduced that interchanging the element in a wedge product mut be accompanied by a change of ign: e e = e e (6 The expreion for an -form in a local coordinate bai i:... Ω = Ω dx... dx (7 To obtain a (p+q form the wedge product of any p-form Ω and a q-form Ω can be formed by the rule Ω Ω = A( Ω Ω (8 which mut accordingly vanih identically if ( p + q n. By definition q Ω Ω = ( Ω e... e ( Ω e... e (9... p p... q where (... p and (... q are trictly decending equence. Since each of the q bai element q e... mut go through p interchange before Ω Ω can be brought to the form required of Ω Ω conequently: pq q... q... p p Ω Ω = ( ( Ω e... e ( Ω e... e (0 pq = ( Ω Ω If we chooe a uitable coordinate it will be een that our electrogravity (EG equation will be appropriate for infinitely mall four dimenional region. Let x x and x 3 be the pace coordinate and x 4 be the time coordinate in appropriate unit. Here the appropriate unit i the coordinate in which the time unit i choen o that the light peed i equal unit (c= in the local coordinate. If a unit meaure i choen the coordinate with a given orientation of the coordinate have direct phyical meaning in the ene of the theory of relativity. In theory of relativity the following expreion ha a value which i independent of the orientation of the local ytem of coordinate: 3 4 d = dx dx dx + dx ( Let d be the magnitude of linear element pertaining to point of the four-dimenional continuum in infinite proximity. To the mentioned linear element or to the two infinitely proximate point event there are correpond definite differential dx dx dx 3 and dx 4. In thi ytem the dx i repreented here by definite linear homogeneou expreion of the dx ơ : e

3 International Journal of Modern Phyic and Application 07; 4(5: dx = α dx ( Inerting thee expreion in above equation we obtain: d = gdx dx (3 where g are function of x. Thee are independent from the orientation and the tate of motion of the local ytem of the coordinate. d i independent of any particular choice of coordinate. If it i poible to chooe the ytem of coordinate in the finite region in uch a way that the g µν ha contant value: g µ = (4 It will be een that a free material point move relatively to thi ytem with uniform motion on a traight line. But if a new pace-time coordinate x x x 3 and x 4 i choen the g µν in the new ytem will not be contant but function of pace and time and the motion of free material point will be a curvilinear motion. Thi motion mut be interpreted a a motion under the influence of the EG field. So it can be found the occurrence of an EG field connected with the pace-time variable of g µ. So the g µ repreenting the EG field at the ame time define the metrical propertie of the pace-time. Now a four-vector can be defined uing g µ. Let ue the following metric for thi purpoe: tt rr θθ θ gϕϕdϕ d = g dt g dr g d (5 A four vector h µ can be defined uing g µ a following: hµ = ( h h h3 h4 where h = gtt h = grr h3 = g θθ h4 = g ϕϕ or hµ ( gµµ A an example for Schwarzchild metric: = (6 m d = ( dt dr r dθ r in θdϕ r m (7 ( r Thi four-vector will be: h m = ( r rin r m r µ θ And for any arbitrary metric like ( d = g dx + g dx dx + g dx dx + g dx dx g dx dx + g dx + g dx dx + g dx dx g dx dx + g dx dx + g dx + g dx dx g dx dx g dx dx g dx dx g dx The four-vector will be: A g R can write: (9 hµ = ( g g g g (0 α µ µα = R where R µ i the Riemann tenor one µα µα α α h R = R ( µα By contracting ( twice it i obtained: Rµα = R where we call curvature four vector to R. Now to find the geometrical form of the antiymmetric tenor which wa mentioned in introduction and it relationhip with gravitational geometry we define S µ uing the following wedge product: S = R h ( µ µ where S µ i an antiymmetric tenor. Now the EG tenor can be defined a following: where R i the Ricci tenor. µ ΓµS Γ S µ + hµ R 3. The EG Field Equation in the Abence of Matter (3 The mathematical importance of the above mentioned EG tenor i that if there i a coordinate ytem with reference to which the g µ are contant ( K 0 coordinate then all component of the EG tenor will vanih. If any new ytem of coordinate ( K i choen in place of the original one ( K 0 the g µ will not be contant but in conequence of it tenor nature the tranformed component of the EG tenor will till vanih in the new ytem ( K. Relatively to thi ytem all component of the EG tenor vanih in any other ytem of coordinate. Thu the required equation of the matter-free EG field in any cae mut be atified if all component of the EG tenor vanih. So uing eq. 3 the equation of matter-free field are: µ µ µ µ Γ S Γ S + h R = 0 (4 In approximation for conidering the electromagnetic field

