Black Holes

With these provisions, the so-called black hole solutions to Einstein's Equations are very important physically for what they predict about the region exterior to the horizon, and very interesting to contemplate. Because all information about the region interior to the horizon is unavailable to observers outside, only the globally measurable quantities are experimentally accessible: mass, angular momentum and charge. The Kerr-Newman Solution to Einstein's Equations describes the spacetime exterior to any massive charged rotating object with axial symmetry, whether it has collapsed inside its horizon radius or not. Because of this importance, we will examine it in 4 dimensions in some detail before considering other dimensions.

The Kerr-Newman metric in D = 4 in (asymptotically flat) Boyer-Lindquist coordinates is [Wald]

ds² = (r² + a² cos(θ)²) dr² / (a² + e² + r² - 2 M r) +

(r² + a² cos(θ)²) dθ² +

((a² + r²)² - a²(a² + e² + r² - 2 M r) sin(θ)²) sin(θ)² dφ² / (r² + a² cos(θ)²) +

a (- e² + 2 M r) sin(θ)² dφ dt / (r² + a² cos(θ)²) -

(a² cos(θ)² + e² + r² - 2 M r) dt² / (r² + a² cos(θ)²)

(our a is the negative of Wald's for compatibility with the Myers-Perry metric below)

• e = 0, called the Kerr metric;
• a = 0, called the Reissner-Nordstrom metric, and
• e = 0 and a = 0, which is of course the Schwarzschild metric.

With the usual coordinate transformation c = cos(θ), the Kerr-Newman metric can be written as

ds² = (r² + a² c²) dr² / (a² + e² + r² - 2 M r) +

(r² + a² c²) dc² / (1 - c²) +

(1 - c²) ((a² + r²)² - a² (1 - c²)(a² + e² -2 M r + r²)) dφ² / (r² + a² c²) +

a (1 - c²) (- e² + 2 M r) dφ dt / (r² + a² c²) -

(a² c² + e² + r² - 2 M r) dt² / (r² + a² c²)

g r r	(a2 + e2 + r2 - 2 M r) / (a2 c2 + r2)
g c c	(1 - c2) / (a2 c2 + r2)
g φ φ	(a2 c2 + e2 + r2 - 2 M r) / ((1 - c2) (a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
g φ t	a (-e2 + 2 M r) / ((a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
g t t	- (a4 c2 - a2 e2 + a2 c2 e2 + 2 a2 M r - 2 a2 c2 M r + a2 r2 + a2 c2 r2 + r4) /
	((a2 c2 + r2) (a2 + e2 + r2 - 2 M r))

Killing Vectors and Null Surfaces

The norm of the asymptotic timelike Killing vector {0,0,0,1} is zero at the degeneracy singularity

r = M +- (M² - a² c² - e²)^1/2,

We will define an event horizon as a closed, null 2-surface generated by a Killing Vector. A null surface is spanned by at least one null vector and no timelike ones. Because it is a null surface, the light cones of all events on the surface are tangent to the surface. Because it is generated by a Killing Vector, it is invariant with respect to the symmetries of the metric. Hence it defines an invariant surface which can be crossed in only one direction. We can use the following Mathematica code (and some intuition) to find a Killing Vector which is timelike in the ergosphere but null on the event horizon:

kvu = { 0, 0, phco, 1};

Numerator[ FullSimplify[ Sum[ Sum[ g[[i,j]] kvu[[i]] kvu[[j]], {j, 4}], {i, 4}]]]

- (a² + e² - 2 M r + r² - (a⁴ phco² + phco² r⁴ + 2 a phco (e² - 2 M r) + a² (1 + 2 phco² r²)) Sin[th]² + a² phco² (a² + e² - 2 M r + r²) Sin[th]⁴)

rsol = Solve[ a² + e² - 2 M r + r² == 0, r]

{{ r -> M - Sqrt[ - a² - e² + M²]}, { r -> M + Sqrt[ - a² - e² + M²]}}

Flatten[ Union[ FullSimplify[ Solve[

((a⁴ phco² + phco² r⁴ + 2 a phco (e² - 2 M r) + a² (1 + 2 phco² r²)) /. rsol[[2]]) == 0, phco]]]]

{phco -> - a / (e² - 2 M (M + Sqrt[ - a² - e² + M²]))}

We have actually not proved that the event horizon can only be crossed in one direction, but an analysis of geodesic motion will confirm our computation of its location. The asymptotic coordinates fail at the horizon, so we have ignored the inner horizon solution. A coordinate extension similar to but much more complicated than the Kruskal Extension is possible here, and we will assume that for those who wish to contemplate the region interior to the horizon, the coordinate r is similarly extendible.

