The Kerr-Newman Black Hole

Schwarzschild's solution to Einstein's free-field equation for gravity describes the field around a spherically symmetric body with no electrodynamic field. Thus the body has mass but no charge, magnetic dipole or higher-order distribution of electromagnetic properties; and it is not even spinning (which would break spherical symmetry by having a preferred axis). During the half century following this first solution's discovery, assorted researchers explored the other properties an isolated body surrounded by empty space might have, and what fields would ensue. Of particular interest was the study of cases where the only non-vacuum place in the system is a singularity – such solutions are known as black holes. Kerr (1963) handled the case of a spinning black hole and Newman (et al., 1965) worked out what happens if it's charged; and it's since been shown that charge, mass and angular momentum (a.k.a. spin) are all the properties a static (i.e. time-independent) black hole can posess. (This is usually expressed by the slogan black holes have no hair for reasons I don't know off hand, aside from their being related to the hairy ball theorem of topology – any continuous vector field on (all of) the surface of a sphere necessarily takes the value zero somewhere on the surface.)

Space-time's metric and the electromagnetic field around the Kerr-Newman black hole with charge, mass and angular momentum encoded by lengths Q, M and an area J are given [I've renamed φ as m, and replaced θ with n, which differs from it by a quarter turn] as

g = R.(dr×dr / Δ +dn×dn) +(

((r.r +a.a).dm −a.dt)×((r.r +a.a).dm −a.dt).cos(n).cos(n)

−Δ.(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm)

)/R
F = (

dr^(dt −a.cos(n).cos(n).dm).(r.r −a.a.sin(n).sin(n))

−2.a.r.sin(n).cos(n).dn^((r.r+a.a).dm −a.dt)

).Q/R/R

wherein t is a time co-ordinate (scaled by the speed of light, c, so as to give it units of length), m is a longitude measured round the spin axis from some fixed half-plane bounded by that axis, n is a lattitude measured relative to the spin axis and a plane, n=0, normal to it, in which the system has reflection symmetry (i.e. it notionally meets the axis at the massive body) and r is a radial distance – we may think of it as distance from a central point at which the n=0 plane meets the axis, though it isn't entirely sensible to discuss this point, since it's where our description of space-time breaks down. The following abbreviations have been used

a = J / M
R = r.r +a.a.sin(n).sin(n)
Δ = r.r −2.M.r +a.a +Q.Q

and √R may be thought of as an adjusted radius (indeed, my source for the above called it ρ; I have, here, replaced ρ.ρ with R). To recover the actual charge, mass, angular momentum and time from Q, M, J and t, we need to scale the latter by assorted combinations of G (Newton's gravitational constant), c (the speed of light) and Z₀ (the impedance of free space); thankfully, Planck's constant is not needed. Our time co-ordinate, measured in units of time, is t/c; the black hole's mass is c.c.M/G, its charge is Q.c.√(c/Z₀/G) and its angular momentum is J.c.c.c/G (with J = M.a having the units of length.length, a.k.a. area). The given conversion factors are:

c = 299792458 metre / second, 1/c ≈ 3.336 × 10⁻⁹ second / metre
c.c/G = 1.3471 × 10²⁷ kilogramme / metre ≈ 0.6772 Solar mass / km
c.√(c/Z₀/G) = c/√(μ₀.G) = c.c.√(ε₀/G) = 32.741 × 10¹⁵ Coulomb / metre
c.c.c/G = 403.8 × 10³³ Joule.second / metre²

(the first due to the definition of the metre, the rest depending on the value of G, which we do not know very accurately). Note that a = J/M is a length while dividing the black hole's angular momentum by its mass will yield a.c, an area/time (a.k.a. length times speed). If we take a=0 and Q=0, making F=0, notice that √R reduces to r and Δ/R reduces to 1−2.M/r, so that g reduces to the Schwarzschild metric.

Verification

To verify that the above is a solution we need to compute its Ricci tensor (which has to yield, when added to a suitably scaled F/g\F, a multiple of the metric), a trace of the electromagnetic field's derivative (which must equal the suitably scaled 4-current; which is zero except at the spatial origin) and the alternating derivative of the electromagnetic field (which must be zero). For all but the last, we need the covariant derivative, which deems the metric constant.

