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
Emai l: garcia@dft .if.u erj.br
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 dy-
namo amplification. The approach discussed here allows us to get rid of the unpleasant photon mass by simply consid-
ering 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 me-
chanism 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 ampli-
fication. Also Kostelecky et al. [7] investigated the
Lorentz violation (LV) of the fermionic sector of a Lag-
rangean, in Riemann-Cartan spacetime the curvature
tensor ijkl and the e.m sector Rij kl
F
F, (i, j = 0, 1, 2, 3)
given by displays the same symmetries of
LV term, where the Riemann-Cartan curvature tensor,
including torsion terms plays the role of the Higgs sector
constants ijkl . 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
ij kl
ijkl
RFF
k
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 coup-
ling 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 o scillation. 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 cur-
vature effects of Einstein gravity sector, second-order
torsion effects are neglected in the torsioned quantum
electrodynamics (QED). Photon and gravity torsion se-
ctors are given by [9]

1
4*
2
2
**
11
=d 4
ij ij
ij ij
il kij klijk
iklijklik j
SxgFFaRFF
m
bRF FcRF FdDF DF

 



 

 (1)
C
opyright © 2012 SciRes. IJAA

L. C. GARCIA DE ANDRADE 181
The physical constants are obtained by
means of the conventional Feynman diagram techniques
[10]. Here the e.m field tensor ij
,,,abc d
F
are given by the
minimal coupling through torsion gravity by the cova-
riant derivative
=
i
ji jijk
k
A
AKA (2)
where K represents the contortion part of the Riemann-
Cartan 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
** 2
=2
ikklmn lm
ikk n
F
FFAKF   (3)
where *
F
comes from the two-differential form of the
e.m field coupled minimally to torsion as [9]
*=2 k
ijijkijn n
F
FAK   (4)
where i
K
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
=
ijj iij
F
AA. To simplify matters, only the axial
part of the contortion tensor ijk
K
in the form
=
i ijkl
j
kl
K
K  (5)
Since the term contains second-order
terms in torsion they reduce to th e usual term .
Of course this is possible since we are using Minkowski
metric which does not allow us to consider post-infla-
tionary models. The Maxwell generalised equations in
Fourier space reduce to
**ij kl
ijkl
RFF
ij kl
ijkl
RFF
===
i
iAiAkA0
0
0
(6)
where k is the wave vector chosen here as k = (0,k,0).
With this choice the remaining Maxwell like equations
read

221 22203
2=kAi kkKA



 
 (7)

223 22201
2=kA ikkKA



 
 (8)
where

is the conformal time given by the Min-
kowskian metric 22
d=dd
2
s
x  . Here the ansatz for
the magnetic vector potential is 0
=i
A
Ae


and 0
K
is
the only contortion component different from zero. Here
also, 0
A
and 2
A
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
 


  
2222 20
222 022
2
2
kikk
ik kKk






 

K





(9)
Thus by considering the two dimensional column
matrix


13
=,
T
A
AA one may write the Maxwell
generalised equation as
=0JA (10)
Therefore the equation allows us to obtain
the dispersion relation for the photon-torsion spacetime
background as
=0DetJ
22 0
=112
k
kK







 




 (11)
Now to compute the components of the magnetic field
in terms of the vector electromagnetic potential one
simply use the expression
==i
B
AkA

(12)
Which yields
32
=BikA
1
3
K
(13)
12
=BikA (14)
Taking now the square root of these expr ession s as our
magnetic field one obtains the following expression
11 0
10B (15)
on a coherent scale of 100
M
pc one obtains
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
6
10BG


017
10 1
K
m


.
3. Discussion and Conclusion
It is shown here that by considering only first order ef-
fects on spacetime torsion, and photon-torsion minimal
coupling the photon mass introduced by the gauge sym-
metry breaking does not appear in the Maxwell gene-
ralised equations which allows us to obtain the galactic
magnetic field without dynamo amplification. On a fu-
ture publication one could address the problem of in-
vestigating post-inflationary models primordial field ge-
neration 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 pa-
per. 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 Pri-
mordial 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

L. C. GARCIA DE ANDRADE 182
Nature of Dark Energy and the Origin of Cosmic Mag-
netic Fields,” Progress of Theoretical Physics Supplement,
in Press.
[3] V. de Sabbata and M. Gasperini, “Gauge Invariance, Semi-
Minimal Coupling, and Propagating Torsion,” Physical
Review D, Vol. 23, 1982, p. 2116.
[4] V. de Sabbata and C. Sivaram, “Spin and Torsion in Gra-
vitation,” 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 Po-
larisation in Background Gravitational Field and Its In-
fluence 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