Numerical evolution of shocks in the interior of Kerr black holes

Paul M. CheslerBlack Hole Initiative, Harvard University, Cambridge, MA 02138, USA∗

Erik CurielMunich Center for Mathematical Philosophy, Ludwig-Maximilians-Universitat,

Ludwigstraß 31, 80539 Munchen, Germany and Black Hole Initiative,Harvard University, Cambridge, MA 02138, USA†

Ramesh NarayanBlack Hole Initiative, Harvard University, Cambridge, MA 02138, USA ‡

(Dated: August 24, 2018)

We numerically solve Einstein’s equations coupled to a scalar field in the interior of Kerr blackholes. We find shock waves form near the inner horizon. The shocks grow exponentially in amplitudeand need not be axisymmetric. Our numerical results are consistent with the geometry inside theinner horizon exponentially collapsing to zero volume, meaning that the Kerr geometry effectivelyends at the inner horizon.

Introduction.—The no-hair theorem postulates thatthe exterior geometry of black holes is completely de-scribed by the black hole’s mass, charge and angular mo-mentum. However, in the interior of a black hole the no-hair theorem doesn’t apply. Kerr-Newman black holescontain an inner and outer horizon, with the geometryinside the inner horizon susceptible to instabilities [1].

The instability of the interior geometry has been mostwidely studied for Reissner-Nordstrom black holes [2–15].This is due to the fact that one can impose spherical sym-metry, simplifying calculations. In this case, shocks formon the inner horizon. Perhaps the most dramatic effectis that the shocks result in the geometry inside the in-ner horizon collapsing to zero volume: the affine distancefrom the inner horizon to the central singularity decreasesexponentially with time [9–11, 16]. For a solar mass blackhole the central singularity lies a Planck distance awayfrom the inner horizon after a time typically on the or-der of milliseconds. This means that the end result ofthe instablity is that the Reissner-Nordstrom geometryeffectively ends at the inner horizon, with the centralsingularity practically coinciding with the inner horizonat late times [6]. Perturbative analyses suggest similarresults should hold for Kerr black holes [15, 17, 18].

In the present work we study the instability of theinterior of Kerr black holes by numerically solving Ein-stein’s equations coupled to a scalar field. Like Reissner-Nordstrom black holes, we find shocks form near the in-ner horizon. Our numerics are consistent with the geom-etry inside the inner horizon exponentially collapsing tozero volume, with the central singularity lying exponen-tially close to the inner horizon at late times. Addition-ally, we find that rotational invariance can be broken inthe vicinity of the inner horizon, with the amplitude ofnon-axisymmetric shocks growing exponentially in time.

Setup.— We numerically solve Einstein’s equationscoupled to a massless real scalar field Ψ. The equations

FIG. 1: A cartoon of our computational setup. The scalarfield Ψ is localized between the inner and outer horizons lo-cated at r− and r+, respectively. The computational domainis limited to rmin(v) ≤ r ≤ rmax(v). As time progresses thescalar wave packet and rmin and rmax all approach r = r−.

of motion read Rµν − 12Rgµν = 8πTµν and ∇2Ψ = 0

where Tµν = ∇µΨ∇νΨ− 12gµν(∇Ψ)2 is the stress tensor.

Our numerical evolution scheme is detailed in [19].Here we outline the salient details. We employ a charac-teristic evolution scheme where the metric takes the form

ds2 = −2Adv2+2dvdr + Σ2hab(dxa−F adv)(dxb−F bdv),

(1)with xa = θ, ϕ where θ is the polar angle and ϕ is theazimuthal angle. The two dimensional angular metric habsatisfies dethab = sin2 θ. Lines of constant time v andangles θ, ϕ are radial null infalling geodesics. The radialcoordinate r is an affine parameter for these geodesics.Correspondingly, the metric (1) is invariant under theresidual diffeomorphism r → r + ξ(v, θ, ϕ) where ξ isarbitrary. We fix ξ such that the inner horizon of thestationary Kerr geometry is located at r = r− = 1.