Let us, then, explore the differential operator which annihilates this metric, along with the Riemann and Ricci tensors it begets. We are using transpose([t,r,n,m]) as a chart of space-time; these beget (at each point) a basis b = [dt,dr,dn,dm] of gradients; let p = [s,q,u,w] be the dual basis (at each point) of tangents (defined by b(i)·p(j) = 0 if i and j differ, else 1). First, let us establish that the inverse of g is

g⁻¹

= ( Δ.q×q +u×u: +(w +a.cos(n).cos(n).s)×(w +a.cos(n).cos(n).s)/cos(n)/cos(n); −(a.w +(r.r +a.a).s)×(a.w +(r.r +a.a).s)/Δ )/R

= ( Δ.q×q +u×u: +w×w/cos(n)/cos(n) −a.a.w×w/Δ; +a.w×s −(r.r +a.a).a.w×s/Δ; +a.s×w −(r.r +a.a).a.s×w/Δ; +a.a.cos(n).cos(n).s×s −(r.r +a.a).(r.r +a.a).s×s/Δ )/R

= ( Δ.q×q +u×u: +(Δ −a.a.cos(n).cos(n)).w×w/Δ/cos(n)/cos(n); +a.(Δ −(r.r +a.a)).(w×s +s×w)/Δ; +( a.a.cos(n).cos(n).Δ −(r.r +a.a).(r.r +a.a) ).s×s/Δ )/R

= ( Δ.q×q +u×u: +(R +Q.Q −2.M.r).w×w/Δ/cos(n)/cos(n); +a.(Q.Q −2.M.r).(w×s +s×w)/Δ; +( a.a.cos(n).cos(n).(Q.Q −2.M.r) −R.(r.r +a.a) ).s×s/Δ )/R

by contracting it (in its first given form – which combines w and s in exactly the ways needed to produce vectors annihilated by the covectors whose squares appear in g – though I'll use the last below) with g:

( R.(dr×dr / Δ +dn×dn) +( ((r.r +a.a).dm −a.dt)×((r.r +a.a).dm −a.dt).cos(n).cos(n) −Δ.(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm) )/R )·( Δ.q×q +u×u +(w +a.cos(n).cos(n).s)×(w +a.cos(n).cos(n).s)/cos(n)/cos(n) −(a.w +(r.r +a.a).s)×(a.w +(r.r +a.a).s)/Δ )/R

= dr×q +dn×u +(: ((r.r +a.a).dm −a.dt)×((r.r +a.a).dm −a.dt).cos(n).cos(n); −Δ.(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm)
)·(: (w +a.cos(n).cos(n).s)×(w +a.cos(n).cos(n).s)/cos(n)/cos(n); −(a.w +(r.r +a.a).s)×(a.w +(r.r +a.a).s)/Δ
)/R/R

= dr×q +dn×u +(: ((r.r +a.a).dm −a.dt)×((r.r +a.a).dm −a.dt)·(w +a.cos(n).cos(n).s)×(w +a.cos(n).cos(n).s); −cos(n).cos(n).((r.r +a.a).dm −a.dt)×((r.r +a.a).dm −a.dt)·(a.w +(r.r +a.a).s)×(a.w +(r.r +a.a).s)/Δ; −Δ.(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm)·(w +a.cos(n).cos(n).s)×(w +a.cos(n).cos(n).s)/cos(n)/cos(n); +(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm)·(a.w +(r.r +a.a).s)×(a.w +(r.r +a.a).s)
)/R/R

= dr×q +dn×u +(: (r.r +a.a −a.a.cos(n).cos(n)).((r.r +a.a).dm −a.dt)×(w +a.cos(n).cos(n).s); +(r.r +a.a −a.a.cos(n).cos(n)).(dt −a.cos(n).cos(n).dm)×(a.w +(r.r +a.a).s)
)/R/(r.r +a.a.sin(n).sin(n))

= dr×q +dn×u +(: (r.r +a.a).dm×w +a.cos(n).cos(n).(r.r +a.a).dm×s; −a.dt×w −a.a.cos(n).cos(n).dt×s; +a.dt×w +(r.r +a.a).dt×s; −a.a.cos(n).cos(n).dm×w −a.cos(n).cos(n).(r.r +a.a).dm×s
)/R

= dr×q +dn×u +(: (r.r +a.a −a.a.cos(n).cos(n)).dm×w; +(r.r +a.a −a.a.cos(n).cos(n)).dt×s
)/(r.r +a.a.sin(n).sin(n))

= dr×q +dn×u +dm×w +dt×s

which is, indeed, the identity on co-vectors. That the contraction the other way round will yield the transpose of this, the identity on vectors, follows from the manifest symmetry of both g and its proposed inverse, along with the symmetry of contraction's restriction to vectors and co-vectors.

