Theory of Dynamic Gravitational. Electromagnetism

Transcription

1 Adv. Studies Theor. Phys., Vol. 6, 0, no. 7, Theory of Dynami Gravitational Eletromagnetism Shubhen Biswas G.P.S.H.Shool, P.O.Alaipur, Pin.-7445(W.B), India Abstrat The hange of gravitational potential due to moving mass partile reates gravitational varying extra potential at the observation point. So for a moving mass partile there will be a dynami hange in extra potential at the observation point. It is postulated that the hanging potential will be transformed into eletromagneti field. In a four dimensional spae time ontinuum the eletromagneti field tensors are onstruted with the help veloity dependents extra potential. The relations are very muh onsistent with dimensions and as well as Maxwell equation. With the help of these relations I have explained the planetary magneti field, and alulation provides somehow a good agreement with the observed value for Earth and then for others. Exluding the ions ore dynamo theory alulation for typial Neutron star (Magnetar) shows magneti field in the order of giga tesla, gives a strong support to the presribed theory of dynami gravitational eletromagnetism. Keywords: Dynami gravitations, Extra potential, Eletromagneti field tensor, Maxwell relations, Earth s magneti field, Neutron star s magneti field, Intodution From speial theory of relativity diminishing of mass is equivalent to the radiation of energy also it is the fat that the energy withdrawn from a body beomes energy of radiation makes no differene[].suppose a rest mass m is moving uniformly with veloity v and it is ompelled to stop suddenly,then it will enounter a instant loss of mass ( )m,where [].If it doesn t allow to evolve in to mehanial or hemial energy then an amount of energy will be radiated due to mass energy equivalene. This is pratially observed in high energy partile aelerators.

2 340 S. Biswas The above fat of radiation an be explained over gravitational field. The general field equation atually onstruted for the stati mass and hene the orresponding metri [5] also. Now if the moving mass ontinues to move uniformly then there will be variation of metri whih ultimately leads to a dynami variation of potential at the observation point. This dynami variation of extra potential generates vetor potentials in four dimensional spae time ontinuums. Hene from onservation of energy this variation of gravitational potential need to be identified with eletromagneti vetor potential, beause at the stopping of mass the eletromagneti radiation must be followed by the sudden unfolding of indued veloity dependent extra potential. Thus there have the sope to onstrut anti symmetri ontra variant seond rank eletromagneti field tensor [] from the rank one veloity dependent extra potential. These fats will be formulated and supported with example in the following setions. The notations used in the following setion are i, j, k, or usually,, 3 for spatial oordinates. αβ,, For four spaes time inertial oordinates. μ, ν, λ Generally for general oordinates.. The variation of metri for moving mass From general theory of relativity around a stati mass m at any point gravitational field is defined by the spae time urvatures. And around any point in a small region in the domain of gravitational field defines own spae time oordinates [5]. Thus domain of entire field onstitutes an inhomogeneous spae time ontinuum. Now for weak stati gravitational field the spae time at any point may be represented by nearly Cartesian Co-ordinate system in whih the metri tensor for spae time are The inequality suggests that we ignore powers of h αβ higher than in the R + g R= G. field equation whih ultimately leads uv uv uv At present it is to be aimed to unfold something new exept the gravitational radiation and quantization of field. Let it to be onsidered what will be the atual happening of uniformly movement of soure point with respet to field points i.e. point of observation. If the soure moves with respet to observation point it is obvious that field point will fae a dynami hange of gravitational field. That is for the uniformly moving soure the observation point will enounter a variation of the metri tensor. ()

