Broken Symmetries in Spacetime with Torsion and Galactic
Transcript Of Broken Symmetries in Spacetime with Torsion and Galactic
International Journal of Astronomy and Astrophysics, 2012, 2, 180-182 http://dx.doi.org/10.4236/ijaa.2012.23022 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
Broken Symmetries in Spacetime with Torsion and Galactic Magnetic Fields without Dynamo Amplification
Luis Carlos Garcia de Andrade
Department of Theoretical Physics, IF-UERJ, Rio de Janeiro, Maracanã Email: [email protected]
Received February 6, 2012; revised March 15, 2012; accepted March 26, 2012
ABSTRACT
Since Kostelecky et al. [Physical Review Letters 100, 111102 (2008)], have shown that there is an intimate connection between spacetime with torsion and the possibility of constraining it to Lorentz violation, a renewed interest in torsion theories of gravity has arised. In this paper, minimal coupling between photons on a torsioned background is shown to allow us to obtain the galactic magnetic field strength μG without dynamo amplification. This agrees with recent results by Jimenez and Maroto (2011) for spiral galaxies, with galactic magnetic field constraints from Dark matter without dynamo amplification. The approach discussed here allows us to get rid of the unpleasant photon mass by simply considering the Lagrangean cut off for second order torsion terms. Therefore though the gauge and Lorentz symmetries are broken here one does not have to deal with photon masses.
Keywords: Star Dynamos; Torsion Theories; Lorentz-Violation; Cosmology; Astro-Particle Physics
1. Introduction
There are many papers where galactic magnetic fields [1] and not only primordial fields can be obtained from a Biermann battery type and not from a usual dynamo mechanism of magnetic field amplification. More recently Jimenez and Maroto cite [2] have shown that in the presence of dark energy, magnetic fields are generated without any amplification. They obtained for spiral galaxies fields of the strength of 10–9 G in accordance with Schuster-Blackett conjecture that the magnetic fields are obtained from angular momentum. Earlier on De Sabbata and Gasperini [3] showed that this conjecture could be placed in the realm of Einstein-Cartan gravity [4], where spin of the elementary particles are the Cartan torsion source [5]. More recent Opher and Wichoski [6] have shown that the same mechanism can be used to obtain galactic magnetic fields without dynamos amplification. Also Kostelecky et al. [7] investigated the Lorentz violation (LV) of the fermionic sector of a Lagrangean, in Riemann-Cartan spacetime the curvature tensor Rijkl and the e.m sector F ij F kl , (i, j = 0, 1, 2, 3) given by Rijkl F ij F kl displays the same symmetries of LV term, where the Riemann-Cartan curvature tensor, including torsion terms plays the role of the Higgs sector constants kijkl . The main difference from previous papers [8] is that one uses here a minimal coupling instead the non-minimal photon-torsion coupling where the magnetic vector potential does not interact directly
with torsion or contortion part in the QED Lagrangean. In this paper, we show that the use of this photon sector coupled with torsion flat mode, via non-minimal coupling yields naturally the breaking of gauge fields and a strength for the galactic magnetic fields of the observed G value without the dynamo amplification. Besides the advantage of the approach here is that by considering only first order terms on torsion, one is able to avoid the uncomfortable photon tiny mass. The paper is organised as follows : In Section 2 we consider the Riemann-Cartan (RC) sector of the Lagrangean obtain a 2D system of Maxwell generalised equation which are similar to the ones obtained for the neutrino mass oscillation. Section 5 contains conclusions and discussions.
2. Flat Photon-Torsion Minimal Coupling and Magnetic Field Amplification
Since the torsion effects are cosmological weak in the galaxy formation post inflationary era, compared to curvature effects of Einstein gravity sector, second-order torsion effects are neglected in the torsioned quantum electrodynamics (QED). Photon and gravity torsion sectors are given by [9]
S = 1
1
d4 x g 2
1
F ij Fij
aR*F ij Fij
m2
4
(1)
bRi*k F il Flk
cRi*jkl F ij F kl
dDi
F
ij
Dk
F
k j
Copyright © 2012 SciRes.
