Fermion Black Hole Similarity

Fermion – black hole similarity

and black holes magnetic field.

Corrado Massa

Via Fratelli Manfredi 55 42124 Reggio Emilia Italia [email protected] [email protected]

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.

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 ).

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

μ / J ( μ = magnetic moment) of a charged black

hole is equal to Q / M just as for an electron [ 1 ].

What follows is a wide speculation that, thinking

such similarity consistently through to the end, gets three interesting consequences.

The first one springs from Dirac’s wave equation in

five dimensions. I remember that Dirac’s equation,

usually written in a four dimensional form, can be

more naturally written in a five – dimensional form

because of the existence of five anticommuting

Dirac matrices.

The five dimensional form, considered in the

context of the Kaluza – Klein unified field theory,

results in an anomalous magnetic moment term

in four dimensions given by μ = G 1 / 2 ( s / c ) where c is the speed of light and G is the Newton

constant [ 2 , 3 , 4 ]

Any fermion with spin s is expected to have this

intrinsic magnetic moment which adds to the

eventual magnetic moment term of ordinary

electromagnetic origin.

If we assume a complete fermion–black hole

similarity, we conclude that any neutral black

hole with intrinsic angular momentum S has

the magnetic moment

μ = G 1 / 2 ( S / c ) ( 1 )

The related dipolar magnetic field near the

horizon of a neutral black hole with mass M is expected to be ( here and below numerical

factor of the order of unity are neglected )

B = μ / R 3 = S c 5 / ( M 3 G 5 / 2 ) ( 2 )

where R = G M / c 2 is the black hole “ radius”.

With S = G M 2 / c = the maximal angular

momentum of a spinning black hole, we have

a magnetic field of strength:

B = A / M ( 3 )

where A = c 4 / G 3 / 2 ~ 5 x 10 52 g 3 / 2 s – 1 cm – 1 / 2 .

For a stellar black hole ( M ~ 10 34 g ) B ~ 10 18 gauss.

The second consequence is the electromagnetic

power output due to the fall of matter into a

black hole. Matter falling into a black hole can be a significant source of gravitational waves,

and if m is the mass of an infalling lump of matter

then the total energy emitted is about ( m c ) 2 / M

if m ~ M [ 1 a ] . A gravitational wave passing through

a magnetic field shakes the magnetic field and

generates electromagnetic waves; the gravitational

radiation is totally converted into electromagnetic

radiation if

B L ~ c 2 / G 1 / 2 ~ 10 2 4 gauss x cm ( 4 )

where L is the length of the path that the

gravitational radiation walks along.

If we put eq ( 3 ) into eq ( 4 ) with L = R = = G M / c 2 we see that condition ( 4 ) is

satisfied, and the infalling matter of mass m

will radiate an electromagnetic power output

( m / M )2 ( c 5 / G ) ~ ( m / M ) 2 x 10 59 erg / s

That for m ~ M equals the maximal power

In the world [ 5 ].

The third consequence is that black holes might

exhibit quantum behavior, with a De Broglie

wavelength λ = S / p ( p = the black hole linear

momentum. This could be related to the observed

quantized redshifts of galaxies, since most galaxies

lodge gigantic ( M ~ 10 40 g ) black holes in

their nuclei. The idea of a macroscopic form of quantum mechanics is not new, see e.g. [ 6 ]

and references therein.

References

[ 1 ] Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation (Freeman and Company, San Francisco

1973) box 33.2, p. 883.

[ 1 a ] ibidem, Ch. 36, Sec. 5, p. 982.

[ 2 ] Pauli, W.: Annalen der Physik vol. 18,

p. 337 (1933) see p. 372

[ 3 ] Barut, A.O. and Gornitz, Th.:

Foundations of Physics vol. 15, p. 433 (1985).

[ 4 ] Hosoya, A. Ishikawa, K., Ohkuwa, Y.

and Yawagishi, K.:

Physics Letters B vol. 134, p. 44 (1984)

[ 5 ] Massa, C.: Astrophysics and Space Science,

vol. 232, p. 143 ( 1995 )

[ 6 ] Massa, C.: Annalen der Physik,

vol 45, p. 391, ( 1988 )