3 Theory of dynami gravitational eletromagnetism 34 From () δ gαβ = δηαβ + δ hαβ () or δgαβ = δhαβ, [ as δηαβ = 0] η represent the metri of inertial Co-ordinates, but δ g αβ αβ an be expressed as the derivatives of the metri. gαβ δ gαβ =. δχ (3) χ From () and (3) gαβ δ hαβ = δχ (4) χ Thus new metri an be represented by the initial metri as the following. g' αβ = ηαβ + hαβ + δ hαβ gαβ = ( ηαβ + hαβ ) + δχ (5) ν χ gαβ = gαβ + δχ ν χ Then from (4) and (5) the variation of metri tensor at the observation point due to relative hange of position of the soure is justδ h αβ. ( ) gαβ g' αβ gαβ = δχ As, χ = δ h αβ Hene when soure is moving at any instant the hange of the metri with respet to proper time δτ at the observation point will be δ h αβ gαβ = δτ (6) χ τ 3. Explanation of relativisti extra potential The line element in presene of gravitational field is given by K. Shwarzshild [4] as Gm Gm =. ds dt dr r r

4 34 S. Biswas rdθ r sin θdφ (7) Representing (7) in quasi Minkowskian o-ordinates[5] as defined by the transformation, and (8) When the field is very weak, suh that Then line element of the above will be just like a Minkowskian line element exept the temporal part. Thus for weak field approximation we an ignore all h αβ part in the metri exept. Then equating metri omponent with (8) Gm g00 = r Gm Or η00 + h00 = r h Gm = r φ g Gm, φ g = r 00 = (9) As the only interested terns is using (6) the variation of the metri for proper time δτ in whih information s of moving soure will reah to the observation point leads δh g δτ χ τ h oo = δτ χ τ oo oo =

5 Theory of dynami gravitational eletromagnetism 343 As = o for r=,,3 τ Here the observation point supposed to be rest with respet to moving soure in its own o-ordinate system. φ gυ = os ψ rδτ r r Putting δτ = Here r is the distane between retarded position of moving soure and the point of observation. or φ g υ δ h oo. osψ r =, in terms of present position and υ π time from Fig.- os ψ r = at fixed ψo = Or φ g υ δ h oo =. φ g δh oo = ( ) (0) Where =. ν / Then (0) is the variation of metri due to moving soure. And at the observation point assoiated varying extra potential δφg = φg () 4. Formation of long rage rank one tensor potential Now when the soure moves uniformly with respet to point of observation it is obvious at the observation δ hoo term will be a funtion of proper time. Taking the derivative of δ hoo with respet to the proper time τ d ( oo) ( hoo) h d δ χ δ = dτ χ dτ = h ( ) oo U ()

6 344 S. Biswas U the four veloity as soure is moving uniformly respet to the observation point, then U an be taken inside the braket. = h ( ) oo U (3) If the observation point has same veloity as of the soure, i.e. both of them are o-moving, and then there will be no variation of as well as of field. In this d ase δ h oo = 0 and hene ( δ hoo) 0, dτ d leads ( 00) 0 dτ δh = for any general o-ordinate system. Then (3) an be written as h00( ) U μ = 0 μ = and priniple of general equivalene φ g μ Or ( ) U = 0 μ From the energy onservation point of view, the four divergene of flow of extra potential gives Or φ g ( ) U μ = 0 μ (4) Here ( ) is not a pure onstant is a funtion of relative veloity so annot be dropped out and the term in the square braket shows the flow of extra potential at the observation point at present time. μ μ φg ( ) U = A Putting (5) μ A 0 μ = Equation (5) shows divergene of four vetors ( A μ ) Thus first order approximation leads is zero. μ υφ μ υ ( A ) = gu, (- )= + higher order (6) 4 4

