Can the Blackett conjecture directly account for the magnetic fields of celestial bodies and galaxies? And, is a lab-based test for the Blackett conjecture feasible?
aa r X i v : . [ phy s i c s . g e n - ph ] S e p Can the Blackett conjecture directly account forthe magnetic fields of celestial bodies and galaxies?And, is a lab-based test for the Blackett conjecture feasible?
Leonardo Campanelli ∗ All Saints University, 5145 Steeles Ave., M9L 1R5, Toronto (ON), Canada (Dated: October 7, 2020)No.
Introduction. – The Blackett hypothesis [1] is a con-jecture according to which any neutral massive rotatingmacroscopic body should possess a magnetic moment µ proportional to its angular momentum J according to µ = β √ G c J . (1)Here, G is the Newton constant, c is the speed of light,and β is a free dimensionless constant of order unity.(In this paper, we use Gaussian-cgs units.) This effectwould have his origin in a putative fundamental unifiedtheory of gravitation and electromagnetism, in which the“gravitational magnetism” would emerge.The plausibility of the Blackett hypothesis reposes en-tirely on empirical “evidences”. Indeed, Blackett ob-served in his 1947-paper [1] that the magnetic field cal-culated from Eq. (1) agrees with its observed value forthe Earth, Sun, and 78 Virginis (a spectral type B2star). About 32 years later, Sirag [2] tested the Blackettconjecture using new available data for Mercury, Venus,Jupiter, Saturn, the Moon, and the pulsar Her X-1.Equation (1), once again, seemed to be relatively success-ful in explaining the observed magnetic fields of celestialbodies.It is important to stress two points. First, at thosetimes there was no satisfactory explanation for the ex-istence of the earth magnetic field and in general ofmagnetic fields of planets and stars. Second, althoughone could expect a correlation between the angular mo-mentum and the magnetic field of a rotating magnetizedbody, the impressive feature is that such a (linear) cor-relation extended over 15 orders of magnitude in both J and µ . Therefore, the Blackett hypothesis, renewed bySirag, was perhaps a legitimate tentative to give a the-oretical explanation for the magnetization of so vastlydifferent celestial bodies.More recently, Opher and Wichoski [3] have appliedthe Blackett conjecture to the study of galactic magneticfields. Their results suggest that the Blackett effect di-rectly accounts for the magnetization of galaxies if theBlackett constant β is in the range 10 − . β . − .Jimenez and Maroto [4], on the other hand, have shownthat the Blackett hypothesis naturally emerges in an elec-tromagnetic theory that includes nonminimal couplingsto the spacetime curvature. These analyses seem, onceagain, not to rule out the Blackett hypothesis. Recently enough, instead, Barrow and Gibbons [5] have somehow“relaxed” the Blackett conjecture by suggesting that theBlackett’s constant is bounded above by a number of or-der unity, and have verified their conjecture for (classical)charged rotating black holes in theories where the exactsolution is known. Limit on Blackett’s constant. – Planets and satellitesof the solar system are neutral rotating systems which,according to the Blackett conjecture, should be magne-tized, and indeed they are, as revealed by the data ofa number of spacecrafts [6]. Approximating such sys-tems as spheres of radius R , the average magnetic field B inside (and on the surface) is proportional to the mag-netization, B = 2 µ /R [7]. Outside the systems, themagnetic field is that of a magnetic dipole with magneticmoment µ . The angular momentum can be written as J = 2 πI/P , where P is the intrinsic rotational period, I = kM R is the moment of inertia, M the mass, and k is the moment-of-inertia parameter (which for an ho-mogeneous and perfectly spherical object is equal to 1).A strong constraint on β is given by the non-observation of a dipolar magnetic field of Mars (yet aresidual crustal magnetization has been detected, whichseems to point towards an extinct dynamo action). Usingthe upper limit on the Martian magnetic dipole moment, µ . × G cm [8], we find β . × − , (2)where we used M = 6 . × kg, R = 3390km, k =0 . P = 1 .