Geometry

Γ rr r	(a2 (c2 (M - r) + r) + r (e2 - M r)) / ((a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
Γ rr c	a2 c / (a2 c2 + r2)
Γ rc c	- r (a2 + e2 + r2 - 2 M r) / ((1 - c2) (a2 c2 + r2))
Γ rφ φ	- (1 - c2) (a2 + e2 + r2 - 2 M r) (r5 + a4 (c2 M + c4 (r - M)) + a2 r (((1 - c2) e2) +
	r (2 c2 r - (1 - c2) M))) / (a2 c2 + r2)3
Γ rφ t	- a (1 - c2) (a2 + e2 + r2 - 2 M r) (a2 c2 M + r (e2 - M r)) / (a2 c2 + r2)3
Γ rt t	- (a2 + e2 + r2 - 2 M r) (a2 c2 M + r (e2 - M r)) / (a2 c2 + r2)3
Γ cr r	- a2 c (1 - c2) / ((a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
Γ cr c	r / (a2 c2 + r2)
Γ cc c	c (a2 + r2) / ((1 - c2) (a2 c2 + r2))
Γ cφ φ	c (1 - c2) (a6 c4 + r6 + a2 r2 (-2 (1 - c2) e2 + r (4 M - 4 c2 M + r + 2 c2 r)) +
	a4 (- (1 - c4) e2 + r (2 (1 - c4) M + c2 (2 + c2) r))) / (a2 c2 + r2)3
Γ cφ t	- a c (1 - c2) (e2 - 2 M r) (a2 + r2) / (a2 c2 + r2)3
Γ ct t	- a2 c (1 - c2) (e2 - 2 M r) / (a2 c2 + r2)3
Γ φr φ	(r3 (e2 + r2 - 2 M r) + a4 (c2 M + c4 (- M + r)) + a2 r (e2 + r (- M - c2 M + 2 c2 r))) /
	((a2 c2 + r2)2 (a2 + e2 + r2 - 2 M r))
Γ φr t	a (a2 c2 M + r (e2 - M r)) / ((a2 c2 + r2)2 (a2 + e2 + r2 - 2 M r))
Γ φc φ	- c (a4 c4 + r4 + a2 (- (1 - c2) e2 + 2 r (M - c2 M + c2 r))) / ((1 - c2) (a2 c2 + r2)2)
Γ φc t	a c (e2 - 2 M r) / ((1 - c2) (a2 c2 + r2)2)
Γ tr φ	- a (1 - c2) (a4 c2 M + r3 (2 e2 - 3 M r) + a2 (1 + c2) r (e2 - M r)) /
	((a2 c2 + r2)2 (a2 + e2 + r2 - 2 M r))
Γ tr t	- (a2 + r2) (a2 c2 M + r (e2 - M r)) / ((a2 c2 + r2)2 (a2 + e2 + r2 - 2 M r))
Γ tc φ	- a3 c (1 - c2) (e2 - 2 M r) / (a2 c2 + r2)2
Γ tc t	- a2 c (e2 - 2 M r) / (a2 c2 + r2)2

The r-c hypersurface is in general geodesic. For the non-rotating cases, r is a geodesic direction, and the r-t, r-c and spacelike hypersurfaces are all geodesic. The fact that the r-t hypersurface is not geodesic in the rotating case reflects the frame dragging effects mentioned earlier: radial timelike geodesics do not exist when a is nonzero.

The nonzero components of the Riemann Tensor are

R r c r c	(- (a2 c2 e2) + 3 a2 c2 M r + e2 r2 - M r3) / ((1 - c2) (a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
R r c φ t	- a c (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)2
R r φ r φ	- (1 - c2) (2 a6 c2 e2 - a6 c4 e2 + a4 c2 e4 - a4 c4 e4 - 9 a6 c2 M r + 6 a6 c4 M r -
	8 a4 c2 e2 M r + 8 a4 c4 e2 M r - 4 a4 e2 r2 + 6 a4 c2 e2 r2 - a4 c4 e2 r2 - 3 a2 e4 r2 +
	3 a2 c2 e4 r2 + 12 a4 c2 M2 r2 - 12 a4 c4 M2 r2 + 3 a4 M r3 - 14 a4 c2 M r3 +
	6 a4 c4 M r3 + 8 a2 e2 M r3 - 8 a2 c2 e2 M r3 - 5 a2 e2 r4 + 4 a2 c2 e2 r4 -
	4 a2 M2 r4 + 4 a2 c2 M2 r4 + 4 a2 M r5 - 5 a2 c2 M r5 - e2 r6 + M r7) /
	((a2 c2 + r2)3 (a2 + e2 + r2 - 2 M r))
R r φ r t	a (1 - c2) (- 2 a4 c2 e2 - a2 c2 e4 + 9 a4 c2 M r + 8 a2 c2 e2 M r + 4 a2 e2 r2 - 2 a2 c2 e2 r2 +
	3 e4 r2 - 12 a2 c2 M2 r2 - 3 a2 M r3 + 9 a2 c2 M r3 - 8 e2 M r3 + 4 e2 r4 + 4 M2 r4 - 3 M r5) /
	((a2 c2 + r2)3 (a2 + e2 + r2 - 2 M r))
R r φ c φ	3 a2 c (1 - c2) (a2 + r2) (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)3
R r φ c t	- a c (- 3 a2 + 2 a2 c2 - r2) (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)3
R r t r t	- (2 a4 c2 e2 - a4 c4 e2 + a2 c2 e4 - 9 a4 c2 M r + 3 a4 c4 M r - 8 a2 c2 e2 M r - 4 a2 e2 r2 +
	2 a2 c2 e2 r2 - 3 e4 r2 + 12 a2 c2 M2 r2 + 3 a2 M r3 - 7 a2 c2 M r3 + 8 e2 M r3 - 3 e2 r4 -
	4 M2 r4 + 2 M r5) / ((a2 c2 + r2)3 (a2 + e2 + r2 - 2 M r))
R r t c φ	- a c (- 3 a2 + a2 c2 - 2 r2) (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)3
R r t c t	3 a2 c (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)3
R c φ c φ	(4 a6 c2 e2 - a6 c4 e2 + a4 c2 e4 - a4 c4 e4 - 9 a6 c2 M r + 3 a6 c4 M r - 5 a4 c2 e2 M r +
	5 a4 c4 e2 M r - 2 a4 e2 r2 + 8 a4 c2 e2 r2 - a4 c4 e2 r2 - a2 e4 r2 + a2 c2 e4 r2 + 6 a4 c2 M2 r2 -
	6 a4 c4 M2 r2 + 3 a4 M r3 - 16 a4 c2 M r3 + 3 a4 c4 M r3 + 3 a2 e2 M r3 - 3 a2 c2 e2 M r3 - 3 a2 e2 r4 +
	4 a2 c2 e2 r4 - 2 a2 M2 r4 + 2 a2 c2 M2 r4 + 5 a2 M r5 - 7 a2 c2 M r5 - e2 r6 + 2 M r7) /
	(a2 c2 + r2)3
R c φ c t	- a (- 4 a4 c2 e2 - a2 c2 e4 + 9 a4 c2 M r + 5 a2 c2 e2 M r + 2 a2 e2 r2 - 4 a2 c2 e2 r2 + e4 r2 -
	6 a2 c2 M2 r2 - 3 a2 M r3 + 9 a2 c2 M r3 - 3 e2 M r3 + 2 e2 r4 + 2 M2 r4 - 3 M r5) / (a2 c2 + r2)3
R c t c t	(4 a4 c2 e2 - 3 a4 c4 e2 + a2 c2 e4 - 9 a4 c2 M r + 6 a4 c4 M r - 5 a2 c2 e2 M r -
	2 a2 e2 r2 + 2 a2 c2 e2 r2 - e4 r2 + 6 a2 c2 M2 r2 +
	3 a2 M r3 - 5 a2 c2 M r3 + 3 e2 M r3 - e2 r4 - 2 M2 r4 + M r5) / ((1 - c2) (a2 c2 + r2)3)
R φ t φ t	- (1 - c2) (a2 + e2 + r2 - 2 M r) (- (a2 c2 e2) + 3 a2 c2 M r + e2 r2 - M r3) / (a2 c2 + r2)3