7 Theory of dynami gravitational eletromagnetism 345 The authentiity of equation (5) an be heked by four divergenes taking μ υφ μ ( A ) gu, 4 = as the following. φ, as. 4 g υ U μ υ μ = t υ = [ g. υ + g. υ] 4 = o 5. Properties and identifiation of veloity dependent four potential with eletromagneti four vetor potential a) From equation () moving mass point indues an extra potential over stati mass at the point of observation so from equation (6) A μ, the veloity dependent four potential is just a four dimensional flow of gravitational extra potential. b) If moving mass point omes to rest instantly then there must be a disappearane of extra potential into ertain types of energy aording to mass energy onservation []. In this situation the disappearane of veloity dependent extra potential might be transformed as eletromagneti energy.so veloity dependant four potential likely to be the eletromagneti four potential. ) The eletromagneti four potential assoiate with a harged partile are λ λ A U, and also the indued veloity dependent four potential due to r μ μ moving mass point from (6) A U. r Both of them are long ranged rank one tensor. d) From Lorenz gauge the four divergene of eletromagneti four potential vanish, [6] also (5) shows same property for the indue veloity dependent four potential. e) Here ( A μ ) is purely rank one tensor so it doesn t ontribute to the gravitational field whih atually haraterized over seond rank metri. All these above physially means that when mass is moving uniformly there will be a flow of extra potential from the observation point with the same veloity as the moving mass point, and the indued gravitational extra potential will be transformed into some kind of four potential similar to eletromagneti four potential. Here let it to be postulated that the indued four potential is nothing but the eletromagneti four potential. Also it is obvious as the relative

8 346 S. Biswas veloity between soure and observer vanishes the indued eletromagneti potential vanish instantly. 6. Representation of gravitational eletromagneti field As eletromagneti field tensors follow the general ovariane formula in four dimensional spaes, our hoie of eletromagneti tensor regarding the veloity dependent extra potential should be followed by the same. μv Sine the Maxwell s eletromagneti field tensor [F ] [] in Relativisti eletrodynamis expressed as v v A A μ μ F = μ v (7) The tensor assoiated with veloity dependent gravitational extra potential using equation given as [0] μv v μ F = ( δφg. U ) ( δφg. U ) μ v v u U s are independent on χ. μv v μ F = ( δφg) U ( δφg) U μ v For uniform veloity the four veloities Using the above equation, the omponent jo ο F = ( δφg) U ( δφg) U j ο j (8) Thus (9) (0) j ο χ represents spae oordinates and χ (=t) represent time oordinate. Here The first partial differentiation in the square braket of must be arried over onstant present time (t 0 ) and the seond partial differentiation with respet to time must be arried over onstant present position ( χ α ). At present time separation between moving point soure and observation point is α α spae like [3], beause, d ( χ χ ) 0 ο ο And d( t t ο ) = 0, () As, t = t ο, at present time. ο dt Now U =, τ is proper time dτ Using equation (4), at onstant present time, () ο

9 Theory of dynami gravitational eletromagnetism 347 U d( t t ) dτ ο ο = = 0 Moving soure υ o Points of observation r Present position o (Present position and time) (0,0,0, α τ ) [ x, t ] ψ r ψ o r o o Retarded position (0,0,0, τ ) Fig. In fig., the retarded and present position of soure in terms of proper oordinates are given by (0,0,0, τ ) and (0,0,0, τ.) Now proper time interval for the moving point soure τ = τ τ dτ = d( τ τ ) = dt ( t ) dτ = d( t t ) Then, But present time related with retarded time [6], t t Now = + ο dr dt ( ο t) = dr dt ( t) = [here present time t = t ] ο dr dτ j j j d( χ χ ) U = ο dτ χ j = Or, Hene = (3) d ο Then at onstant present position, 0 r

10 348 S. Biswas Or U j j dχ d j = χ dτ = dτ (4) At present position implying (), (3) and equation (4) in equation (0), and υχ j using [ υ ] j r = we have j0 j F 4 g υ υ = (5) Equating the tensor omponent with the eletromagneti tensor omponents [] derived from the Maxwell s eletromagneti relations. F = j0 E j j0 j [in S.I units] F = E [ in C.G.S units] (6) Where E E E3 0 E 0 B3 B uv [ F ] = E In S.I system (7) B3 0 B E 3 B B 0 Equation (5) and (6) gives 3 Gm E= 4g υ υ υ υ 4 r = [In S.I units] 0 [In C.G.S units] (8) Equation (8) is the required representation of eletri field at the observation point due to a moving point mass. The field tensor omponent F Here also putting omponent. ( ) ( ) j υ χ υ 4 φg d j 4 φg = ij d dχ i τ dτ dr φg i dτ = and ontravariantly,, i g = i (9) = the gravitational field = But F ij = [in C.G.S and S.I units] B κ