Covariant Derivative

We must first compute the derivatives of the components of the metric, with respect to our given chart. Our metric has diagonal elements for all four basis vectors plus off-diagonal elements for dm×dt and dt×dm; all other co-ordinates are zero, hence have zero derivative (w.r.t. co-ordinates). Thus we have:

d(u·g·u)

= dR

= d(r.r +a.a.sin(n).sin(n))

= 2.r.dr +2.a.a.sin(n).cos(n).dn

d(q·g·q)

= d(R/Δ)

= dR/Δ −R.dΔ/Δ/Δ

= 2.(r.dr +a.a.sin(n).cos(n).dn)/Δ −2.(r−M).dr.R/Δ/Δ

= 2.a.a.sin(n).cos(n).dn/Δ +2.(r.Δ −r.R +M.R).dr/Δ/Δ

d(s·g·s)

= d((−Δ +a.a.cos(n).cos(n))/R)

= d((2.M.r −Q.Q)/R −1)

= 2.M.dr/R −(2.M.r−Q.Q).(2.r.dr +2.a.a.sin(n).cos(n).dn)/R/R

= 2.(dr.(M.R +(Q.Q −2.M.r).r) +(Q.Q −2.M.r).a.a.sin(n).cos(n).dn)/R/R

d(s·g·w) = d(w·g·s)

= a.d(cos(n).cos(n).(Δ −r.r −a.a)/R)

= a.d(cos(n).cos(n).(Q.Q −2.M.r)/R)

= a.( −2.sin(n).cos(n).dn.(Q.Q −2.M.r)/R: −2.cos(n).cos(n).M.dr/R; −2.cos(n).cos(n).(Q.Q −2.M.r).(r.dr +a.a.sin(n).cos(n).dn)/R/R )

= −2.a.( sin(n).cos(n).(Q.Q −2.M.r).R.dn: +cos(n).cos(n).M.R.dr +cos(n).cos(n).(Q.Q −2.M.r).r.dr; +a.a.sin(n).cos(n).cos(n).cos(n).(Q.Q −2.M.r).dn )/R/R

= −2.a.cos(n).( sin(n).(r.r +a.a).(Q.Q −2.M.r).dn +cos(n).(M.R +(Q.Q −2.M.r).r).dr )/R/R

d(w·g·w)

= d(cos(n).cos(n).( (r.r +a.a).(r.r +a.a) −a.a.cos(n).cos(n).Δ )/R)

= d(cos(n).cos(n).( (r.r +a.a).(r.r +a.a) −a.a.cos(n).cos(n).(r.r +a.a +Q.Q −2.M.r) )/R)

= d(cos(n).cos(n).( (r.r +a.a).(r.r +a.a.sin(n).sin(n)) −a.a.cos(n).cos(n).(Q.Q −2.M.r) )/R)

= d(cos(n).cos(n).( r.r +a.a −a.a.cos(n).cos(n).(Q.Q −2.M.r)/R ))

= 2.cos(n).cos(n).( r.dr: +a.a.sin(n).cos(n).dn.(Q.Q −2.M.r)/R +M.a.a.cos(n).cos(n).dr/R; +a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.dr +a.a.sin(n).cos(n).dn)/R/R
) −2.sin(n).cos(n).dn.(r.r +a.a −a.a.cos(n).cos(n).(Q.Q −2.M.r)/R)

= 2.sin(n).cos(n).dn.((r.r +a.a).(r.r +a.a).(Q.Q −2.M.r)/R/R −Δ): +2.cos(n).cos(n).dr.(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R)

yielding (as the tensor with which each b(i) contracts to yield D(b(i))),

g \ ( (τ[0,1,2] −τ[1,0,2] −τ[2,1,0])(sum(: d(p(j)·g·p(k))×b(j)×b(k) ←[j,k] :)) ) / 2

= g \ ( (τ[0,1,2] −τ[1,0,2] −τ[2,1,0])(: d(s·g·w)×(dt×dm +dm×dt); +d(s·g·s)×dt×dt; +d(w·g·w)×dm×dm; +d(q·g·q)×dr×dr; +d(u·g·u)×dn×dn ) ) / 2

= g \ ( (τ[0,1,2] −τ[1,0,2] −τ[2,1,0])(: −a.cos(n).( sin(n).(r.r +a.a).(Q.Q −2.M.r).dn +cos(n).(M.R +(Q.Q −2.M.r).r).dr )×(dt×dm +dm×dt)/R/R; +( dr.(M.R +(Q.Q −2.M.r).r) +(Q.Q −2.M.r).a.a.sin(n).cos(n).dn )×dt×dt/R/R; +cos(n).( cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).dr +sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a)).dn )×dm×dm; +(a.a.sin(n).cos(n).dn.Δ +(r.Δ −r.R +M.R).dr)×dr×dr/Δ/Δ; +(r.dr +a.a.sin(n).cos(n).dn)×dn×dn ) )

