3.3 Horizon-penetrating solutions

We noted in Section 3.1.1 that the time-independent maximal slicing of Schwarzschild with isotropic coordinates covers only the exterior of the black hole. This is because the time independence of this gauge requires that the lapse vanish on the horizon. It is possible to evolve into the black hole’s interior when starting from initial data constructed in this gauge, but it requires a choice for the lapse that yields a time-dependent solution [94]. The time dependence of such a solution is purely gauge, of course, since the spacetime is static.

It is possible to cover all, or part, of the interior of a single black hole with a time-independent slicing. However, doing so seems to require that we give up the maximal-slicing condition. To cover the interior of the black hole, we need a slicing that passes smoothly through the event horizon. A convenient way to generate such solutions is to begin with the metric in standard ingoing-null coordinates. If we want to consider a rotating and charged black hole, then we use the Kerr–Newman geometry in Kerr coordinates:

( 2M r − Q2 ) 2M r − Q2 ds2 = − 1 − -----2----- dV&tidle;2 + 2dV&tidle;dr − 2 -----2----asin2πœƒd &tidle;V d&tidle;Ο• ρ ρ [ ] (83 ) + ρ2dπœƒ2 − 2asin2πœƒdrd &tidle;Ο• + -1- (r2 + a2)2 − Δa2 sin2πœƒ sin2 πœƒd&tidle;Ο•2, ρ2
where ρ and Δ are defined by Eq. (79View Equation), and &tidle; V is the ingoing-null coordinate. This metric is regular at ∘ --2----2-----2 r = r± ≡ M ± M − a − Q, where r+ and r− are the locations of the event horizon and the Cauchy horizon, respectively.

This metric can be put into a form suitable for producing time-independent Cauchy initial data by making coordinate transformations of the general form

dt = d&tidle;V + f(r)dr, d Ο• = d&tidle;Ο• + g(r)dr, (84 )
where f and g are suitably chosen functions of the radial coordinate r. There are a few particularly significant solutions for the general Kerr–Newman geometry, and I will outline these below, listing the nonzero components of the metric in the standard 3 + 1 format.

3.3.1 Kerr–Schild coordinates

A spherical coordinate version of the standard Kerr–Schild coordinate system is obtained from (83View Equation) by using the coordinate choice (cf. Refs. [7442Jump To The Next Citation Point])

t = &tidle;V − r and Ο• = &tidle;Ο•. (85 )
The nonzero components of the lapse, shift, and 3-metric are then given by:
− 2 2M r − Q2 α = 1 + -----2----, (86 ) ρ
r 22M--r −-Q2- β = α ρ2 , (87 )
2 γ = 1 + 2M--r −-Q--, (88 ) rr ρ2
[ ] 2M r − Q2 2 γrΟ• = − 1 + -----2----- asin πœƒ, (89 ) ρ
γπœƒπœƒ = ρ2, (90 )
[ 2 ] γϕϕ = r2 + a2 + 2M--r −-Q-a2 sin2 πœƒ sin2 πœƒ. (91 ) ρ2
Cartesian coordinate components can be obtained from these via the standard Kerr–Schild coordinate transformations [80]
x + iy = (r + ia)eiΟ•sinπœƒ and z = rcos πœƒ. (92 )
This yields the implicit definition of r from
r4 − r2(x2 + y2 + z2 − a2) − a2z2 = 0, (93 )
with r > 0 and r = 0 on the disk described by z = 0 and x2 + y2 ≤ a2.

3.3.2 Harmonic coordinates

Harmonic time slicing is integral to some hyperbolic formulations of general relativity, and a time-independent harmonic slicing of the Kerr–Newman geometry does exist [1742Jump To The Next Citation Point]. The harmonic time slicing condition is β–‘t = 0, which can be written