Note that this metric is the first we have considered in which curvature components with more than 2 different indices are nonzero; this is an indication of the complexity of the metric. The nonzero components of the Ricci Tensor are

R r r	- e2 / ((a2 c2 + r2) (a2 + e2 + r2 - 2 M r))
R c c	e2 / ((1 - c2) (a2 c2 + r2))
R φ φ	- (1 - c2) e2 (- 2 a4 + a4 c2 - a2 e2 + a2 c2 e2 + 2 a2 M r - 2 a2 c2 M r -
	3 a2 r2 + a2 c2 r2 - r4) / (a2 c2 + r2)3
R φ t	a (1 - c2) e2 (2 a2 + e2 - 2 M r + 2 r2) / (a2 c2 + r2)3
R t t	e2 (2 a2 - a2 c2 + e2 + r2 - 2 M r) / (a2 c2 + r2)3

The vector potential for the charged metric is

A_EM = - a (1 - c²) e r dφ / (a² c² + r²) - e r dt / (a² c² + r²)

F_{a b} = d_a A_b - d_b A_a :

and the following components of the stress-energy tensor

T_{a b} = F_a^c F_{c b} - g_{a b} F_{c d} F^{c d} / 4 :

The invariants we have chosen to examine are

R	0
Ra b Ra b	4 e4 / (a2 c2 + r2)4
Ra b c d Ra b c d	-8 ( -7 a4 c4 e4 + 6 a6 c6 M2 + 60 a4 c4 e2 M r + 34 a2 c2 e4 r2 -
	90 a4 c4 M2 r2 - 120 a2 c2 e2 M r3 -
	7 e4 r4 + 90 a2 c2 M2 r4 + 12 e2 M r5 - 6 M2 r6) / (a2 c2 + r2)6
Ra b Rca Rb c	0
Ra b c d Ra c Rb d	16 e4 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) / (a2 c2 + r2)7
Ra b c d Rea Rb c d e	16 e4 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) / (a2 c2 + r2)7
Ra b c d Re fa b Rc d e f	-96 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) ( - 2 a4 c4 e4 +
	3 a6 c6 M2 + 18 a4 c4 e2 M r + 12 a2 c2 e4 r2 -
	27 a4 c4 M2 r2 - 44 a2 c2 e2 M r3 - 2 e4 r4 +
	33 a2 c2 M2 r4 + 2 e2 M r5 - M2 r6) / (a2 c2 + r2)9
Ra b c d Reafc Rb e d f	-48 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) (- a4 c4 e4 +
	3 a6 c6 M2 + 18 a4 c4 e2 M r + 14 a2 c2 e4 r2 -
	27 a4 c4 M2 r2 - 44 a2 c2 e2 M r3 - e4 r4 +
	33 a2 c2 M2 r4 + 2 e2 M r5 - M2 r6) / (a2 c2 + r2)9
Ra b c d Reafc Rb f d e	-24 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) (3 a6 c6 M2 +
	18 a4 c4 e2 M r + 16 a2 c2 e4 r2 - 27 a4 c4 M2 r2 -
	44 a2 c2 e2 M r3 + 33 a2 c2 M2 r4 + 2 e2 M r5 - M2 r6) / (a2 c2 + r2)9
Ra b; c Ra b; c	-16 e4(-4 a4 c2 + 5 a4 c4 + a2 c2 e2 - 2 a2 c2 M r -
	4 a2 r2 - 5 e2 r2 + 10 M r3 - 5 r4) / (a2 c2 + r2)7
Ra b; c Ra c; b	-8 e4(-8 a4 c2 + 7 a4 c4 - a2 c2 e2 + 2 a2 c2 M r -
	8 a2 r2 - 7 e2 r2 + 14 M r3 - 7 r4) / (a2 c2 + r2)7
Ra b; a Rcb; c	0
Ra b c d; e Ra b c d; e	16 (44 a8 c6 e4 - 76 a8 c8 e4 - 32 a6 c6 e6 +
	45 a10 c10 M2 + 45 a8 c8 e2 M2 - 180 a8 c6 e2 M r +
	720 a8 c8 e2 M r + 604 a6 c6 e4 M r - 90 a8 c8 M3 r - 156 a6 c4 e4 r2 +
	712 a6 c6 e4 r2 + 588 a4 c4 e6 r2 -
