Magnetic field of GRBs in MWT

Transcription

1 Adv. Studies Theo. Phys., Vol. 6, 0, no. 3, Magnetic field of GRBs in MWT A. Eid, Depatment of Physics, College of Science, Al-Imam Muhammad Ibn Saud Islamic Univesity, Kingdom of Saudi Aabia. Pemanent addess: Astonomy Depatment, Faculty of Science Caio Univesity, Egypt. Abstact The pape discusses the possibility of intepeting the magnetic fields of GRBs in the famewok of a unified field theoy. Using one of the solutions of the genealized field theoy, a diect elation between the pola magnetic field, the angula velocity and the gavitational potential of the body consideed, is obtained. The geometic model used fo applications has spheical symmety and is of the type (FIGI). The theoetical fomula gives a possible intepetation of a seed magnetic field which will develop and poduce the lage-scale magnetic field obseved fo GBRs. The fomula shows that the field is geneated as a esult of otation of a massive object. The Schuste-Blackett fom of NMGEC was obtained fom the Mikhail and Wanas s tetad theoy of gavitation(mwt). The NMGEC-MW fomalism investigate the oigin of the intense magnetic fields of GRBs (Gamma-ay busts). Keywods: Geneal Relativity, Unified Field Theoy, Magnetic Fields, Gamma Ray Busts, Astophysics Intoduction One of the inteesting and vivid poblems of astophysics is the oigin and evolution of the magnetic fields associated with most celestial bodies. The Schuste-Blackett (S-B) conjectue, suggests that the magnetic fields in planets and stas aise due to thei otation. In 947 Blackett[] suggested that thee may be a fundamental elation between the magnetic field of a otating massive body

2 38 A. Eid and its angula momentum. In this scenaio, neutal mass cuents geneate magnetic fields, implying the existence of a non-minimal coupling between gavitational and electomagnetic fields (NMGEC). Blackett himself declaed that thee may be some hope to discove this elation though one of the unified field theoies, which discusses gavitational and electomagnet -ic effects on the same equal footing. An ealy attempt to make a theoy that encompasses the S-B conjectue was made by Pauli[]. Latte, attempts wee made by Bennet [3], Papapetou[4], Luchack[5], Tonnelat [6], McCea [7], Ahluwalia[8] and Baut[9]. The majoity of these studies wee based on the five dimensional Kaluza- Klein fomalism. Vaious authos have suggested that the gamma-ay bust (GRB) cental engine is a apidly otating, stongly magnetized,( G) compact object. The stong magnetic field can acceleate and collimate the elativistic flow and the otation of the compact object can be the enegy souce of the GRB. The majo poblem in this scenaio is the difficulty of finding an astophysical mechanism fo obtaining such intense fields. Vaious authos have investigated this magneta-grb connection [0,,, 3,4, 5]. Mikhail et al.[6] showed that Mikhail and Wanas tetad theoy of gavitation (MWT) [7, 8, 9] pedicts the S-B conjectue of NMGEC. We call this the NMGEC-MWT theoy. It is the pupose of the pesent pape to exploe this possibility. We investigate hee the possibility that NMGEC-MWT is the oigin of the intense magnetic fields G, connected with GRBs poduced by otating neuton stas o black holes. In section, we discuss a solution of MWT and in section 3, NMGEC. In section 4, we apply NMGEC-MWT to GRBs. Ou conclusions and discussion ae pesented in section 5. A Solution of MWT Having Spheical Symmety The field equations to be solved ae of the fom (Mikhail and Wanas [7]): E = 0, whee E is a second-ode non-symmetic tenso given by: E def α β α αβ = g L L g C α C μcν g μαc βν + C g ν μ and L def def α β = βμ αν C μcν L g L, =, C def α μ = μα, + β α α def α α = Γ Γ νμ, def β α = g αβ. The vetical ba denotes absolute diffeentiation using the non-symmetic