4 47 Sadegh Mouavi: General Field Equation for Electromagnetim and Gravitation behavior it i better to focu on microcopic cale. If ome electric charge are placed in the media it i clear that in thi cale the effect of gravitational field are o much maller than electromagnetic field o the gravitational field can be ignored here. A the exiting ma (the ma of electric charge m i very mall ( m 0 then T 0 where T i the energy-momentum tenor thu according to the Eintein field equation: R 0 and expreion (3 turn into: µ Γ S µ Γ Sµ (5 Uing 5 in addition the two following tenor can be written: µ Γ µs Γ µs µ Γ S µ Γ µ S ν (6 (7 From eq. 4 it i een that all component of the EG tenor vanihed o the ummation of the above 3 tenor (5 6 and 7 mut be vanihed alo. Now adding 5 6 and 7 it i obtained: and uing S µ µ + + Γ µs Γ S µ µ µ µ µ µ Γ S Γ S Γ S Γ S = 0 = S in equation (8: µ µ µ µ + + = 0 µ 4. The General Form of the EG Field Equation (8 (9 The field equation (eq. 4 which are obtained for matter free pace-time are to be compared with the field equation ϕ = 0 of Newton theory or R µ = 0 of Eintein gravity field equation in vacuum. We require the equation correponding to Poion equation: ϕ = 4k πρ or Rµ gµ R = ktµ (30 of Eintein general form of the gravitational field equation. For thi purpoe T µ i defined a following: α α µ α µ T = g ( kt + kt h (3 Where k andk are two contant related to the gravity and electromagnetim repectively and T = g T α (3 α i the energy-momentum tenor and T α i the electromagnetic energy tenor. Thu intead of eq. 4 we write: Γ S Γ S + h R g h R = T µ µ µ µ µ µ (33 Therefore equation (33 i the required general form of the EG field equation. Equation (33 in it pecial form eaily give u the general form of the Eintein field equation in general relativity and all Maxwell equation a following. A Eintein ha mentioned in hi paper (The Foundation of the General Theory of Relativity The Chritoffel ymbol are ued a the field component of gravitation in general relativity. So in the abence of matter by approximation if the gravitational field i very mall then: Γ 0 and according to the Eintein vacuum equation: R 0. For example it can be reached to thi approximation in a region without any matter unle ome electrical charge like electron that have a very mall ma. So in the mentioned approximation eq. 33 turn into: µ = T µ (34 Let uppoe that we are working in a region that the total electric charge denity of the area i ρ q and the current denity i j i. In one pecial cae we uppoe that all component of T µνơ are zero except the four component of T4. Now a 4_vector Now from eq. 34 and 35: T can be defined a: T = k( ρ j j j = j ( q 3 = k j j j = j = j µ ( ρq 3 µ µ µ (36 where S µν i an antiymmetric tenor. Therefore from eq. 9 and 36 on identifying S µ with the electromagnetic tenor F µ one will recognize eq. 9 and 36 a the Maxwell equation. Therefore S µ can be identified a the electromagnetic tenor (F µ. Moreover the equation 33 in it pecial form eaily give u the general form of the Eintein field equation for gravitation. 5. Concluion In thi manucript a unified field equation (eq. 33 for

5 International Journal of Modern Phyic and Application 07; 4(5: both electromagnetim and gravity (Electrogravity field equation i preented. From general relativity it i wellknown that the Chritoffel ymbol are conidered a the field component of gravitation. So in thi reearch it ha been found a relationhip between the gravitation and electromagnetim. We have found F αβ in term of Γ µ a following: µ Γµ Γµ k k F = g h ( Γk Γ + Γk Γ h αβ α µ µ β Acknowledgement (37 The author wihe to thank Profeor Youef Sobouti and Ehagh Soufi for fruitful dicuion and uggetion. Reference [] A. Eintein. Letter to V. Bargmann July AE [] A. Eintein. Äther und Relativitättheorie. Springer 90. [3] A. Eintein and N. Roen. The particle problem in the General theory of relativity. Phyical Review 48 (: [4] Subrahmanyan Chandraekhar The mathematical theory of blackhole New York Oxford Univerity Pre 983. [5] A. Eintein The foundation of the General theory of relativity Doc [6] Paul Adrien Maurice Dirac General Theory of Relativity Princeton Univerity Pre 975. [7] Malcolm A. H. MacCallum George F. R. Elli and Roy Maarten Relativitic Comology Cambridge Univerity Pre 0. [8] W. M. Boothby An Introduction to Differentiable Manifold and Riemannian Geometry Academic Pre. New York 986. [9] M. P. Hobon G. P. Eftrathiou and A. N. Laenby. General relativity Cambridge Univerity 006. [0] Steven Weinberg Gravitation and Comology: Principle and Application of the General Theory of Relativity. John Wiley & Son New York 97. [] J. D. Jackon Claical Electrodynamic Univerity of California Berkeley 974. [] Sean M. Carroll Spacetime and Geometry: An Introduction to General Relativity. Addion Weley San Francico 004.