√-1--∂ μ(√ −-gg0μ) = 0. (94 ) − g
This equation is satisfied by using the coordinate choice
|| || t = V&tidle; − r + 2M ln |-2M---| , Ο• = Ο•&tidle;. (95 ) |r − r−|
The nonzero components of the lapse, shift, and 3-metric are then given by:
2 ( ) 2 2( ) α −2 = 1 + 2M-r-−-Q--- -r +-r+ + r+ +-a-- --2M--- , (96 ) ρ2 r − r− ρ2 r − r−
2 2 βr = α2 r+-+-a-, (97 ) ρ2
( ) βΟ• = − α2 a- --2M--- , (98 ) ρ2 r − r−
[ ( 2 ) ] γrr = 2 − 1 − 2M--r −-Q-- r +-r+- r-+-r+-, (99 ) ρ2 r − r− r − r−
[ 2( )] γrΟ• = − 1 + 2M-r-−-Q--- r-+-r+- a sin2 πœƒ, (100 ) ρ2 r − r−
γπœƒπœƒ = ρ2, (101 )
[ ] 2 2 2M r − Q2 2 2 2 γϕϕ = r + a + ----ρ2----a sin πœƒ sin πœƒ. (102 )
Cartesian coordinate components can be obtained from these via the standard Kerr–Schild coordinate transformations (92View Equation) and (93View Equation). However, for the harmonic slicing, the t = const. hypersurface is spacelike only outside the Cauchy horizon at r > r −.

Fully harmonic coordinates (μ β–‘x = 0) can be defined when Cartesian spatial coordinates are used by employing a variation of the standard Kerr–Schild coordinate transformations [42]

x + iy = (r − m + ia)eiΟ•sin πœƒ and z = (r − m )cosπœƒ. (103 )
This yields the implicit definition of r from
4 2 2 2 2 2 2 2 (r − m ) − (r − m ) (x + y + z − a ) − a z = 0. (104 )
Fully harmonic coordinates are useful because applying a boost to a harmonically sliced black hole yields a solution that satisfies (94View Equation) only if the black hole is written in fully harmonic coordinates. In this case, the boosted solution also satisfies the fully harmonic coordinate conditions.

3.3.3 Generalized Painlevé–Gullstrand coordinates

The Painlevé–Gullstrand gauge choice for the Schwarzschild geometry has been rediscovered many times because of its simple form (cf. Refs. [8659896568]). It is another time-independent solution, but the 3-geometry is completely flat (not simply conformally flat). The lapse is one in this gauge, and all of the information regarding the curvature of spacetime is contained in the shift. The Painlevé–Gullstrand gauge also has an intuitive physical interpretation [75]. An observer starting at rest at infinity and freely falling will trace out a world line that is everywhere orthogonal to the t = const. hypersurfaces in Painlevé–Gullstrand coordinates.

A generalization of the Painlevé–Gullstrand gauge derived by Doran [48] includes the Kerr spacetime, and the extension of this solution to the full Kerr–Newman spacetime is trivial. In the limit that a and Q vanish, this solution reduces to the Painlevé–Gullstrand gauge. The coordinate transformation is written most easily as

dr a dr dt = d &tidle;V − ----∘------------, dΟ• = d &tidle;Ο• − -----------∘------------. (105 ) 2M-r-−-Q2-- r2 + a2 2M-r-−-Q2-- 1 + r2 + a2 1 + r2 + a2
The nonzero components of the lapse, shift, and 3-metric are then given by:
α−2 = 1, (106 )
∘ --------∘ ----------- r 2 r2 +-a2 2M--r −-Q2- β = α ρ2 ρ2 , (107 )
2 γ = --ρ----, (108 ) rr r2 + a2
[∘ ----2--∘ ---------2] γ = − ---ρ--- 2M-r-−-Q--- a sin2 πœƒ, (109 ) rΟ• r2 + a2 ρ2
γπœƒπœƒ = ρ2, (110 )
[ ] 2 2 2M r − Q2 2 2 2 γϕϕ = r + a + -----2----a sin πœƒ sin πœƒ. (111 ) ρ
Notice that the lapse remains one, but the 3-geometry is no longer flat when the black hole is spinning. Cartesian coordinate components can be obtained from these via the standard Kerr–Schild coordinate transformations (92View Equation) and (93View Equation). Like the Kerr–Schild time slicing, a t = const. slice of the generalized Painlevé–Gullstrand gauge remains spacelike for all r ≥ 0.
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