3 Magnetic field of GRBs in MWT 39 α connection Γ α ( = λ λ ). The (+) and the ( ) signs ae used in the usual i i μ,ν manne to distinguish between the two types of absolute deivatives, and g ( = λ λ ) is a symmetic tenso. The stuctue of spaces admitting absolute i μ i ν paallelism with spheical symmety has been fistly studied by Einstein and Maye [0]. Howeve, Robetson [] has used goups of motions fo a detailed study of such spaces. The tetad vectos defining the stuctue of such spaces ae given in the following matix, in spheical pola coodinates (Mikhail []): A D 0 0 B B sin φ μ 0 B sinθ cosφ cosθ cosφ sin θ λ = cos 0 sinθ sinφ cosθ sinφ (.) φ i B B B sin θ B 0 B cosθ sinθ 0 whee A, B and D ae unknown functions of only. We ae going to use one of these solutions (Wanas [9]) fo ou pupose. This solution peseves the type of the space (FIGI). It is given in the fom 3 α α α ( + α) + α( + α) A = ( + ), B = ( + ), D = (.) 4 ( + α) whee α,α ae constants. Fom the solution (.) we get the metic, ds = γ ( R) dt dr R ( dθ + sin θdφ ) (.3) γ ( R) whee α 4α α γ ( R) = +. (.4) 4 R R R Hence the metic (.3) is educed to a fom identical with the well-known Schwazschild exteio metic with α = m, whee m is the geometic mass of the object (the mass in elativistic units, C = G = ). Fo sufficiently lage values 4 of R, we can neglect tems of ode ( R ), then the expession (.4) can be witten in the fom α 4α γ ( R) =, (.5) R R which can be compaed with the well-known Reissne-Nodstom metic fo a chaged point mass, viz., ds = γ ( R) dt dr R ( dθ + sin θdφ ), γ ( R) whee m Ke γ ( R) =, (.6) R R

4 40 A. Eid whee m is the geometic mass, e is the electic chage of the paticle. Compaing γ,γ fom (.5) and (. 6), we get K α = m, α = e. (.7) 8 So, fo (.4), γ (R) will take the fom 4 m Ke K e γ ( R ) = +. 4 R R 3R (.8) Taking GM 8πG m =, K =, we get ( M is the mass in gams) 4 c c 4 GM 4πGe π G e γ ( R) = c R c R c R (.9) Thus α epesents the geometic mass, and α epesents the chage of the souce in elativistic units (we ae going to call it the geometic chage). Now, we ae going to study the effect of setting 0 in the solution (.). Setting the constant α. α = 0 (.0) into the electomagnetic potential, we get 9α F0 =, 5 4 (.) 0 9α F =, 5 4 (.) and 0 9α F =, (.3) α f = sinθ. (.4) 4 R Substituting fom (.0) into (.3) and (.4) we get ds = γ 3 ( R) dt dr R ( dθ + sin θdφ ) (.5) γ 3 ( R) whee ( α γ 3 R) =. R We note that the vanishing of α will not lead to the vanishing of the electomagnetic field. The condition α = 0 (o e = 0) will not affect the type of the tetad (.) i.e. FIGI. It is to be consideed that solution (.) will give ise to the Schwazschild metic, but we have found that F 0. Due to exta tems of α in (.4), thee exists a contibution of the electomagnetic field to the gavitational field. Also fom (.) - (.4) it is clea that gavitation has a contibution to electomagnetism as the constant α is

5 Magnetic field of GRBs in MWT 4 identified with the mass of the souce. This type of inteaction may give ise to an effect simila to Blackett s effect when the model (with e = 0) is otated. We study the effect of tanslation on the pesent model, to show whethe tanslation is capable of giving any magnetic field o not. We noted that, the tanslational motion will not give ise to any magnetic components. This esult is in ageement with Wilson s expeiment. By using the angula velocity ω, then the non-vanishing components of the electomagnetic field tenso ae ~ 0 9α F = ~. (.6) 5 ~ 3 9ωα F = ~. 5 ~ Also, the non-vanishing components of the tenso density f, ae ~ 9α 0 ~ f = ~ sin θ (.7) and ~ 3 9ωα ~ f = ~ sinθ. (.8) Fom (.6) we note that in a otating model with e = 0, the component of the ~ electomagnetic field tenso will change (i.e. F 3 0 ), that is a magnetic field is poduced as a esult of otation of the body. Discussion and Results Thee ae two geometic quantities that may be used to epesent the esulting 3 magnetic field. These ae: F (the component of electomagnetic field tenso) ~ defined by (.6), f 3 (the component of electomagnetic tenso density) defined by (.8): ~ 3 ~ 3 9 m F = F = ω in elativistic units, R ~ 3 ~ 3 9 M F = F = ω in cgs units, (.9) R whee M =mass of body in gams, ~ 3 ~ 3 9 m f = f = ω sin θ in elativistic units, 4 R ~ 3 ~ 3 9 M f = f = ω sinθ in cgs units, (.0) 4 R ~ To distinguish between these quantities we found that the dimensions of F 3 3 is cm in elativistic units and 5 g cm sec in cgs units. The dimensions of