IJAA
L. C. GARCIA DE ANDRADE
181
The physical constants a,b, c, d are obtained by
means of the conventional Feynman diagram techniques [10]. Here the e.m field tensor Fij are given by the minimal coupling through torsion gravity by the cova-
riant derivative
i Aj = i Aj Kijk Ak
(2)
where K represents the contortion part of the RiemannCartan connection. By using this expression for the covariant derivative of the magnetic potential vector in spacetime, in the expression for the Maxwell sector of the Lagrangean one obtains
Fik* F *ik = F 2 2 Ak klmn Kn F lm
(3)
where F* comes from the two-differential form of the e.m field coupled minimally to torsion as [9]
Fij* = Fij 2 Ak kijn Kn
(4)
where Ki is the Cartan torsion vector and kijn is the totally skew-symmetric Levi-Civita symbol. The mag-
netic field tensor non-miminally coupled would be F ij = i A j j Ai . To simplify matters, only the axial
part of the contortion tensor Kijk in the form
K i = ijkl K jkl
(5)
Since the term Rijkl F *ij F *kl contains second-order terms in torsion they reduce to the usual term Rijkl F ij F kl . Of course this is possible since we are using Minkowski metric which does not allow us to consider post-inflationary models. The Maxwell generalised equations in Fourier space reduce to
i Ai = A = ik A = 0
(6)
where k is the wave vector chosen here as k = (0,k,0). With this choice the remaining Maxwell like equations read
2 k 2 A1 i 2k 2 2 k 2 K 0 A3 = 0 (7)
2 k 2 A3 i 2k 2 2 k 2 K 0 A1 = 0 (8)
where is the conformal time given by the Minkowskian metric ds2 = d 2 dx2 . Here the ansatz for the magnetic vector potential is A = A ei and K 0 is
0
the only contortion component different from zero. Here also, A0 and A2 components vanish. The frequency is to be determined by the determinant of the Jaco-
bian J part of the matrix obtained from the last two
equations
2 k2
i 2k2 2 k2K 0
i 2k 2 2 k 2 K 0 (9)
2 k 2
Thus by considering the two dimensional column
matrix AT = A1, A3 one may write the Maxwell
generalised equation as
JA = 0
(10)
Therefore the equation DetJ = 0 allows us to obtain the dispersion relation for the photon-torsion spacetime background as
=
k
1
1
2k2
K
0
(11)
2
Now to compute the components of the magnetic field
in terms of the vector electromagnetic potential one
simply use the expression
B = A = ik A
(12)
Which yields
B3 = ik 2 A1
(13)
B1 = ik 2 A3
(14)
Taking now the square root of these expressions as our magnetic field one obtains the following expression
B 1011 K 0
(15)
on a coherent scale of 100Mpc one obtains B 106 G which is the observed galactic magnetic field. This is obtained for a torsion field compatible to the one obtained from Hughes-Drever experiment estimates [11] as K 0 1017 m1 .
3. Discussion and Conclusion
It is shown here that by considering only first order effects on spacetime torsion, and photon-torsion minimal coupling the photon mass introduced by the gauge symmetry breaking does not appear in the Maxwell generalised equations which allows us to obtain the galactic magnetic field without dynamo amplification. On a future publication one could address the problem of investigating post-inflationary models primordial field generation by considering the extension of the equations here to include a Friedmann cosmology background.
4. Acknowledgements
I would like to express my gratitude to D. Sokoloff and R. Soldati for helpful discussions on the subject of this paper. Financial support from CNPq. and University of State of Rio de Janeiro (UERJ) are grateful acknowledged.
REFERENCES
[1] A. M. Howard and R. Kulsrud, “The Evolution of a Primordial Galactic Magnetic Field,” The Astrophysical Journal, Vol. 483, No. 2, 1997, p. 483. doi:10.1086/304262
[2] J. B. Jimenez and A. L. Maroto, “On the Electromagnetic
Copyright © 2012 SciRes.
IJAA
182
L. C. GARCIA DE ANDRADE
Nature of Dark Energy and the Origin of Cosmic Magnetic Fields,” Progress of Theoretical Physics Supplement, in Press.
