<p>Fermion – black hole similarity </p><p> and black holes magnetic field. </p><p>Corrado Massa</p><p>Via Fratelli Manfredi 55 42124 Reggio Emilia Italia [email protected] [email protected]</p><p>Poki ciacer: Disen kal bus nigher angamia camp magnetic, mo l’ é mia veira: al bus nigher al ga al sò bel camp, e po’ anka grooss: 101 8 gauss. </p><p>Abstract: The impressive similarities between fermions and black holes suggest that any neutral black hole with intrinsic angular momentum J has the intrinsic magnetic moment ( J / c ) √ G ( c is the speed of light and G is the gravitational constant ).</p><p>Elementary particles are characterized uniquely by three parameters: electric charge, mass, and spin. Black holes too are characterized uniquely by three parameters: Q = their electric charge, M = their mass and J = their intrinsic angular momentum (analogous to the spin). Furthermore, the gyromagnetic ratio </p><p>μ / J ( μ = magnetic moment) of a charged black </p><p> hole is equal to Q / M just as for an electron [ 1 ].</p><p>What follows is a wide speculation that, thinking </p><p> such similarity consistently through to the end, gets three interesting consequences. </p><p>The first one springs from Dirac’s wave equation in</p><p> five dimensions. I remember that Dirac’s equation, </p><p> usually written in a four dimensional form, can be </p><p> more naturally written in a five – dimensional form </p><p> because of the existence of five anticommuting </p><p>Dirac matrices. </p><p>The five dimensional form, considered in the </p><p> context of the Kaluza – Klein unified field theory, </p><p> results in an anomalous magnetic moment term </p><p> in four dimensions given by μ = G 1 / 2 ( s / c ) where c is the speed of light and G is the Newton </p><p> constant [ 2 , 3 , 4 ] </p><p>Any fermion with spin s is expected to have this </p><p> intrinsic magnetic moment which adds to the </p><p> eventual magnetic moment term of ordinary </p><p> electromagnetic origin.</p><p>If we assume a complete fermion–black hole </p><p> similarity, we conclude that any neutral black </p><p> hole with intrinsic angular momentum S has </p><p> the magnetic moment</p><p>μ = G 1 / 2 ( S / c ) ( 1 ) </p><p>The related dipolar magnetic field near the</p><p> horizon of a neutral black hole with mass M is expected to be ( here and below numerical </p><p> factor of the order of unity are neglected )</p><p>B = μ / R 3 = S c 5 / ( M 3 G 5 / 2 ) ( 2 ) </p><p> where R = G M / c 2 is the black hole “ radius”.</p><p>With S = G M 2 / c = the maximal angular </p><p> momentum of a spinning black hole, we have</p><p> a magnetic field of strength:</p><p>B = A / M ( 3 ) </p><p> where A = c 4 / G 3 / 2 ~ 5 x 10 52 g 3 / 2 s – 1 cm – 1 / 2 . </p><p>For a stellar black hole ( M ~ 10 34 g ) B ~ 10 18 gauss. </p><p>The second consequence is the electromagnetic </p><p> power output due to the fall of matter into a </p><p> black hole. Matter falling into a black hole can be a significant source of gravitational waves,</p><p> and if m is the mass of an infalling lump of matter</p><p> then the total energy emitted is about ( m c ) 2 / M </p><p> if m ~ M [ 1 a ] . A gravitational wave passing through </p><p> a magnetic field shakes the magnetic field and </p><p> generates electromagnetic waves; the gravitational </p><p> radiation is totally converted into electromagnetic </p><p> radiation if </p><p>B L ~ c 2 / G 1 / 2 ~ 10 2 4 gauss x cm ( 4 ) </p><p> where L is the length of the path that the </p><p> gravitational radiation walks along. </p><p>If we put eq ( 3 ) into eq ( 4 ) with L = R = = G M / c 2 we see that condition ( 4 ) is </p><p> satisfied, and the infalling matter of mass m </p><p> will radiate an electromagnetic power output </p><p>( m / M )2 ( c 5 / G ) ~ ( m / M ) 2 x 10 59 erg / s </p><p>That for m ~ M equals the maximal power </p><p>In the world [ 5 ]. </p><p>The third consequence is that black holes might</p><p> exhibit quantum behavior, with a De Broglie </p><p> wavelength λ = S / p ( p = the black hole linear </p><p> momentum. This could be related to the observed</p><p> quantized redshifts of galaxies, since most galaxies</p><p> lodge gigantic ( M ~ 10 40 g ) black holes in </p><p> their nuclei. The idea of a macroscopic form of quantum mechanics is not new, see e.g. [ 6 ]</p><p> and references therein. </p><p>References </p><p>[ 1 ] Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation (Freeman and Company, San Francisco </p><p>1973) box 33.2, p. 883.</p><p>[ 1 a ] ibidem, Ch. 36, Sec. 5, p. 982.</p><p>[ 2 ] Pauli, W.: Annalen der Physik vol. 18,</p><p> p. 337 (1933) see p. 372 </p><p>[ 3 ] Barut, A.O. and Gornitz, Th.: </p><p>Foundations of Physics vol. 15, p. 433 (1985).</p><p>[ 4 ] Hosoya, A. Ishikawa, K., Ohkuwa, Y. </p><p> and Yawagishi, K.: </p><p>Physics Letters B vol. 134, p. 44 (1984) </p><p>[ 5 ] Massa, C.: Astrophysics and Space Science, </p><p> vol. 232, p. 143 ( 1995 ) </p><p>[ 6 ] Massa, C.: Annalen der Physik, </p><p> vol 45, p. 391, ( 1988 ) </p>