	1215 a8 c8 M2 r2 - 2340 a6 c6 e2 M2 r2 + 180 a6 c4 e2 M r3 -
	2700 a6 c6 e2 M r3 - 4236 a4 c4 e4 M r3 +
	2520 a6 c6 M3 r3 - 156 a4 c2 e4 r4 - 744 a2 c2 e6 r4 +
	1890 a6 c6 M2 r4 + 9270 a4 c4 e2 M2 r4 +
	324 a4 c2 e2 M r5 - 1332 a4 c4 e2 M r5 + 3540 a2 c2 e4 M r5 -
	6300 a4 c4 M3 r5 + 44 a2 e4 r6 -
	712 a2 c2 e4 r6 + 76 e6 r6 + 1890 a4 c4 M2 r6 -
	5364 a2 c2 e2 M2 r6 - 36 a2 e2 M r7 + 1980 a2 c2 e2 M r7 -
	260 e4 M r7 + 2520 a2 c2 M3 r7 + 76 e4 r8 - 1215 a2 c2 M2 r8 +
	261 e2 M2 r8 - 108 e2 M r9 - 90 M3 r9 +
	45 M2 r10) / (a2 c2 + r2)9
Ra b c d; a Reb c d; e	-48 e4 (a c - r) (a c + r) (a2 c2 + e2 - 2 M r + r2) / (a2 c2 + r2)7
Euler class	(-5 a4 c4 e4 + 6 a6 c6 M2 + 60 a4 c4 e2 M r + 38 a2 c2 e4 r2 -
	90 a4 c4 M2 r2 - 120 a2 c2 e2 M r3 -
	5 e4 r4 + 90 a2 c2 M2 r4 + 12 e2 M r5 - 6 M2 r6) / (4 π2 (a2 c2 + r2)6)
εa b c i ... εe f gi ... Rb c eh Rf g a h	-96 (- a2 c2 e2 - a3 c3 M - 2 a c e2 r + 3 a2 c2 M r + e2 r2 + 3 a c M r2 - M r3)
	(a2 c2 e2 - a3 c3 M - 2 a c e2 r - 3 a2 c2 M r - e2 r2 + 3 a c M r2 + M r3) /
	(a2 c2 + r2)6

Finally, the first Pontrjagin Class (p_{1r c φ t}) is

- a c (a² c² M + 2 e² r - 3 M r²) (- a² c² e² + 3 a² c² M r + e² r² - M r³) /

(π² (a² c² + r²)⁵).

It is instructive to examine the invariants computed by Greenberg, at least to third order:

C. C	-48 (- a2 c2 e2 + a3 c3 M + 2 a c e2 r + 3 a2 c2 M r + e2 r2 - 3 a c M r2 - M r3)
	(a2 c2 e2 + a3 c3 M + 2 a c e2 r - 3 a2 c2 M r - e2 r2 - 3 a c M r2 + M r3) / (a2 c2 + r2)6
* C. C	192 a c (a2 c2 M + 2 e2 r - 3 M r2) (- a2 c2 e2 + 3 a2 c2 M r + e2 r2 - M r3) / (a2 c2 + r2)6
C. C. C	96 (- a2 c2 e2 + 3 a2 c2 M r + e2 r2 - M r3) (- a4 c4 e4 + 3 a6 c6 M2 + 18 a4 c4 e2 M r + 14 a2 c2 e4 r2 - 27 a4 c4 M2 r2 - 44 a2 c2 e2 M r3 - e4 r4 + 33 a2 c2 M2 r4 + 2 e2 M r5 - M2 r6) / (a2 c2 + r2)9
* C. C. C	192 a c (a2 c2 M + 2 e2 r - 3 M r2) (-3 a4 c4 e4 + a6 c6 M2 + 22 a4 c4 e2 M r + 10 a2 c2 e4 r2 - 33 a4 c4 M2 r2 - 36 a2 c2 e2 M r3 - 3 e4 r4 + 27 a2 c2 M2 r4 + 6 e2 M r5 - 3 M2 r6) / (a2 c2 + r2)9
S. S	4 e4 / (a2 c2 + r2)4
S. S. S	0
C. E. E	32 e4 (a2 c2 e2 - 3 a2 c2 M r - e2 r2 + M r3) / (a2 c2 + r2)7
* C. E. E	-64 a c e4 (a2 c2 M + 2 e2 r - 3 M r2) / (a2 c2 + r2)7

Note that S S is equal to the square of the Ricci Tensor, and S S S is equal to its cube because, of course, R = 0. Note too that C E E is equal to 2 R^{a b c d} R_{a c} R_{b d} and C C C is equal to the cube of the Riemann Tensor. Finally, *C C is proportional to the first Pontrjagin Class. In the Kerr limit, of course, the Riemann Tensor is equal to the Weyl Tensor with obvious implications for these invariants.