[3] V. de Sabbata and M. Gasperini, “Gauge Invariance, SemiMinimal Coupling, and Propagating Torsion,” Physical Review D, Vol. 23, 1982, p. 2116.
[4] V. de Sabbata and C. Sivaram, “Spin and Torsion in Gravitation,” World Scientific, Singapore, 1995.
[5] L. C. G. de Andrade and C. Sivaram, “Torsion, Massive Electrodynamics and Primordial Magnetic Fields in the Early Universe,” Astrophysics and Space Science, Vol. 209, No. 1, 1993, pp. 109-111. doi:10.1007/BF00666441
[6] R. Opher and U. F. Wichoski, “Origin of Magnetic fields in the Universe and due to Non-Minimal Coupling of Gravitational and Electromagnetic Coupling,” Physical Review Letters, Vol. 78, No. 5, 1997, p. 787. doi:10.1103/PhysRevLett.78.787
[7] A Kostelecky, N. Russell and J. Tasson, “Constraints on
Torsion from Lorents Violation,” Physical Review Letters,
Vol. 100, 2008, Article ID: 111102.
[8] L. C. G. de Andrade, “Galactic Dynamos from Torsion Flat Modes and Lorents Violation,” Physical Letters B, 2011.
[9] L. C. G. de Andrade, “Constraints on Torsion from the Bosonic Sector of Lorents Violation and Magnetogenesis Data,” Nuclear Physics B, Vol. 847, No. 3, 2011, p. 485.
[10] I. T. Drummond and S. J. Hathrell, “QED Vacuum Polarisation in Background Gravitational Field and Its Influence in the Velocity of Photon,” Physical Review D, Vol. 22, No. 2, 1980, p. 343. doi:10.1103/PhysRevD.22.343
[11] B. R. Heckel, E. G. Adelberger, C. E. Cramer, T. S. Cook, S. Schlamminger and U. Schmidt, “Preffered Frame and CP Violation Tests with Polarised Electrons,” Physical Review D, Vol. 78, 2008, Article ID: 092006. doi:10.1103/PhysRevD.78.092006
Copyright © 2012 SciRes.
IJAA
Broken Symmetries in Spacetime with Torsion and Galactic Magnetic Fields without Dynamo Amplification
Luis Carlos Garcia de Andrade
Department of Theoretical Physics, IF-UERJ, Rio de Janeiro, Maracanã Email: [email protected]
Received February 6, 2012; revised March 15, 2012; accepted March 26, 2012
ABSTRACT
Since Kostelecky et al. [Physical Review Letters 100, 111102 (2008)], have shown that there is an intimate connection between spacetime with torsion and the possibility of constraining it to Lorentz violation, a renewed interest in torsion theories of gravity has arised. In this paper, minimal coupling between photons on a torsioned background is shown to allow us to obtain the galactic magnetic field strength μG without dynamo amplification. This agrees with recent results by Jimenez and Maroto (2011) for spiral galaxies, with galactic magnetic field constraints from Dark matter without dynamo amplification. The approach discussed here allows us to get rid of the unpleasant photon mass by simply considering the Lagrangean cut off for second order torsion terms. Therefore though the gauge and Lorentz symmetries are broken here one does not have to deal with photon masses.
