Sphere r = a

E.B=0

˜˜

E>B Singular Ring

E>B

Cut

E=B

E<B

E>B

Fig. 25.1. Planar cut through the origin, orthogonal to the z = 0 plane, showing

the delineation of regions of E > B and E < B, for the potential given by eq.

(28.1).

Lynden-Bell

372

On the cut itself we have R < a and z = 0+.

3/2 3/2

E + iB = q (R ’ ia) i a2 ’ R2 = ’q (a + iR) a2 ’ R2 .

This gives an electric ¬eld vertically down into the disc and a magnetic ¬eld

parallel to the disk surface for R < a as though the disk has a Meissner

e¬ect. The corresponding charge density on the symmetry plane is

’3/2

σ = ’ (q /2π ) a a2 ’ R2 .

This charge density gives a divergent total charge but that divergence is

cancelled by a ring of opposite charge on the edge which leaves the total

charge not ˜negative™ but ˜positive™ +q. The total charge at axial distance

less than R is Q(< R) = ’q a(a2 ’ R2 )’1/2 ’ 1 , R < a. From the dis-

continuity in the B ¬eld across the cut we ¬nd 4πJφ = ’2qR(a2 ’ R2 )’3/2 .

This corresponds to the charge density given above rotating with angular

velocity „¦ = c/a, reaching the velocity of light at the singularity. Again its

e¬ect is reversed by a ring current at the edge. The ¬elds are illustrated in

Figures 25.2 and 25.3.

25.2 The connection to Kerr™s metric and the electron

A much more complicated but more intriguing derivation of the above re-

sults is to take the Kerr (1963) metric of a black hole of mass m and

angular momentum mac. Then complexify it following Newman (1973) to

get the Kerr-Newman metric of charge q, (Newman et al. 1965). Finally, take

the limit with G ’ 0 leaving the charge and the moment corresponding to

˜a™ but now in ¬‚at space. The resultant electromagnetic ¬eld is exactly that

derived and discussed above, (Pekeris & Frankowski 1987). Carter (1968a)

showed that all the Kerr-Newman metrics had the same gyromagnetic ra-

tio as the Dirac electron. Does this mean that there is some relationship

between the charge distribution of the Kerr-Newman metric and the charge

distribution of the quantum electrodynamic ¬eld of a point electron?

Classical models of the electron had a problem over the gyromagnetic

ratio. Even if all the charge were con¬ned to a ring rotating at close to

the velocity of light the magnetic moment generated gives a gyromagnetic

ratio of one rather than the electron™s value of 2.0023193044. It is of some

interest to gain an understanding as to how the Kerr-Newman metric does

it. The answer is that the charge distribution is not all of one sign. In fact

a circular current dipole of two rings of opposite charge rotating uniformly

about their common axis gives a net magnetic moment but no net charge.

A magic electromagnetic ¬eld 373

Fig. 25.2. A plot of electric ¬eld lines for the potential given by eq. (28.1).

Fig. 25.3. A plot of magnetic ¬eld lines for the potential given by eq. (28.1).

Lynden-Bell

374

The way our electromagnetic ¬eld gets its large magnetic dipole moment

per unit net charge is that its much larger internal charges are of opposite

signs but rotate together giving a magnetic dipole with relatively little net

charge. We show elsewhere that this is a characteristic of relativistically

rotating conductors!

25.3 Separability of motion in the ¬eld

Studies of separability of wave equations in the Kerr and Kerr-Newman

metrics (Carter 1968b, Teukolsky 1972, 1973, Chandrasekhar 1976, Page

1976) have shown that Dirac™s equation is separable in these metrics. This

of course implies that it is still separable in their ¬‚at space limit as G ’

0. The criterion for the separability of Schr¨dinger™s equation in a real

o

potential in spheroidal coordinates is ¦ = [ζ(») ’ ·(µ)] /(» ’ µ) (Morse

and Feshback 1953). Here » and µ are spheroidal coordinates and ζ, · are

arbitrary functions of their arguments.

The ¬eld that we derived so simply above is rewritten in spheroidal coor-

dinates as follows: » and µ are the roots for „ of the quadratic

x2 + y 2 z 2

+ =1,

a2 + „ „

where x2 + y 2 = R2 = » + a2 µ + a2 /a2 and the metric is

»’µ »’µ

ds2 = dx2 + dy 2 + dz 2 = d»2 + dµ2 + R2 dφ2 .

2) 2)

4» (» + a 4µ (µ + a

To compare to Kerr™s metric one uses the quasi-spherical form of spheroidal

√

coordinates r = » , µ = ’a2 cos2 ‘, z = r cos ‘. Note however that r is

constant on spheroids and r = 0 is the disc z = 0, R ¤ a. Also ‘ is not the

θ of spherical polar coordinates but is constant in hyperboloids. Thus

ds2 = r 2 + a2 cos2 ‘ / r 2 + a2 dr 2 + r 2 + a2 cos2 ‘ d‘2 +

+ r 2 + a2 sin2 ‘dφ2 .

In spheroidal coordinates our potential Ψ = q/ (r ’ ia)2 takes the simple

forms

√ √

√ √ » + i ’µ q

» ’ i ’µ = q

Ψ=q = .

»’µ r ’ ia cos ‘

The second of these forms is exactly of the right type for separability of the

Schr¨dinger equation but the similarity is partly misleading for Schr¨dinger™s

o o

equation only separates in an electrostatic potential of that form. When

A magic electromagnetic ¬eld 375

the imaginary (magnetic) part is added Schr¨dinger™s equation no longer

o

separates although the Klein-Gordon equation now does separate (which it

does not with only the electrostatic part). For a derivation and explanation

of these results see Lynden-Bell (2000).

Systems with the same charge distribution but less magnetic ¬eld are

given by taking ψ = ±Ψ + (1 ’ ±)Ψ— for ± < 1. The magnetic ¬elds are then

multiplied by 2± ’ 1. These are weighted superpositions of discs rotating

forwards and backwards so the net rotation is less fast and ± = 1/2 is static.

These ¬elds lose the magic of separability. For the other charge & current

distributions with that property see Lynden-Bell (2000).

25.4 Eulogy

In closing, let me say that I still do not know the answer to the problem