All invariants involving the dual of the Weyl tensor are proportional to a, and of course all invariants involving S and E vanish in the Kerr limit. C C has real zeroes in the Kerr limit at

r = +- a c, and (+-2 +- 3^1/2) a c.

r = (e² +- (e⁴ + 3 a² c² M²)^1/2) / (3 M)

= +- a c / 3^1/2 in the Kerr limit

r = 0 and +- 3^1/2 a c.

θ ∈ (0, π / 2)

θ_i ∈ (0, π),

φ and φ₂ ∈ (0, 2π)

ρ² = r² + a² cos² (θ)

Δ(D) = μ / (r^D-5 ρ²)

ψ(D) = ρ² / ((r² + a²) - μ / r^D-5)

ds² = r² cos² (θ) dΩ² + ψ(D) dr² + ρ² dθ² +

((r² + a²) sin² (θ) + Δ(D) a² sin⁴ (θ)) dφ² +

2 Δ(D) a sin² (θ) dφ dt + (Δ(D) - 1) dt²

Normalizing the Komar integrals for J and M by 4 Ω (where Ω is the area of a solid angle in D dimensions), we find that

J = a μ / 2

M = (D - 3) J / a

D	Horizons exist at zeroes of	Ergospheres are located at zeroes of	volume element	area of a surface of constant r
4	a2 + r (-μ + r)	-a2 c2 + μ r - r2	a2 c2 + r2	4 π (a2 + 3 r2) / 3
5	a2 - μ + r2	-a2 c2 + μ - r2	c r (a2 c2 + r2)	π2 r (a2 + 2 r2)
6	-μ + a2 r + r3	μ - r (a2 c2 + r2)	c2 r2 (a2 c2 + r2)	8 π2 r2 (3 a2 + 5 r2) / 15
7	-μ + a2 r2 + r4	μ - r2 (a2 c2 + r2)	c3 (1 - c12)1/2 r3	π3 r3 (2 a2 + 3 r2) / 3
			(a2 c2 + r2)
8	-μ + a2 r3 + r5	μ - r3 (a2 c2 + r2)	c4 (1 - c12) (1 - c22)1/2	16 π3 r4 (5 a2 + 7 r2) / 105
			r4 (a2 c2 + r2)
9	-μ + a2 r4 + r6	μ - r4 (a2 c2 + r2)	c5 (1 - c12)3/2 (1 - c2)2	π4 r5 (3 a2 + 4 r2) / 12
			(1 - c32)1/2 r5 (a2 c2 + r2)
10	-μ + a2 r5 + r7	μ - r5 (a2 c2 + r2)	c6 (1 - c12)2 (1 - c22)3/2	32 π4 r6 (7 a2 + 9 r2) / 945
			(1 - c32) (1 - c42)1/2
			r6 (a2 c2 + r2)
11	-μ + r6 (a2 + r2)	μ - r6 (a2 c2 + r2)	c7 (1 - c12)5/2 (1 - c22)2	π5 r7 (4 a2 + 5 r2) / 60
			(1 - c32)3/2 (1 - c42)
			(1 - c52)1/2 r7 (a2 c2 + r2)

In the limit a -> 0, the area of a surface of constant r goes to that of flat space, the locations of the horizons go to those of the Schwarzschild metric in D dimensions and the curvature invariants go to those of the Schwarzschild metric in D dimensions, but the metric itself does NOT go to the Schwarzschild metric in D-dimensional asymptotically spherical coordinates.

We would like to find a coordinate system in which the Kerr^D metrics go to the Schwarzschild^D metrics in that limit. The key is to examine the different parameterizations of the D-2 spherical portion of each metric. By explicitly embedding each parameterization into a D-1 Euclidean space and setting the resultant metrics equal to each other, we can find the appropriate coordinate transformation using Mathematica:

sphcart[ n_, args_] := Block[ {},(

k = 1;

retab = {};

Do[ (prod = 1;

Do[ prod *= If[ i == k, Cos, Sin][args[[i]]], {i, k}];

AppendTo[ retab, prod];), {k, n}];

AppendTo[ retab, Last[retab] /. Cos[ args[[n]]] -> Sin[ args[[n]]]];

Return[ retab];

)];

dim = 5;

xs = Flatten[ {r, Table[ Symbol[ "t" <> ToString[i]], {i, dim - 3}], phs, t}]

{r,t1,t2,phs,t}

xk = Flatten[ {Table[ Symbol[ "tk" <> ToString[i]], {i, dim - 5}],

If[ dim >= 5, ph2, {}], r, th, phk, t}]