03d [6]. To our knowledge, this is thestrongest constraint on the Blackett’s constant. (In themodel of Jimenez and Maroto, the model-dependent limiton the Blackett’s constant comes from the constraints onthe parameterized post-Newtonian parameters and turnsto be of order of β . − [4].)With such a low value for the Blackett’s constant, plan-etary magnetic fields and magnetic fields in stars andgalaxies cannot be directly explained by the Blackett con-jecture (the magnetic field produced by the Blackett ef-fect could act, eventually and at most, as a “seed” forthose fields). Moreover, the above limit on β makes notfeasible, at the present time, a direct lab-based test ofthe Blackett conjecture, as we show below. Barnett effect vs. Blackett effect. – It is well knownthat any (neutral) body rotating at an angular velocity ω acquires a magnetic dipole moment. This effect of “mag-netization by rotation” is known as Barnett effect [9].For a homogeneous diamagnetic or paramagnetic solidoccupying a volume V , the magnetic dipole µ is [7] µ = 2 m e ce χ g − V ω , (3)where m e and e are the mass and electric charge of theelectron, χ is the volume susceptibility, and g is the gy-roscopic g -factor.For a sphere of radius R (the main results do notchange if we consider different shapes), the ratio betweenthe magnetic moment given by the Blackett conjectureand the one given by the Barnett effect is then µ ( Blackett) µ ( Barnett) ∼ − (cid:18) β − (cid:19)(cid:18) − cm / g χ m (cid:19)(cid:18) R (cid:19) , (4)where χ m = χ/ρ is the mass susceptibility and ρ the den-sity. To our knowledge, there are not known solid ma-terials with mass susceptibility below 10 − cm / g. Equa-tion (4), then, shows that the Blackett effect is alwayssubdominant with respect to the Barnett one for lab-scaleobjects. Our conclusion is that, at the present time, theBlackett effect cannot be tested in a laboratory.The Blackett effect could eventually be tested if a ma-terial with a mass susceptibility as low as 10 − cm / gwere synthesized. This is in principle possible if onecombines two or more inert materials with different mag-netic properties. According to the Wiedemann’s addi-tivity law [10], the mass susceptibility of a mixture ofits paramagnetic and diamagnetic components would be χ m = P i m i χ ( i ) m / P i m i , where m i and χ ( i ) m are the massand mass susceptibility of the component i . Thus, anappropriate choice of the mass percentage of each con-stituent in powder form would give the possibility of ob-taining a material with magnetic susceptibility as low asdesired. The resulting powder could be then sinteredand made into a solid. It is worth noticing that sucha procedure has been already applied by Khatiwada etal. to produce a solid material with very low (volume)susceptibility composed by tungsten and bismuth [11].However, even if the resulting solid pellets were compactenough to stay together they were delicate. Accordingto Khatiwada et al., the pressing procedure could be fur-ther enhanced by using higher pressures and tempera-tures to produce strong solids. Even if this were possible,however, the resulting solid material should have a suf-ficiently large volume, and be dense and strong enoughin order to produce a detectable magnetic field once it isput into rotational motion, as we discuss below. Blackett-type experiment. – Let us consider a homo-geneous rotating sphere made of a hypothetical materialwhose mass susceptibility is such that the Blackett ef-fect is dominant with respect to the Barnett one. Themaximum safe angular speed ω can be found as follows. The stress tensor in spherical coordinates r, θ, φ can bewritten as σ ij = c ij ( ν, θ, r ) ρ ω R , where c ij ( ν, θ, r ) isa dimensionless tensor, ν is the Poisson’s ratio [12], and i, j = r, θ, φ . Here, σ rr is the radial stress, σ rθ is the shearstress, and σ θθ and σ φφ are the angular normal stresses(all the other components of the stress tensor are zeroby symmetry). Using the results of [13] we find that,for given density, angular speed, and radius, the maxi-mum stress corresponds to the angular normal stressesand in particular max θ,r c θθ = max θ,r c φφ = c ν , where c ν = (5 ν − ν − / (25 ν + 10 ν − ≤ ν ≤ /
2, which is certainly true for all metalsand known alloys. Theoretically, the Poisson’s ratio isin the range − ≤ ν ≤ / c ν is anincreasing function of ν such that c = 12 / ≃ .