Keywords: Star Dynamos; Torsion Theories; Lorentz-Violation; Cosmology; Astro-Particle Physics
1. Introduction
There are many papers where galactic magnetic fields [1] and not only primordial fields can be obtained from a Biermann battery type and not from a usual dynamo mechanism of magnetic field amplification. More recently Jimenez and Maroto cite [2] have shown that in the presence of dark energy, magnetic fields are generated without any amplification. They obtained for spiral galaxies fields of the strength of 10–9 G in accordance with Schuster-Blackett conjecture that the magnetic fields are obtained from angular momentum. Earlier on De Sabbata and Gasperini [3] showed that this conjecture could be placed in the realm of Einstein-Cartan gravity [4], where spin of the elementary particles are the Cartan torsion source [5]. More recent Opher and Wichoski [6] have shown that the same mechanism can be used to obtain galactic magnetic fields without dynamos amplification. Also Kostelecky et al. [7] investigated the Lorentz violation (LV) of the fermionic sector of a Lagrangean, in Riemann-Cartan spacetime the curvature tensor Rijkl and the e.m sector F ij F kl , (i, j = 0, 1, 2, 3) given by Rijkl F ij F kl displays the same symmetries of LV term, where the Riemann-Cartan curvature tensor, including torsion terms plays the role of the Higgs sector constants kijkl . The main difference from previous papers [8] is that one uses here a minimal coupling instead the non-minimal photon-torsion coupling where the magnetic vector potential does not interact directly
with torsion or contortion part in the QED Lagrangean. In this paper, we show that the use of this photon sector coupled with torsion flat mode, via non-minimal coupling yields naturally the breaking of gauge fields and a strength for the galactic magnetic fields of the observed G value without the dynamo amplification. Besides the advantage of the approach here is that by considering only first order terms on torsion, one is able to avoid the uncomfortable photon tiny mass. The paper is organised as follows : In Section 2 we consider the Riemann-Cartan (RC) sector of the Lagrangean obtain a 2D system of Maxwell generalised equation which are similar to the ones obtained for the neutrino mass oscillation. Section 5 contains conclusions and discussions.
2. Flat Photon-Torsion Minimal Coupling and Magnetic Field Amplification
Since the torsion effects are cosmological weak in the galaxy formation post inflationary era, compared to curvature effects of Einstein gravity sector, second-order torsion effects are neglected in the torsioned quantum electrodynamics (QED). Photon and gravity torsion sectors are given by [9]
S = 1
1
d4 x g 2
1
F ij Fij
aR*F ij Fij
m2
4
(1)
bRi*k F il Flk
cRi*jkl F ij F kl
dDi
F
ij
Dk
F
k j
Copyright © 2012 SciRes.
IJAA
L. C. GARCIA DE ANDRADE
181
The physical constants a,b, c, d are obtained by
means of the conventional Feynman diagram techniques [10]. Here the e.m field tensor Fij are given by the minimal coupling through torsion gravity by the cova-
riant derivative
i Aj = i Aj Kijk Ak
(2)
where K represents the contortion part of the RiemannCartan connection. By using this expression for the covariant derivative of the magnetic potential vector in spacetime, in the expression for the Maxwell sector of the Lagrangean one obtains
Fik* F *ik = F 2 2 Ak klmn Kn F lm
(3)
where F* comes from the two-differential form of the e.m field coupled minimally to torsion as [9]
Fij* = Fij 2 Ak kijn Kn
(4)
where Ki is the Cartan torsion vector and kijn is the totally skew-symmetric Levi-Civita symbol. The mag-
netic field tensor non-miminally coupled would be F ij = i A j j Ai . To simplify matters, only the axial
part of the contortion tensor Kijk in the form
K i = ijkl K jkl
(5)
Since the term Rijkl F *ij F *kl contains second-order terms in torsion they reduce to the usual term Rijkl F ij F kl . Of course this is possible since we are using Minkowski metric which does not allow us to consider post-inflationary models. The Maxwell generalised equations in Fourier space reduce to
i Ai = A = ik A = 0
(6)
where k is the wave vector chosen here as k = (0,k,0). With this choice the remaining Maxwell like equations read
2 k 2 A1 i 2k 2 2 k 2 K 0 A3 = 0 (7)
2 k 2 A3 i 2k 2 2 k 2 K 0 A1 = 0 (8)
where is the conformal time given by the Minkowskian metric ds2 = d 2 dx2 . Here the ansatz for the magnetic vector potential is A = A ei and K 0 is
0
the only contortion component different from zero. Here also, A0 and A2 components vanish. The frequency is to be determined by the determinant of the Jaco-
bian J part of the matrix obtained from the last two
equations
2 k2
i 2k2 2 k2K 0
i 2k 2 2 k 2 K 0 (9)
2 k 2
Thus by considering the two dimensional column
matrix AT = A1, A3 one may write the Maxwell
generalised equation as
JA = 0
(10)
Therefore the equation DetJ = 0 allows us to obtain the dispersion relation for the photon-torsion spacetime background as
=
k
1
1
2k2
K
0
(11)
2
Now to compute the components of the magnetic field
in terms of the vector electromagnetic potential one
simply use the expression
B = A = ik A
(12)
Which yields
B3 = ik 2 A1
(13)
B1 = ik 2 A3
(14)
Taking now the square root of these expressions as our magnetic field one obtains the following expression
B 1011 K 0
(15)
on a coherent scale of 100Mpc one obtains B 106 G which is the observed galactic magnetic field. This is obtained for a torsion field compatible to the one obtained from Hughes-Drever experiment estimates [11] as K 0 1017 m1 .