= ( Δ.q×q +u×u: +(R +Q.Q −2.M.r).w×w/Δ/cos(n)/cos(n); +a.(Q.Q −2.M.r).w×s/Δ; +a.(Q.Q −2.M.r).s×w/Δ; +( a.a.cos(n).cos(n).(Q.Q −2.M.r) −R.(r.r +a.a) ).s×s/Δ
)·(: −a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dn×dt×dm −dt×dn×dm −dm×dt×dn +dn×dm×dt −dm×dn×dt −dt×dm×dn)/R/R; −a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dr×dt×dm −dt×dr×dm −dm×dt×dr +dr×dm×dt −dm×dr×dt −dt×dm×dr)/R/R; +(M.R +(Q.Q −2.M.r).r).(dr×dt×dt −dt×dr×dt −dt×dt×dr)/R/R; +(Q.Q −2.M.r).a.a.sin(n).cos(n).(dn×dt×dt −dt×dn×dt −dt×dt×dn)/R/R; +cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).(dr×dm×dm −dm×dr×dm −dm×dm×dr); +cos(n).sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a)).(dn×dm×dm −dm×dn×dm −dm×dm×dn); +a.a.sin(n).cos(n).(dn×dr×dr −dr×dn×dr −dr×dr×dn)/Δ; −(r.Δ −r.R +M.R).dr×dr×dr/Δ/Δ; +r.(dr×dn×dn −dn×dr×dn −dn×dn×dr); −a.a.sin(n).cos(n).dn×dn×dn
)/R

= ( Δ.q×q +u×u: +( (R +Q.Q −2.M.r).w/cos(n)/cos(n) +a.(Q.Q −2.M.r).s )×w/Δ; +( a.(Q.Q −2.M.r).w +a.a.cos(n).cos(n).(Q.Q −2.M.r).s −R.(r.r +a.a).s )×s/Δ
)·(: (M.R +(Q.Q −2.M.r).r).dr×(dt×dt)/R/R; −a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).dr×(dt×dm +dm×dt)/R/R; +cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).dr×(dm×dm); −(r.Δ −r.R +M.R).dr×dr×dr/Δ/Δ; −a.a.sin(n).cos(n).dr×(dn×dr +dr×dn)/Δ; +r.dr×(dn×dn); +(Q.Q −2.M.r).a.a.sin(n).cos(n).dn×(dt×dt)/R/R; −a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).dn×(dt×dm +dm×dt)/R/R; +cos(n).sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a)).dn×(dm×dm); +a.a.sin(n).cos(n).dn×(dr×dr)/Δ; −r.dn×(dr×dn +dn×dr); −a.a.sin(n).cos(n).dn×dn×dn; +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).dm×(dt×dn +dn×dt)/R/R; +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).dm×(dt×dr +dr×dt)/R/R; −cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).dm×(dr×dm +dm×dr); −cos(n).sin(n).( a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a) ).dm×(dn×dm +dm×dn); +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).dt×(dn×dm +dm×dn)/R/R; +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).dt×(dr×dm +dm×dr)/R/R; −(M.R +(Q.Q −2.M.r).r).dt×(dr×dt +dt×dr)/R/R; −(Q.Q −2.M.r).a.a.sin(n).cos(n).dt×(dn×dt +dt×dn)/R/R
)/R

= Δ.q×(: (M.R +(Q.Q −2.M.r).r).(dt×dt)/R/R; −a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dt×dm +dm×dt)/R/R; +cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).dm×dm; −(r.Δ −r.R +M.R).dr×dr/Δ/Δ; −a.a.sin(n).cos(n).(dn×dr +dr×dn)/Δ; +r.dn×dn )/R
+u×(: (Q.Q −2.M.r).a.a.sin(n).cos(n).dt×dt/R/R; −a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dt×dm +dm×dt)/R/R; +cos(n).sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a)).dm×dm; +a.a.sin(n).cos(n).dr×dr/Δ; −r.(dr×dn +dn×dr); −a.a.sin(n).cos(n).dn×dn )/R
+( (R +Q.Q −2.M.r).w/cos(n)/cos(n) +a.(Q.Q −2.M.r).s )×(: +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dt×dn +dn×dt)/R/R; +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dt×dr +dr×dt)/R/R; −cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).(dr×dm +dm×dr); −cos(n).sin(n).( a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a) ).(dn×dm +dm×dn) )/Δ/R
+( a.(Q.Q −2.M.r).w +a.a.cos(n).cos(n).(Q.Q −2.M.r).s −R.(r.r +a.a).s )×(: +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dn×dm +dm×dn)/R/R; +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr)/R/R; −(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr)/R/R; −(Q.Q −2.M.r).a.a.sin(n).cos(n).(dn×dt +dt×dn)/R/R )/Δ/R

from which we obtain

R.D(dr)

(M.R +(Q.Q −2.M.r).r).dt×dt.Δ/R/R: −a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dt×dm +dm×dt).Δ/R/R; +cos(n).cos(n).(r +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)/R/R).Δ.dm×dm; −(r.Δ −r.R +M.R).dr×dr/Δ; −a.a.sin(n).cos(n).(dn×dr +dr×dn); +r.dn×dn.Δ