{ph2, r, th, phk, t}

fns = Table[ Apply[ xs[[i]], Join[ {th,phk}, Take[ xk, {1, dim - 4}]]], {i, 2, dim - 1}]

{t1[th, phk, ph2], t2[th, phk, ph2], phs[th, phk, ph2]}

xkcart = Flatten[ {r * Cos[th] sphcart[ dim - 4, Take[ xk, dim - 4]],

r * Sin[th] Cos[phk], r * Sin[th] Sin[phk]}]

{r Cos[ph2] Cos[th], r Cos[th] Sin[ph2], r Cos[phk] Sin[th],

r Sin[phk] Sin[th]}

xscart = r * sphcart[ dim - 2, fns]

{r Cos[t1[th, phk, ph2]], r Cos[t2[th, phk, ph2]] Sin[t1[th, phk, ph2]],

r Cos[phs[th, phk, ph2]] Sin[t1[th, phk, ph2]] Sin[t2[th, phk, ph2]],

r Sin[phs[th, phk, ph2]] Sin[t1[th, phk, ph2]] Sin[t2[th, phk, ph2]]}

sol = PowerExpand[ FullSimplify[

Solve[ Take[ Thread[ xkcart == xscart], dim - 2], fns][[28]]]]

{phs[th, phk, ph2] -> phk,

t2[th, phk, ph2] -> ArcCos[Cos[th] Sin[ph2] / Sqrt[1 - Cos[ph2]² Cos[th]²]],

t1[th, phk, ph2] -> ArcCos[Cos[ph2] Cos[th]]}

This process can be repeated in any dimension, and from it we can abstract the required transformation as follows: consider a coordinate transformation to {r, θ'_i, φ, t}:

θ = cos^-1 ( Sqrt (

cos² (θ'₁) + sin² (θ'₁) (

cos² (θ'₂) + sin² (θ'₂) (

... + sin² (θ'_D-4) cos² (θ'_D-3) ))))

θ_i = cos^-1 ( cos (θ'_i) / Sqrt (

cos² (θ'_i) + sin² (θ'_i) (

... + sin² (θ'_D-4) cos² (θ'_D-3) ))))

φ₂ = cos^-1 ( cos (θ'_D-4) / Sqrt (

cos² (θ'_D-4) + sin² (θ'_D-4) cos² (θ'_D-3) ))))

ds'² = dS'² + f(r, θ'_i) dr² + h(r, θ'_i) dφ² + 2 p(r, θ'_i) dφ dt + q(r, θ'_i) dt²

The following Mathematica code can be used to implement the coordinate transformation in any dimension, and show the equivalence in the limit a -> 0:

xku = Flatten[ {Table[ Symbol["tk" <> ToString[i]], {i,dim - 5}], If[dim >= 5, ph2, {}], r, th, phk, t}];

rho = Sqrt[ r^2 + a^2 Cos[th]^2];

delta = mu / (r^(dim - 5) rho^2);

psi = r^(dim - 5) rho^2 / (r^(dim - 5) (r^2 + a^2) - mu);

gkll = 0*IdentityMatrix[dim];

Do[

gkll[[i, i]] = r^2 Cos[th]^2 Product[ Sin[ xku[[j - 1]]]^2, {j, 2, i}],

{i, dim - 4}];

gkll[[dim - 3, dim - 3]] = psi;

gkll[[dim - 2, dim - 2]] = rho^2;

gkll[[dim - 1, dim - 1]] = (r^2 + a^2) Sin[th]^2 + delta * a^2 Sin[th]^4;

gkll[[dim - 1, dim]] = delta * a * Sin[th]^2;

gkll[[dim, dim - 1]] = gkll[[dim - 1, dim]];

gkll[[dim, dim]] = -1 + delta;

gkll = simpler[gkll];

gksll = Simplify[ gkll /. a -> 0];

xsu = Flatten[ {r, Table[ Symbol["t" <> ToString[i]], {i,dim - 3}], phs, t}];

f[r] = 1 - mu / r^(dim - 3);

gsll = 0*IdentityMatrix[dim];

Do[

gsll[[i, i]] = r^2 Product[ Sin[xsu[[j]]]^2, {j, 2, i - 1}],

{i, 2, dim - 1}];

gsll[[1, 1]] = 1 / f[r];

gsll[[dim, dim]] = -f[r];

gsll = simpler[gsll];

fns = Flatten[ {

Table[ Apply[xku[[i]], Take[xsu, {2, dim - 1}]], {i, dim - 4}],

Table[ Apply[xku[[i]], Take[xsu, {2, dim - 1}]], {i, dim - 2, dim - 1}]}];

xkcart = Flatten[ {

r * Cos[fns[[dim - 3]]] sphcart[dim - 4, fns],

r * Sin[fns[[dim - 3]]] Cos[fns[[dim - 2]]],

r * Sin[fns[[dim - 3]]] Sin[fns[[dim - 2]]]}];

Print["Check: ", Simplify[ Sum[xkcart[[i]]^2, {i, dim - 1}] == r^2]];

xscart = r * sphcart[dim - 2, Take[xsu, {2,dim - 1}]];

Print["Check: ", Simplify[ Sum[xscart[[i]]^2, {i, dim - 1}] == r^2]];

csexpr[nav_, stop_, ii_] :=

If[ii == stop, Cos[xsu[[nav - ii + stop + 1]]]^2,

csexpr[nav, stop, ii - 1] Sin[xsu[[nav - ii + stop + 1]]]^2 + Cos[xsu[[nav - ii + stop + 1]]]^2];

rules[] := Flatten[{ Table[

fns[[i]] -> ArcCos[ Cos[xsu[[i + 1]]] / Sqrt[ csexpr[dim - 3, i, dim - 3]]],

{i, dim - 4}],

fns[[dim - 3]] -> ArcCos[ Sqrt[ csexpr[dim - 3, 1, dim - 3]]],

fns[[dim - 2]] -> phs}];

Print["Using ", rules[]];

Print["Check: ", PowerExpand[ Simplify[

((xkcart == xscart) /. rules[]) //.