11 Theory of dynami gravitational eletromagnetism 349 Then the required relation for the magneti field due to a moving point mass is (30) Equation (3) represents the magneti field at the observation point due to a moving point mass. 7. Reonstrution of relations in the unit systems of eletri and magneti field and Maxwell s relations Table-I Relations between C.G.S and S.I unit systems S.I C.G.S υm/ s υ 0 m/ s gm. / s g 0 m/ s 8 0 = 3 0 m/ s = 3 0 m/ s Using these values in equation (8) and (30), I have ome to the relation for both the eletri and magneti field in the two units systems 4 E [S.I= 3 0 CGS.. ] 4 B [ S.I =0 CGS.. ] The above relations give the first essene about the similarity with the pratially observed relations between two unit systems (C.G.S and S.I) in ase of eletri and magneti field. Exept this the onsisteny of the Maxwell s equation regarding equation (8) and (30) an be proved as the following. E = g υ ν And ( 4 ) Gm = 4 υ g r0, t ν ( ) ν τ 0 υ g = ( r 0 ν ) as [ r0, tν = 0] rο υ = r ( g v) (3) ο

12 350 S. Biswas = ( [ ν ] B υ t t 4 g υ t g υ rο ) ν 4 [ ν ] ( t ) rο = 0 = ( g ν) (3) = But [ ] Equation (3) and (3) implies E= B. (33) μv Also as [F ] in (7) is anti symmetri, it is obvious B = O (34) t 8. Explanation of the Earth-Magnetism The most reently aepted theory to explain Earth magnetism is Dynamo theory [8] of molten ion ore. This theory requires irulation of urrents but aording to Lenz s law the indued magneti field must oppose the very ause of generating the urrent up till now the energy needed to sustain irulating urrent are not very well understood.exept this there are number of disrepanies in dynamo theory [].Here we need to onsider about highly massive bodies whih will be moving with very high speed to have the signifiant magneti field using the equation just have been developed. Then we have the sope to treat the Earth as an experimental objet for alulating its magneti field as the following. For simpliity let us onsider the Earth is made of with two hemispherial lobes and the entre of mass eah of the lobe is at distane ' r ' from the Earth s enter (Fig.). B' = υ 4 g' ν (35) ' υ B = 4 ( g' ν ) osθ υ (36) ' υ BH = ( g ' ν )sin θ 4 (37) ' For horizontal omponent B H total magneti field along horizontal diretion for two lobes is zero. Only the vertial omponent exists and added up. The total vertially down word field from fig.

13 Theory of dynami gravitational eletromagnetism 35 ω B ' H B P θ R g ' ' B θ v v M/ o r M/ Fig. Fig represents Polar magneti field of Earth for two hemispherial lobes. ' B= B υ υ = g ' ν os θ =.. os,[ ] (38) B υ GM 4 υ θ ( R + r ) r r ω. ω. r., osθ (39) R + r R + r 3 3 ω Rα GM = Putting r = Rα 4 R ( + α ). R( + α )/' r GM = = Hene 4 ( R + r ) 3 ωαgm = 4 ( + α ) 3/ 3 GM ω R α =... 4 R ( + α ). 3/ 3 g( ωr) α B =. 4 [ + α ] 3/ (40)