= (M.R +(Q.Q −2.M.r).r).(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm).Δ/R/R: +cos(n).cos(n).r.Δ.dm×dm; −(r.Δ −r.R +M.R).dr×dr/Δ; −a.a.sin(n).cos(n).(dn×dr +dr×dn); +r.dn×dn.Δ

R.D(dn)

= (Q.Q −2.M.r).a.a.sin(n).cos(n).dt×dt/R/R: −a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dt×dm +dm×dt)/R/R; +cos(n).sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R)/R/R −(r.r +a.a)).dm×dm; +a.a.sin(n).cos(n).dr×dr/Δ; −r.(dr×dn +dn×dr); −a.a.sin(n).cos(n).dn×dn

= (Q.Q −2.M.r).sin(n).cos(n).(a.dt −(r.r +a.a).dm)×(a.dt −(r.r +a.a).dm)/R/R: −Δ.cos(n).sin(n).dm×dm; +a.a.sin(n).cos(n).dr×dr/Δ; −r.(dr×dn +dn×dr); −a.a.sin(n).cos(n).dn×dn

R.R.R.Δ.D(dt)

= a.(Q.Q −2.M.r).(: +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dt×dn +dn×dt); +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dt×dr +dr×dt); −cos(n).cos(n).(r.R.R +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)).(dr×dm +dm×dr); −cos(n).sin(n).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R) −(r.r +a.a).R.R).(dn×dm +dm×dn)
) +(a.a.cos(n).cos(n).(Q.Q −2.M.r) −R.(r.r +a.a)).(: +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dn×dm +dm×dn); +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr); −(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −(Q.Q −2.M.r).a.a.sin(n).cos(n).(dn×dt +dt×dn)
)

= a.a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(Q.Q −2.M.r).(dt×dn +dn×dt): +a.a.cos(n).cos(n).(Q.Q −2.M.r).(M.R +(Q.Q −2.M.r).r).(dt×dr +dr×dt); −a.cos(n).cos(n).(Q.Q −2.M.r).(r.R.R +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)).(dr×dm +dm×dr); −a.cos(n).sin(n).(Q.Q −2.M.r).(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R) −(r.r +a.a).R.R).(dn×dm +dm×dn); +a.a.a.cos(n).cos(n).cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(Q.Q −2.M.r).(dn×dm +dm×dn); +a.a.a.cos(n).cos(n).cos(n).cos(n).(Q.Q −2.M.r).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr); −a.a.cos(n).cos(n).(Q.Q −2.M.r).(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −a.a.a.a.cos(n).cos(n).cos(n).sin(n).(Q.Q −2.M.r).(Q.Q −2.M.r).(dn×dt +dt×dn); −a.cos(n).sin(n).R.(r.r +a.a).(r.r +a.a).(Q.Q −2.M.r).(dn×dm +dm×dn); −a.cos(n).cos(n).R.(r.r +a.a).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr); +R.(r.r +a.a).(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); +R.(r.r +a.a).(Q.Q −2.M.r).a.a.sin(n).cos(n).(dn×dt +dt×dn)

= ( (r.r +a.a).(Q.Q −2.M.r) −a.a.cos(n).cos(n).(Q.Q −2.M.r) +R.(r.r +a.a): ).a.a.cos(n).sin(n).(Q.Q −2.M.r).(dn×dt +dt×dn); +( a.a.cos(n).cos(n).(Q.Q −2.M.r) −a.a.cos(n).cos(n).(Q.Q −2.M.r) +R.(r.r +a.a); ).(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); +( a.a.cos(n).cos(n).(Q.Q −2.M.r).(M.R +(Q.Q −2.M.r).r); −(Q.Q −2.M.r).(r.R.R +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)); −R.(r.r +a.a).(M.R +(Q.Q −2.M.r).r); ).a.cos(n).cos(n).(dr×dm +dm×dr); +( a.a.cos(n).cos(n).(r.r +a.a).(Q.Q −2.M.r) −R.(r.r +a.a).(r.r +a.a); −(a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R) −(r.r +a.a).R.R); ).a.cos(n).sin(n).(Q.Q −2.M.r).(dn×dm +dm×dn)

= R.Δ.a.a.cos(n).sin(n).(Q.Q −2.M.r).(dn×dt +dt×dn): +R.(r.r +a.a).(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −R.a.cos(n).cos(n).( r.(r.r +a.a +R).(Q.Q −2.M.r) +(r.r +a.a).M.R ).(dr×dm +dm×dr); −R.Δ.a.a.a.cos(n).cos(n).cos(n).sin(n).(Q.Q −2.M.r).(dn×dm +dm×dn)

R.R.R.Δ.D(dm)

