Can magnetic fields of astrophysical objects be fundamental

a rXiv:g r-qc/97354v 121Mar1997CAN MAGNETIC FIELDS OF ASTROPHYSICAL OBJECTS BE FUNDAMENTAL?Tonatiuh Matos ∗,Instituto de F´ısica y Matem´a ticas,Universidad Michoacana de San Nicolas de Hidalgo,PO.Box 2-82,58040Morelia,Michoac´a n,M´e xico February 7,2008Abstract We analyze a new class of static exact solutions of Einstein-Maxwell-Dilaton gravity with arbitrary scalar coupling constant α,representing a gravitational body endowed with electromagnetic dipole moment.This class possesses mass,dipole and scalar charge parameters.A dis-cussion of the geodesic motion shows that the scalar field interaction is so weak that it cannot be measured in gravitational fields like thesun,but it could perhaps be detected in gravitational fields like pul-sars.The scalar force can be attractive or repulsive.This gives rise to the hypothesis that the magnetic field of some astrophysical objects could be fundamental.PACS numbers:04.50.+h,04.20Jb,04.90+eA great amount of astrophysical objects in Cosmos are gravitational bodies with magnetic dipolefields.One should suppose that the Einstein-Maxwell(EM)theory predicts the existence of gravitational objects endowed with magnetic dipoles.In fact there is a set of exact solutions of the EM equations representing exteriorfields of gravitational objects endowed with magnetic dipoles[1].Some of them are reasonably small,but they do not have the right behavior of the gravitationalfield far away from the sources; the other ones are acceptable in their behavior at infinity[2],but the num-ber of terms of them is so enormous that it makes them unmanageable.On the other hand,EM theory actually is not an unification theory,but rather a superposition one,Einstein plus Maxwell.Here the electromagneticfield appears as energy-momentum tensor,there is in fact no explanation of its existence,the electromagneticfield appears like a model.For other theories like Kaluza-Klein(KK)and Low Energy Superstring(LESS)theories,the electromagneticfield is a component of a more generalfield,the existence of gravitation and electromagnetism follows from its decomposition.In these theories the electromagneticfield is a consequence of a more general uni-fiedfield,it is not a model.In[3]and[4]it is shown that the existence of electromagnetic dipoles is natural for LESS and KK but not so natural for EM.A class of solutions given in[3]possesses a gravitationalfield with the behavior of the Schwarzschild solution coupled with a magnetic dipole. They are reasonably small,but they possess a scalarfield interaction,the so-called dilaton.Of course we have not observed astrophysical objects with a scalarfield interaction,but its prediction in KK and LESS theories should be established at classical level if such theories could be taken as realistic.In fact there are enough classical objects in nature with manifest gravitational-electromagnetical interactions.KK and LESS predict the existence of the dilaton at this level.In this work we will show that the dilaton interaction cannot be measured in weak gravitationalfields like the sun,even if the sun would posses one,but it will be perhaps possible to measure it in stronger gravitationalfields like a pulsar.According to KK and LESS theories,since these solutions posses a magnetic dipole moment parameter and a newtonian behavior at infinity,this gives rise to the hypothesis that the magneticfield of some astrophysical objects could be of fundamental origin,i.e.,the magnetic field could be a consequence of a more general scalar-gravito-electromagnetic field.In a previous work[3]we presented a method forfinding exact solutions of the KKfield equations.These solutions represent exteriorfields of a gravi-2tational body,endowed with arbitrary electromagneticfields like monopoles, dipoles,etc.or the superposition of them,from thefive-dimensional point of view.Here there exists a coupling between the electromagnetic and a scalar field,parametrized by a coupling constantα2=3.In other work[4]we generalized this method for arbitraryαin order to incorporate all the most important theories unifying gravitation and electromagnetism;KK,LESS and EM.The solutions found in[4]are naked singularities in r=2m for α=0.Nevertheless,we will investigate them for r>>2m supposing that they represent the exteriorfield of some astrophysical body.In the present work we analyze explicit solutions of thefield equations of the LagrangianL=√r1−2m(1−2m(λ,ζ−τ0τ,ζ)2,k e,ζ=−ργq1q2(τζ)2,τ0=q1+q2,2α2and the subclass b)ρg=a1τ+1,k s,ζ=whereζ=ρ+i z=√1+α2,β=2α2r ).τdetermines uniquely theelectromagnetic potential.Two examples are the magnetic monopole τ=ln(1−2m(r−m)2−m2cos2θ,A3=Q(r−m)sin2θIn this work we are interested in extracting some physics from dilatontheories.In order to do so,we study the geodesic motion of test particles traveling around the space time(2).Since e2(k s+k e)−1∼10−11for a star like the sun,metric(2)is spherically symmetric in this aproximation.We startfrom the LagrangianL=e2(k s+k e)gγ(dr1−2m ds)2−1−2m gγ(dt2,inthis case the functionτfor the dipolefield does not contribute and g=1. But in general,for any value ofθ,the function g changes only very near to the Schwarzschild radius r s=2m,but it tends very rapidly to one far away from r s,for any value ofθ.In any case,in the following we will set g in all the equations where it appears.Following any standard text book on gravitation,wefirst write the motion equations.We have two constants of motion,δLrds)=AδLds=Bso dtdscan be put in terms of A and ing the equation of motionPµPµ=−c2one obtains−ǫ=e2(k e+k s)gγ(dr1−2m ds)2−1−2m gγ(dtds )2+e−2(k s+k e)r2gγ+ǫ](1−2mvariation with respect to the coordinate t .Let us define an effective potential byV eff =e −2(k s +k e )r 2g γ+ǫ](1−2m2e −2(k e +k s )A 2.