14 35 S. Biswas Equation (40) is appliable for any large spherial rotating bodies. 9. Calulation of magneti field for Earth and other terrestrial bodies The matter density is not homogeneous in the Earth. And r =.375R as usual for the homogeneous body The Earth has an average mantle density 3gm/ and the average ore density 3gm/ [8] and taking the density as a linear funtion of distaner. ( ignoring the Weihert-Gutenberg [7] Disontinuity at R =.55 R from the entre.) Introduing R=radius of the earth 0r And density, ρ ( r) = (3 ) R For a elementary one, entre of mass X ' = R rρ( τπ ) r tan θdr 0 R ρ( τπ ) r tan θdr 0 X' =.68R For the hemispherial part entre of mass will be situated at r = X ' =.34R α =.34 In S.I units for earth 8 3 g= 9.8 m/ s = 3 0 m/ s,, R = m π ω = / rad s Using equation (40), the alulated value of magneti field at the pole, B 4 =.58 0 Tesla shows a good agreement with the order of observed value [8]. With the help of equation (40) the alulated values of polar magneti field for the planets and a typial Neutron star (Magnetar) are as following. Calulated and observed values for polar magneti field of the terrestrial bodies [7]-[0]. Here for all the bodies assuming value of α to the same to the earth and taking α g( ωr) = / 4 ( + α ) 3 5 = Tesla

15 Theory of dynami gravitational eletromagnetism 353 Table-II Terrestrial bodies g(m/s ) R( m) w( rad / s) alulated (T) observed (T) Merury.38 g.389 R.069 w Venus.904 g.9499 R.004w No signifiant field Earth g R w Mars.37 g.53 R.97w Neutron Star 0 g r 8640w In Table-, T stands for magneti field in tesla. * The other planet and sun are exluded from the table as they are gaseous in nature and ontribute in various ways to their magneti field. In the above table only merurial magneti field is 00 times high enough in the nano order alulated value. This ould be happened due to solar wind and proximity to suns magneti field whih somehow enhaned Merurial magneti field. The others also show good agreement with the order of observed value. Speially the Neutron star,-very highly dense with temperature 0 Kelvin and mainly onsist of neutron and few perentage of proton and eletron, although as a whole globally eletrially neutral forbids one to set up ion-ore-dynamo theory of magnetism. The very high value of magneti field of a neutron star thus an be explained over the theory of dynami gravitational eletromagnetism, beause more massive objet with higher angular veloity produes higher magneti field.. Conlusion Here I wish to say that the eletri and magneti field are evolved due to dynami hange of gravitational field. And the expression (8) and (30) represents the nature of eletri and magneti field orresponding to the dynamial gravitational field aording to the theory whih has been developed. The expressions (33), (34), and table, have introdued in supporting the Theory of dynami gravitational eletromagnetism. From equation (39) we an immediately have B 3, i.e., the dipolar nature of planetary magneti field. R Espeially the expression (40) may be put to the test of the theory for large massive rotating spherial body. Aknowledgements For writing this artile I am grateful to Dr. Abhijit De and Prof. Bikash Chakroborty for their suggestions.

16 354 S. Biswas Referenes [] A. Einstein, The priniple of Relativity Einstein. A olletion of original papers as the speial and general theory of Relativity. Dover Publiation (95),69-7, 33, 53 [] David J. Griffiths. Introdution to eletrodynamis, third edition. Printie hall of India (999) 536, 537 [3] Herbert Goldstein, Classial Mehanis, nd edition, Narosa Publishing House (997) 5, [4] J.V.Narlikar Introdution to osmology, seond edn.(993),6-65 [5] S. Weinberg, Gravitation and osmology. John Wiley and son (Asia), 004 hapter 3-7, 80-8 [6] Wolfgang K.H. Panefsky, Melba Phillips, Classial Eletriity and Magnetism Dover publiation, nd edition,58-59, , [7] Earth magneti field, wikipedia, the free enylopedia-en. wikipedia. org/wiki/earth magnetifield ited on nd April 00. [8] Merury magneti field, en.wikipedia.org/wiki/merury-[planet] ited on 0 th April 00. [9]Mars magnetiifield and magnetosphere on 6 th April 00. [0] Neutron-star-en.wikipedia.org/wiki/neutron-star-ited on 9 th Marh 00. [] Soure of earth magneti field-mb_soft.om/publi/teto.html ited on April 00. Reeived: Otober, 0