= (R +Q.Q −2.M.r).(: +a.tan(n).(r.r +a.a).(Q.Q −2.M.r).(dt×dn +dn×dt); +a.(M.R +(Q.Q −2.M.r).r).(dt×dr +dr×dt); −(r.R.R +a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r)).(dr×dm +dm×dr); −tan(n).( a.a.cos(n).cos(n).(Q.Q −2.M.r).(r.r +a.a +R) −(r.r +a.a).R.R ).(dn×dm +dm×dn)
) +a.(Q.Q −2.M.r).(: +a.cos(n).sin(n).(r.r +a.a).(Q.Q −2.M.r).(dn×dm +dm×dn); +a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr); −(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −(Q.Q −2.M.r).a.a.sin(n).cos(n).(dn×dt +dt×dn) )

= ( (r.r +a.a).(R +Q.Q −2.M.r) −a.a.cos(n).cos(n).(Q.Q −2.M.r): ).(Q.Q −2.M.r).a.tan(n).(dn×dt +dt×dn); +( (R +Q.Q −2.M.r) −(Q.Q −2.M.r); ).a.(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −r.R.R.(R +Q.Q −2.M.r).(dr×dm +dm×dr); +( Q.Q −2.M.r −(R +Q.Q −2.M.r); ).a.a.cos(n).cos(n).(M.R +(Q.Q −2.M.r).r).(dr×dm +dm×dr); +tan(n).(r.r +a.a).R.R.(R +Q.Q −2.M.r).(dn×dm +dm×dn); +( (r.r +a.a).(Q.Q −2.M.r) −(r.r +a.a +R).(R +Q.Q −2.M.r); ).a.a.cos(n).sin(n).(Q.Q −2.M.r).(dn×dm +dm×dn)

= Δ.R.a.tan(n).(Q.Q −2.M.r).(dn×dt +dt×dn): +a.R.(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −R.(r.R.R +a.a.cos(n).cos(n).M.R +(Q.Q −2.M.r).(r.r +a.a).r).(dr×dm +dm×dr); +Δ.R.tan(n).(R.R −a.a.cos(n).cos(n).(Q.Q −2.M.r)).(dn×dm +dm×dn)

These suffice to entirely determine the covariant derivative.

Maxwell's Equations

Maxwell's Equations, expressed in space-time form, reduce to two equations: d^F must be zero and τ[*,*,0](DF) must be equal to the suitably scaled 4-current density (whose time component is charge density, give or take a factor of the speed of light), which is zero in our case (except at the origin's singularity). The first of these doesn't depend on the details of D (because d^ annihilates all its own outputs), but the second does. We obtain:

d^F

= d^( (: dr^(dt −a.cos(n).cos(n).dm).(r.r −a.a.sin(n).sin(n)); −2.a.r.sin(n).cos(n).dn^((r.r+a.a).dm −a.dt)
).Q/R/R )

= Q.d((r.r −a.a.sin(n).sin(n))/R/R)^dr^dt: −a.Q.d(cos(n).cos(n).(r.r −a.a.sin(n).sin(n))/R/R)^dr^dm; −2.a.Q.d(r.sin(n).cos(n).(r.r+a.a)/R/R)^dn^dm; +2.a.a.Q.d(r.sin(n).cos(n)/R/R)^dn^dt

= Q.(: (2.r.dr −2.a.a.cos(n).sin(n).dn)/R/R; −2.(r.r −a.a.sin(n).sin(n)).(2.r.dr +2.a.a.cos(n).sin(n).dn)/R/R/R
)^dr^dt
−a.Q.(: cos(n).cos(n).(2.r.dr −2.a.a.cos(n).sin(n).dn)/R/R; −2.sin(n).cos(n).dn.(r.r −a.a.sin(n).sin(n))/R/R; −2.cos(n).cos(n).(r.r −a.a.sin(n).sin(n)).(2.r.dr +2.a.a.sin(n).cos(n).dn)/R/R/R
)^dr^dm
−2.a.Q.(: dr.sin(n).cos(n).(r.r+a.a)/R/R; +r.(cos(n).cos(n) −sin(n).sin(n)).dn.(r.r+a.a)/R/R; +2.r.r.dr.sin(n).cos(n)/R/R; −2.r.sin(n).cos(n).(r.r+a.a).(2.r.dr +2.a.a.sin(n).cos(n).dn)/R/R/R
)^dn^dm
+2.a.a.Q.(: dr.sin(n).cos(n)/R/R; +r.(cos(n).cos(n) −sin(n).sin(n)).dn/R/R; −2.r.sin(n).cos(n).(2.r.dr +2.a.a.sin(n).cos(n).dn)/R/R/R
)^dn^dt