Sqrt[x_] Sqrt[y_] -> Sqrt[ Simplify[x * y]]] //.

Sqrt[x_] / Sqrt[y_] -> Sqrt[ Simplify[x / y]]]];

Print["Xform check: ", Simplify[ (

(gksll = Simplify[ coordxform[gkll, xku, (xku /. Thread[Head /@ fns -> fns]) /. rules[], xsu]]) /. a -> 0)

== gsll]];

As mentioned above, all invariants depending on R_{a b} and R^{a b c d}_{; a} are zero. All invariants depend only on r, c and the Kerr parameters, and R^{a b c d; e} R_{a b c d; e} vanishes on the ergosphere for D = 4 only. The last invariant examined is an integer multiple of the Kretschmann Invariant. All of the invariants demonstrate the essential singularity at the zero of r² + a² c², commonly called the "ring singularity", and for D > 5, r = 0 is an essential singularity as well. We suspect that this means that the Kruskal Extensions for these metrics are topologically different from those for Kerr⁴ and Kerr⁵.

We note that

R^{a b c d} R^{e f}_{a b} R_{c d e f} + 4 (R^{a b c d} R^e_a^f_c R_{b f d e} - R^{a b c d} R^e_a^f_c R_{b e d f}) = 0,

ρ² = r² + a² cos² (θ)

Δ(D) = (1 - 2 Λ r² / ((D - 1) (D - 2))) (r² + a²) - μ / r^D-5

ψ(D) = 1 + 2 a² Λ cos² (θ) / ((D - 1) (D - 2))

Σ(D) = 1 + 2 a² Λ / ((D - 1) (D - 2))

ds² = r² cos² (θ) dΩ² + ρ² dr² / Δ(D) + ρ² dθ² / ψ(D) +

(ψ(D) (r² + a²)² - Δ(D) a² sin² (θ)) sin² (θ) dφ² / (Σ²(D) ρ²) +

2 a sin² (θ) (ψ(D) (r² + a²) - Δ(D)) dφ dt / (Σ²(D) ρ²) -

(Δ(D) - a² ψ(D) sin² (θ)) dt² / (Σ²(D) ρ²)

R
R^{a b} R_{a b}
R^{a b c d} R_{a b c d}
R^{a b} R^c_a R_{b c}
R^{a b c d} R_{a c} R_{b d}
R^{a b; c} R_{a b; c}
R^{a b; c} R_{a c; b}
R^{a b}_{; a} R^c_{b; c} and
R^{a b c d}_{; a} R^e_{b c d; e}

R^{a b c d} R^e_a R_{b c d e} = that for (A)dS^D + Λ R^{a b c d} R_{a b c d (Λ=0)} / (D - 2)

R^{a b c d} R^{e f}_{a b} R_{c d e f} = that for (A)dS^D + that for Kerr^D_(Λ=0) +

12 Λ R^{a b c d} R_{a b c d (Λ=0)} / ((D - 1) (D - 2))

R^{a b c d} R^e_a^f_c R_{b e d f} = that for (A)dS^D + that for Kerr^D_(Λ=0) -

3 Λ R^{a b c d} R_{a b c d (Λ=0)} / ((D - 1) (D - 2))

R^{a b c d} R^e_a^f_c R_{b f d e} = that for (A)dS^D + that for Kerr^D_(Λ=0) -

6 Λ R^{a b c d} R_{a b c d (Λ=0)} / ((D - 1) (D - 2))

R^{a b c d; e} R_{a b c d; e} = that for Kerr^D_(Λ=0) (1 + Λ f(r, c, μ, a))

ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h} = that for (A)dS^D + that for Kerr^D_(Λ=0)

R^{a b c d; e} R_{a b c d; e} = - 2 (D + 1) (D - 1) (D - 2) (D - 3) μ² ((D - 1) (D - 2) (μ - r^{D - 3}) / 2 +

Λ r^{D - 1}) / r^{3 (D - 1)}

that for (A)dS⁴ + that for Kerr⁴_(Λ=0)

that for (A)dS⁶ + that for Kerr⁶_(Λ=0) + 12 ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^h R_{f g a h (Λ=0)} / 5