(10)in order to obtain the familiar form for the motion equation 1ds )2+V eff =E eff .This interpretation is suggested by performing a series expansion for r >>2m .Nevertheless,definitions(9)and (10)have not necessarily a physical meaning in general.Figure 1:The effective potentials for the magnetized Schwarzschild solutions for the KK (a =1m=4.In the following we will take only the subcase b)of (2),here the function k e =0and the constant τ0=0as well.If θ=πr)2r a (ǫr+B 2r 3)6E eff = (1−m 1−2m 2where a =0for the Schwarzschild space time and a =1rr )2 a as the contribution of thedilatonfield to the effective potential V eff and to the effective energy E eff ,and thefunction g as the contribution of the electromagnetic field.In figure 1we have plotted the effective potential for the different theories.The qualitative behavior is very similar in all of them.In figure 2we see the effective energy for the same values of α,the behavior is here very violent;not so far away from the Schwarzschild radius,the effective energy isconstant.Figure 2:The effective energy for the magnetized Schwarzschild solutions.The plot is made in m unities on the horizontal axis and in E 0unities on the vertical axis.In order to obtain the trajectories of a test particle travelling around of a star of sun’s size,we make the standard transformation u (ϕ)=11−2mu a [(1−2mu )(B 2u 2+ǫ)−A 2]=0,(11)where a prime means derivative with respect to ϕ.This is a first order differential equation of the form 1which define naturally the function V(u).After derivation with respect toϕ,equation(12)transforms into a equation of the form u′′+∂u V(u)=0. This differential equation is very difficult to solve and we will not try tosolve it here.But for a trajectory around a star like the sun,the massparameter m∼1.5Km.,while r∼106Km.,therefore u3∼0is a good approximation,conserving the rest of the terms.In that case the geodesicequation transforms intou′′+ω2u=mǫ1−am2B2.The difference with theSchwarzschild geodesic equation is that for the Schwarzschild caseω=K= 1.Following the standard procedure,wefind that the trajectories are ellipses with a perihelia precession given by∆ϕp=6πm2c2Kb(1−e2)KB2m2A2<<a 1+am2B2m2c2=ac2M2⊙=a2.4810−8.Here G is the universalgravitation’s constant,B=2.781015m2/seg is the angular momentum of Mercure per unity of mass and M⊙is the mass of the sun.This means that the difference between the Schwarzschild geodesics and the(11)geodesics for stars like the sun is then too small to be measured.Let us assume for a moment that we could take these theories as realistic,then we conclude that if a star of the size of the sun contains a scalarfield inherent in it,we could8not know it because its interaction with the rest of the world is too small to be detected.Nevertheless,for a pulsar of mass M=2M⊙,which matter is typicaly contained in a radius of r=10Km∼3m,the scalar interaction cannot be neglected.Thus,such interactions should be detectable in stronger gravitationalfields like pulsars,where the gravitationalfield is much more stronger.We have seen that the KK and the LESS theories predict the existence of magnetic dipoles coupled with gravitational objects naturally,here the electromagneticfield is a consequence of the natural coupling predicted by the theory.If we would like to model a pulsar by such a theory,we would not need to explain the origin of the magnetic dipole in it using internal hypothesis, since this magnetic dipole would be then a consequence of some more general interaction between gravitation and electromagnetism.The price we must pay is the existence of a scalarfield which has not been measured till now. Nevertheless,the KK and the LESS theories predict that even if the magnetic dipolefield can be felt around the body,the scalarfield interaction is so weak that it can be measured only near to a distance of order of the Schwarzschild radius r s.This is so because of the behavior of the scalarfield(see Fig.3)Φ=1r)+βln(Q cosθr s=2m,and it is attractive or repulsive depending onαis positive or neg-ative.Hence,according to these theories,there exist objects which posses a fundamental magnetic dipole moment,which is a consequence of a more general gravito-electromagnetic interaction which posses a scalarfield.Oth-erwise,according to these theories,even if an astrophysical object like the sun would posses a scalarfield inherent in it,we would not be able to measure it because of the small force provoked by it.Nevertheless,this attractive or repulsive scalar force could have effects in stronger gravitationalfields that we should see in astrophysical bodies,but to predict them,we must solve the geodesic equation(8)near to r s.This work is partially supported by CONACYT-Mexico. 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