= −2.Q.(R +2.(r.r −a.a.sin(n).sin(n))).a.a.cos(n).sin(n).dn^dr^dt/R/R/R: +2.a.Q.sin(n).cos(n).(R.(r.r −a.a.sin(n).sin(n)) +(R +2.(r.r −a.a.sin(n).sin(n)) ).a.a.cos(n).cos(n)).dn^dr^dm/R/R/R; −2.a.sin(n).cos(n).Q.((r.r+a.a).R +2.r.r.(R −2.(r.r+a.a))).dr^dn^dm/R/R/R; +2.a.a.sin(n).cos(n).Q.(R −4.r.r).dr^dn^dt/R/R/R

= 2.Q.a.a.cos(n).sin(n).( R +2.(r.r −a.a.sin(n).sin(n)) +R −4.r.r ).dr^dn^dt/R/R/R: +2.Q.a.sin(n).cos(n).( R.(r.r −a.a.sin(n).sin(n)) +(3.r.r −a.a.sin(n).sin(n)).a.a.cos(n).cos(n) +(r.r+a.a).R −2.r.r.(2.r.r +2.a.a −R) ).dn^dr^dm/R/R/R

= 0

as it should.

τ[*,*,0](g\D(F))

= τ[*,*,0](g\D( (: (r.r −a.a.sin(n).sin(n)).(dr×dt −dt×dr)/2; +a.cos(n).cos(n).(r.r −a.a.sin(n).sin(n)).(dm×dr −dr×dm)/2; +a.r.(r.r+a.a).sin(n).cos(n).(dm×dn −dn×dm); +a.a.r.sin(n).cos(n).(dn×dt −dt×dn)
).Q/R/R ))

= Q.τ[*,*,0](g\( (τ[2,0,1] −τ[1,0,2])(: (r.r −a.a.sin(n).sin(n)).(dt×D(dr) −dr×D(dt))/2; +a.cos(n).cos(n).(r.r −a.a.sin(n).sin(n)).(dr×D(dm) −dm×D(dr))/2; +a.r.(r.r+a.a).sin(n).cos(n).(dn×D(dm) −dm×D(dn)); +a.a.r.sin(n).cos(n).(dt×D(dn) −dn×D(dt)) )/R/R; +(r.dr −a.a.sin(n).cos(n).dn)×(dr×dt −dt×dr)/R/R; +a.(cos(n).cos(n).(r.dr −a.a.sin(n).cos(n).dn) −cos(n).sin(n).dn.(r.r −a.a.sin(n).sin(n)))×(dm×dr −dr×dm)/R/R; +a.(dr.(3.r.r+a.a).sin(n).cos(n) +r.(r.r+a.a).(cos(n).cos(n) −sin(n).sin(n)).dn)×(dm×dn −dn×dm)/R/R; +a.a.(dr.sin(n).cos(n) +r.(cos(n).cos(n) −sin(n).sin(n)).dn)×(dn×dt −dt×dn)/R/R
−4.(r.dr +a.a.sin(n).cos(n).dn)×(: (dr×dt −dt×dr).(r.r −a.a.sin(n).sin(n))/2; +a.cos(n).cos(n).(dm×dr −dr×dm).(r.r −a.a.sin(n).sin(n))/2; +a.r.(r.r+a.a).sin(n).cos(n).(dm×dn −dn×dm); +a.a.r.sin(n).cos(n).(dn×dt −dt×dn)
)/R/R/R ))

= Q.τ[*,*,0](g\(

R.(τ[2,0,1] −τ[1,0,2])(: a.r.sin(n).cos(n).(a.dt −(r.r+a.a).dm)×D(dn); +(r.r −a.a.sin(n).sin(n)).(dt −a.cos(n).cos(n).dm)×D(dr)/2; −(a.a.r.sin(n).cos(n).dn +(r.r −a.a.sin(n).sin(n)).dr)×D(dt)/2; +a.cos(n).(cos(n).(r.r −a.a.sin(n).sin(n)).dr +r.(r.r+a.a).sin(n).dn)×D(dm); )

+(: R.(r.dr −a.a.sin(n).cos(n).dn); −2.(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dr×dt −dt×dr)

+a.cos(n).(: R.(cos(n).(r.dr −a.a.sin(n).cos(n).dn) −sin(n).(r.r −a.a.sin(n).sin(n)).dn); −2.cos(n).(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dr −dr×dm)

+a.(: R.(dr.(3.r.r+a.a).sin(n).cos(n) +r.(r.r+a.a).cos(2.n).dn); −4.r.(r.r+a.a).sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dn −dn×dm)

+a.a.(: R.(sin(n).cos(n).dr +r.cos(2.n).dn); −4.r.sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dn×dt −dt×dn)