Collect[Expand[#], lambda, simpler]&

D	P1r c φ t
4	(- 9 a c μ2 r (a2 c2 - 3 r2) (3 a2 c2 - r2)) / (4 (3 + a2 Λ)2 π2 (a2 c2 + r2)5)
5	(72 a c μ2 (a c - r) r (a c + r)) / ((6 + a2 Λ)2 π2 (a2 c2 + r2)5)
6	(- 25 a c μ2 (a2 c2 + 5 r2) (a4 c4 - a2 c2 r2 + 10 r4)) / (3 (10 + a2 Λ)2 π2 r5 (a2 c2 + r2)5)
7	(- 225 a c μ2 (a2 c2 + 3 r2) (a4 c4 + 2 a2 c2 r2 + 5 r4)) /
	(2 (15 + a2 Λ)2 π2 r7 (a2 c2 + r2)5)
8	(- 441 a c μ2 (3 a2 c2 + 7 r2) (2 a4 c4 + 5 a2 c2 r2 + 7 r4)) /
	(4 (21 + a2 Λ)2 π2 r9 (a2 c2 + r2)5)
9	(- 1568 a c μ2 (a2 c2 + 2 r2) (5 a4 c4 + 13 a2 c2 r2 + 14 r4)) /
	(3 (28 + a2 Λ)2 π2 r11 (a2 c2 + r2)5)
10	(- 324 a c μ2 (5 a2 c2 + 9 r2) (5 a4 c4 + 13 a2 c2 r2 + 12 r4)) /
	((36 + a2 Λ)2 π2 r13 (a2 c2 + r2)5)
11	(- 2025 a c μ2 (3 a2 c2 + 5 r2) (7 a4 c4 + 18 a2 c2 r2 + 15 r4)) /
	(2 (45 + a2 Λ)2 π2 r15 (a2 c2 + r2)5)

Note that all are proportional to both M and J, and vanish both on the equatorial hyperplane and in the Schwarzschild limit.

Schwarzschild/Anti-deSitter in D=4

An interesting question arises when considering a black hole in an anti-deSitter background. We take D=4, a=0 and e=0 for simplicity.

Λ for this background can be written as -1/L_AdS², where L_AdS is the anti-deSitter length scale. The metric is then

ds² = dr² / (1 - 2 M / r + (r / L_AdS)² / 3) + r² dθ² + r² sin² θ - dt² (1 - 2 M / r + (r / L_AdS)² / 3),

and the asymptotically flat Schwarzschild solution is obtained in the limit L_AdS→∞. The scalar curvature for this metric is -4/L_AdS: spacelike hypersurfaces have constant negative curvature, which means that the neighborhood of every point is a "3-dimensional saddle". L_AdS determines the relative size of those saddles.

For a spacelike hypersurface, the volume element is

r² sin θ / √(1 - 2 M / r + (r / L_AdS)² / 3),

so the volume grows more slowly with r when L_AdS is finite, and goes to zero as L_AdS→0. Setting η=L_AdS/M, the horizon is located at

r_hor = η^2/3 (η^2/3 - (√(η² + 9) - 3) M / (√(η² + 9) - 3)^1/3

so that when:

• L_AdS = 3 M / 2, r_hor = L_AdS;
• L_AdS < 3 M / 2, r_hor > L_AdS;
• L_AdS > 3 M / 2, r_hor < L_AdS.

Nothing in the equations of motion prohibit the first two possibilities, but what does it mean physically to suggest that the AdS length scale is smaller than the black hole horizon? Small values of L_AdS imply that space is highly curved over short distances; how does this affect, for instance, geodesic motion?

Denoting the timelike Killing vector as ξ, the energy as measured by an observer with 4-velocity u is

E = - g_{μ ν} ξ^μ u^ν

= (1 - 2 M / r + (r / L_AdS)² / 3) dt / dτ

and is of course a constant of the motion (τ is the proper time). We see that the vacuum energy increases the energy measured by the observer, more for small L_AdS.

There is a second constant of the motion associated with the azimuthal Killing vector: the angular momentum

L = r² dφ / dτ.

If we consider the null geodesic equations in the equatorial plane (which of course is perfectly general for a spherical system), we can divide the equations for r(τ) and φ(τ) to obtain

dr / dφ = √(r (2M - r + (E² / L² - 1 / (3 L_AdS²)) r³).

Writing L in terms of the apparent impact parameter b (as seen by a distant observer; see Wald, section 6.3), we have

dr / dφ = √(r (2M - r + (1 / b² - 1 / (3 L_AdS²)) r³).

The effect of finite L_AdS is to increase the effective value of the impact parameter; light rays do not appear to be bent as much as they "should" be by the black hole.

To find the distance of closest approach, we require dr/dφ=0 and

d²r / dφ² = M - r + 2 r³ (1 / b² - 1 / (3 L_AdS²)) > 0.

Using the Mathematica function Reduce to find conditions for such solutions, we find

b < √3 L_AdS

b > 3√3 M L_AdS / √(L_AdS² + 9 M²)

r_closest = the real root of the polynomial equation

-6 M b² L_AdS² + 3 b² L_AdS² x + (b² - 3 L_AdS²) x³ = 0

Because the effective potential has not been modified by L_AdS, the location of the potential barrier (and closest unstable circular orbit) has not changed; the root of the polynomial remains 3M. But the minimum apparent impact parameter below which light rays are captured by the black hole decreases with L_AdS:

So as L_AdS → 0, light rays can appear to come arbitrarily close to the center of the horizon without falling in, even though they never get closer than 3M; the intense curvature allows them to bend completely around the black hole and appear to come out the other side. As L_AdS goes to ∞, the closest apparent impact parameter approaches its value for the asymptotically flat Schwarzschild solution of 3√3.

The next section examines inhomogeneous collapsing dust.

• Table of Contents
• Index:
  A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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