6 4 A. Eid ~ f 3 ae cm in elativistic units and g cm sec in cgs units. Howeve, the equied quantity (the stength of the magnetic field ) should have the dimensions cm in elativistic units o g cm sec in cgs units (cf. Eid 3 3 ~ 3 [3]). Theefoe F ( = F ) should be uled out, and we have to use f to epesent the obseved magnetic field since it has the pope dimensions. So, the suface ( R = R0 ) pola ( θ = π ) magnetic field fo a otating body of mass M gams, adius R 0 cm, and angula velocity ω sec is given, using (.0), by 9 m B p = ω in elativistic units, 4 R 0 9 M B p = ω in cgs units, (.) 4 R Thus, magnetic fields ae geneated as a esult of the otation of the body, as in the S-B conjectue. As shown above, fomula (.) may give a possible intepetation of a seed magnetic field which will develop to poduce the lage scale magnetic field obseved fo celestial objects. This fomula shows that the field is geneated as a esult of otation of the massive object. 3 NMGEC in MWT 3. NMGEC Wilson [4] assumed that a mass m moving with velocity v, even though electically neutal, poduces a magnetic field given by the empiical elation ( G) m [ B = ν ]. C This hypothesis is cetainly untue, as stated by Wilson, if ν is intepeted as a tanslational velocity, but pobably, thee exists a magnetic field due to a otating body. Blackett [] intoduced a hypothesis about the magnetic field of massive otating bodies, but this hypothesis was ejected by Blackett himself, as a esult of some expeiments. The Schuste-Blackett conjectue elates the angula momentum U to the magnetic dipole moment P : G N P = β U (3.) c whee β is appoximately a constant, on the ode of unity, G N the Newtonian constant of gavitation, and c is the speed of light. The angula momentum U is U = Iω, (3.)

7 Magnetic field of GRBs in MWT 43 whee ω = πp is the angula velocity, p the otational peiod, and I is the moment of inetia. The dipole moment P is elated to the magnetic field B by 3( P ) P B =, (3.3) 5 whee is the distance fom P to the point at which B is measued. 3. NMGEC-MWT Compaing elation (.) and Blackett s fomula (3.), we get 4β G B p = Mω, G (3.4) 5R C o whee, 45C β =. (3.5) 8 Gφ So β is not a constant but a paamete depending on the gavitational potential ( φ = M R ) of the body. In units of sola mass ( M 0 )and sola adius ( R 0 ), we have R M R0 M β. (3.6) Fo a black hole, M/R is appoximately a constant and β 5. Fo a sola mass neuton sta, β The Gamma-Ray Bust in MWT NMGEC-MW is applied to a otating poto-magneta. The otational enegy of a otating sta, E ot: J E ot =, (4.) I Whee J = MωR is the angula momentum, with I popotional to MR fo a spheoid. This otational enegy of a black hole with angula momentum J is a faction of the black hole mass M, E ot = f ( λ) Mc, (4.) Whee λ = Jc M GN is the otation paamete(is dimensionless)[5,6]. Fom Eq.(3.4), we get the magnetic field of the black hole at hoizon: 9 G N λω 8 M R G B = (4.3) 4 p M λ ω R c M 0 R0 3

8 44 A. Eid This equation has the same dimension of equation (.). The adius of a otating Ke black hole is m( + a m ) =, whee a = J m is the angula momentum pe unit mass, and a = 0 is a Schwazschild metic. Fo a maximally otating black hole a = m, and (λ = ). This hoizon adius can be witten in the fom = ( + λ ) Sh, (4.4) whee Sh = GN M c is the Schwazschild adius. The pediction fo magnetic fields in GRBs, using Eq. (4.3), is B p G. We conclude that the NMGEC-MW theoy pedicts the equied magnetic fields in otating black holes. Fo a apidly otating poto- magneta, we obtain fom Equations (3.)-(3.3)and (3.6): B p = βp - G, (4.5) whee β 0 and P is the peiod in seconds. Fo P<< s, we obtain B p 0 5 G. 5. Conclusions and Discussion Magnetic fields G have been suggested to be connected with cental engines of GRBs. We consideed the possibility that GRBs ae poweed by a apidly otating highly magnetized cental compact object, a black hole o a neuton sta. Since these magnetic fields ae difficult to be obtained by astophysical mechanisms, we consideed the possibility that the fields ae ceated by NMGEC-MWT. Fo GRBs a magnetic field G is equied to poduce the Poynting flux needed to supply the enegy obseved within the equied time. The fields pedicted by NMGEC-MWT in Eq.(4.3) fo a black hole and Eq.(4.5) fo a neuton sta, ae in ageement with models equiing cental engines with 0 5 G fields. We conclude that NMGEC-MWT is a possible mechanism fo ceating 0 5 G fields in the cental engine of GRBs due to a otating neuton sta o black hole. Acknowledgments. A. Eid thanks Deanship of Academic Reseach, Al- Imam Muhammad Ibn Saud Islamic Univesity, KSA, fo the financial suppot of the poject numbe (34).

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