))/R/R/R

= Q.τ[*,*,0](g\(

R.(τ[2,0,1] −τ[1,0,2])(

a.r.sin(n).cos(n).(a.dt −(r.r +a.a).dm)×(: (Q.Q −2.M.r).sin(n).cos(n).(a.dt −(r.r +a.a).dm)×(a.dt −(r.r +a.a).dm)/R/R; −Δ.cos(n).sin(n).dm×dm; +a.a.sin(n).cos(n).dr×dr/Δ; −r.(dr×dn +dn×dr); −a.a.sin(n).cos(n).dn×dn; )/R

+(r.r −a.a.sin(n).sin(n)).(dt −a.cos(n).cos(n).dm)×(: (M.R +(Q.Q −2.M.r).r).(dt −a.cos(n).cos(n).dm)×(dt −a.cos(n).cos(n).dm).Δ/R/R; +cos(n).cos(n).r.Δ.dm×dm; −(r.Δ −r.R +M.R).dr×dr/Δ; −a.a.sin(n).cos(n).(dn×dr +dr×dn); +r.dn×dn.Δ; )/2/R

−(a.a.r.sin(n).cos(n).dn +(r.r −a.a.sin(n).sin(n)).dr)×(: R.Δ.a.a.cos(n).sin(n).(Q.Q −2.M.r).(dn×dt +dt×dn); +R.(r.r +a.a).(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −R.a.cos(n).cos(n).( r.(r.r +a.a +R).(Q.Q −2.M.r) +(r.r +a.a).M.R ).(dr×dm +dm×dr); −R.Δ.a.a.a.cos(n).cos(n).cos(n).sin(n).(Q.Q −2.M.r).(dn×dm +dm×dn); )/2/R/R/R/Δ

+a.cos(n).(cos(n).(r.r −a.a.sin(n).sin(n)).dr +r.(r.r +a.a).sin(n).dn)×(: Δ.R.a.tan(n).(Q.Q −2.M.r).(dn×dt +dt×dn); +a.R.(M.R +(Q.Q −2.M.r).r).(dr×dt +dt×dr); −R.(r.R.R +a.a.cos(n).cos(n).M.R +(Q.Q −2.M.r).(r.r +a.a).r).(dr×dm +dm×dr); +Δ.R.tan(n).(R.R −a.a.cos(n).cos(n).(Q.Q −2.M.r)).(dn×dm +dm×dn); )/R/R/R/Δ

)

+(: R.(r.dr −a.a.sin(n).cos(n).dn); −2.(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dr×dt −dt×dr)

+a.cos(n).(: R.(cos(n).(r.dr −a.a.sin(n).cos(n).dn) −sin(n).(r.r −a.a.sin(n).sin(n)).dn); −2.cos(n).(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dr −dr×dm)

+a.(: R.(dr.(3.r.r+a.a).sin(n).cos(n) +r.(r.r +a.a).cos(2.n).dn); −4.r.(r.r +a.a).sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dn −dn×dm)

+a.a.(: R.(sin(n).cos(n).dr +r.cos(2.n).dn); −4.r.sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dn×dt −dt×dn)

))/R/R/R

Note that (τ[2,0,1] −τ[1,0,2]) annihilates any tensor product with equal first and last factors, while τ[*,*,0](g\(…)) annihilates any term in (…) whose first two tensor factors – when expressed in terms of dt, dr, dm and dn – are distinct, unless they are dt×dm or dm×dt.

= Q.τ[*,*,0](g\(

R.(τ[2,0,1] −τ[1,0,2])(

a.a.r.sin(n).sin(n).cos(n).cos(n).(: a.(a.dt −(r.r +a.a).dm)×(dr×dr/Δ −dn×dn); −Δ.dt×dm×dm; )/R

+(r.r −a.a.sin(n).sin(n)).(: (dt −a.cos(n).cos(n).dm)×(r.dn×dn.Δ −(r.Δ −r.R +M.R).dr×dr/Δ); +cos(n).cos(n).r.Δ.dt×dm×dm; )/2/R

)

+(: R.(r.dr −a.a.sin(n).cos(n).dn); −2.(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dr×dt −dt×dr)

+a.cos(n).(: R.(cos(n).(r.dr −a.a.sin(n).cos(n).dn) −sin(n).(r.r −a.a.sin(n).sin(n)).dn); −2.cos(n).(r.r −a.a.sin(n).sin(n)).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dr −dr×dm)

+a.(: R.(dr.(3.r.r+a.a).sin(n).cos(n) +r.(r.r +a.a).cos(2.n).dn); −4.r.(r.r +a.a).sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dm×dn −dn×dm)

+a.a.(: R.(sin(n).cos(n).dr +r.cos(2.n).dn); −4.r.sin(n).cos(n).(r.dr +a.a.sin(n).cos(n).dn); )×(dn×dt −dt×dn)

))/R/R/R

Riemann Tensor

Written by Eddy.