34 and c / = 9 / ≃ . σ max , | σ ij | < σ max . This, inturn, determines the maximum possible value for the an-gular speed, ω max = ( σ max /c ν ρ ) / /R . Inserting thisvalue of ω in Eq. (1), we find the maximum magneticfield that can be generated by a rotating sphere, B max ∼ − (cid:18) β − (cid:19)(cid:16) σ max (cid:17) / (cid:18) ρ / cm (cid:19) / (cid:18) R (cid:19) G . (5)For a given radius R , then, the maximum magnetic fieldis large for materials with high density and ultimate ten-sile stress, such as metals and alloys (the dependence of B max and ω max on the Poisson’s ratio is very week).In order to detect B max , or to put a limit on the Black-ett’s constant more stringent than the one in Eq. (1),the hypothetical material must have a sufficiently highultimate tensile stress and density. Indeed, taking β = 10 − , a 2-meter sphere ( R = 1m) would pro-duce a maximal magnetic field of order of B max ∼ − ( σ max / / ( ρ/
1g cm − ) / T. The most sen-sitive magnetometers are SQUID magnetometers, withmaximum sensitivities of order of 1fT / √ Hz [14], andSERF magnetometers, with maximum sensitivities ofabout 0 . / √ Hz [15]. Even taking the maximum theo-retical sensitivity of a SERF magnetometer, estimated tobe 2aT [16], the hypothetical material must satisfy themechanical condition ( σ max / / ( ρ/
1g cm − ) / & . ρ ≃ . / cm and σ max ≃ . σ max ≃ ρ ≃ . / cm [18].) Conclusions. – The Blackett effect is a hypothetical ef-fect consisting in the magnetization by rotation of a rigidneutral body that should emerge from a unified theoryof gravitation and electromagnetism.We have derived a stringent constraint on the Black-ett’s constant, the dimensionless constant of proportion-ality between the magnetization and the angular momen-tum of a body, by using the data on the dipolar magneticfield of Mars. This constraint excludes the possibilitythat the Blackett effect could directly account for plane-tary, stellar, and galactic magnetic fields.We have also pointed out that the Blackett effect issimilar but subdominant for lab-scale objects with re-spect to the well-known and experimentally tested Bar-nett effect, according to which any rotating object ac-quires a magnetic moment proportional to its angularvelocity. The Blackett effect, then, cannot be tested in alaboratory.We would like to thank M. Giannotti for useful discus-sions. ∗ Electronic address: [email protected][1] P. M. S. Blackett, Nature (London) , 658 (1947).[2] S. P. Sirag, Nature , 535 (1979).[3] R. Opher and U. F. Wichoski, Phys. Rev. Lett. , 787(1997).[4] J. B. Jimenez and A. L. Maroto, JCAP , 025 (2010).[5] J. D. Barrow and G. W. Gibbons, Phys. Rev. D , 064040 (2017).[6] G. Schubert and K. M. Soderlund, Phys. Earth PlanetInt. , 92 (2011).[7] L. D. Landau and E. M. Lifshitz, Electrodynamics ofContinuous Media (Pergamon Press, Oxford, England,1984).[8] M. H. Acuna et al. , J. Geophys. Res. , 23403 (2001).[9] S. J. Barnett, Phys. Rev. , 239 (1915).[10] P. W. Kuchel, B. E. Chapman, W. A. Bubb,P. E. Hansen, C. J. Durrant, and M P. Hertzberg, Conc.Magn. Reson. A , 56 (2003).[11] R. Khatiwada, L. Dennis, R. Kendrick, M. Khosravi,M. Peters, E. Smith, and W. M. Snow, Meas. Sci. Tech-nol. , 025902 (2015).[12] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon Press, Oxford, England, 1970).[13] W. H. Muller and P. Lofink, “The movement of theEarth: modelling the flattening parameter,” in:
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