3. Discussion and Conclusion
It is shown here that by considering only first order effects on spacetime torsion, and photon-torsion minimal coupling the photon mass introduced by the gauge symmetry breaking does not appear in the Maxwell generalised equations which allows us to obtain the galactic magnetic field without dynamo amplification. On a future publication one could address the problem of investigating post-inflationary models primordial field generation by considering the extension of the equations here to include a Friedmann cosmology background.
4. Acknowledgements
I would like to express my gratitude to D. Sokoloff and R. Soldati for helpful discussions on the subject of this paper. Financial support from CNPq. and University of State of Rio de Janeiro (UERJ) are grateful acknowledged.
REFERENCES
[1] A. M. Howard and R. Kulsrud, “The Evolution of a Primordial Galactic Magnetic Field,” The Astrophysical Journal, Vol. 483, No. 2, 1997, p. 483. doi:10.1086/304262
[2] J. B. Jimenez and A. L. Maroto, “On the Electromagnetic
Copyright © 2012 SciRes.
IJAA
182
L. C. GARCIA DE ANDRADE
Nature of Dark Energy and the Origin of Cosmic Magnetic Fields,” Progress of Theoretical Physics Supplement, in Press.
[3] V. de Sabbata and M. Gasperini, “Gauge Invariance, SemiMinimal Coupling, and Propagating Torsion,” Physical Review D, Vol. 23, 1982, p. 2116.
[4] V. de Sabbata and C. Sivaram, “Spin and Torsion in Gravitation,” World Scientific, Singapore, 1995.
[5] L. C. G. de Andrade and C. Sivaram, “Torsion, Massive Electrodynamics and Primordial Magnetic Fields in the Early Universe,” Astrophysics and Space Science, Vol. 209, No. 1, 1993, pp. 109-111. doi:10.1007/BF00666441
[6] R. Opher and U. F. Wichoski, “Origin of Magnetic fields in the Universe and due to Non-Minimal Coupling of Gravitational and Electromagnetic Coupling,” Physical Review Letters, Vol. 78, No. 5, 1997, p. 787. doi:10.1103/PhysRevLett.78.787
[7] A Kostelecky, N. Russell and J. Tasson, “Constraints on
Torsion from Lorents Violation,” Physical Review Letters,
Vol. 100, 2008, Article ID: 111102.
[8] L. C. G. de Andrade, “Galactic Dynamos from Torsion Flat Modes and Lorents Violation,” Physical Letters B, 2011.
[9] L. C. G. de Andrade, “Constraints on Torsion from the Bosonic Sector of Lorents Violation and Magnetogenesis Data,” Nuclear Physics B, Vol. 847, No. 3, 2011, p. 485.
[10] I. T. Drummond and S. J. Hathrell, “QED Vacuum Polarisation in Background Gravitational Field and Its Influence in the Velocity of Photon,” Physical Review D, Vol. 22, No. 2, 1980, p. 343. doi:10.1103/PhysRevD.22.343
[11] B. R. Heckel, E. G. Adelberger, C. E. Cramer, T. S. Cook, S. Schlamminger and U. Schmidt, “Preffered Frame and CP Violation Tests with Polarised Electrons,” Physical Review D, Vol. 78, 2008, Article ID: 092006. doi:10.1103/PhysRevD.78.092006
Copyright © 2012 SciRes.
IJAA