Requisite initial data at v = 0 consists of the scalarfield Ψ and the angular metric hab. The remaining com-ponents of the metric are determined by initial value con-straint equations [19]. Perhaps the most natural initial

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FIG. 2: Evolution of the scalar field Ψ in the equatorialplane for spin a = 0.9. The inner and outer boundaries ofthe shaded region represent rmin(v) and rmax(v). As timeprogresses the scalar field becomes localized at r = 1.

data is that where a rotating black hole is formed dynam-ically via gravitational collapse. Another option would beto start with a Kerr black hole and allow infalling radi-ation to perturb the geometry inside the event horizonat r = r+. A third option is to start with a Kerr blackhole and add a perturbation inside the event horizon. Itis reasonable to expect each of these choices to yield dif-ferent evolution at intermediate times, but for universalfeatures to emerge at late times. As we are primarilyinterested in the late time behavior of the system, inthis Letter we choose the last option, as it offers severalcomputational advantages. First, limiting the perturba-tions to the interior of the black hole means that onecan restrict the computational domain to the interior ofthe black hole. Second, since no energy or angular mo-mentum can be radiated to infinity, the mass and spinof the black hole remain constant. Because the geome-try outside the inner horizon is believed to be stable, thismeans that at late times the position of the inner horizonmust approach that of the unperturbed Kerr geometry atr = 1. In our coordinate system this ultimately meansthat at late times one must have A → 0 at r = 1. Hav-ing the inner horizon approach constant r is useful, sinceshocks are expected to form there.

We employ the Kerr metric for initial hab. For initialscalar data we choose

Ψ = 150e−(r−r0)2/2σ2

1 + ζ Re[y10(θ, ϕ)+y11(θ, ϕ)] , (2)

where y`m are spherical harmonics and ζ is a parame-ter controlling the degree of non-axisymmetry in the ini-tial data. We choose r0 and σ such that Ψ is localizedbetween the inner and outer horizons and exponentiallysmall at our outer computational boundary. See Fig. 1for a cartoon of our initial data and computational do-main.

We employ a time dependent radial computational do-main rmin(v) ≤ r ≤ rmax(v). rmin will lie inside the

FIG. 3: Σ′ in the equatorial plane at several times for a = 0.9.The ∗ denote location of the maximum of Ψ at the corre-sponding time. A shock in Σ′ is evident. Outside the shock,Σ′ approaches its Kerr value while inside Σ′ grows in time.

inner horizon and rmax will lie between the inner andouter horizons. At r = rmax, where the scalar fieldis exponentially small, we impose the boundary condi-tion that the geometry is that of Kerr. This is allowedsince there are no outwards propagating modes betweenthe inner and outer horizons of Kerr. Since the scalarfield will propagate inwards, it is convenient to let rmax

propagate inwards as well, so as to not waste compu-tational resources. A choice which accomplishes this isdrmax/dv = min

θ,ϕA |r=rmax

.

In the Kerr geometry outgoing geodesics can propagatefrom the singularity to the inner horizon, leading to abreakdown of predictability. In order to avoid this issuewe choose rmin(v) to be moving outwards at or fasterthan the speed of light. As such, no information from thesingularity can influence dynamics in the computationaldomain. A choice which accomplishes this is drmin/dv =maxθ,ϕ

A |r=rmin. Since at late times A → 0 at r = 1, it is

reasonable to expect rmin and rmax to approach r = 1from below and above, respectively. Indeed, we see thisin our numerical simulations presented below.

Our discretization scheme is nearly identical to that in[20] and is outlined in the Supplementary Material. Wefix the Kerr mass parameter M = 1 and spin a = 0.9, 0.95and 0.99. For a = 0.9, 0.95 we set (r0, σ) = (1.05, 1/150)while for a = 0.99 we set (r0, σ) = (1.01, 1/500). Foraxisymmetric initial data we set ζ = 0 and for non-axisymmetric initial data we set ζ = 1/4.

Results and discussion.— We begin by presenting re-sults for axisymmetric simulations. In Fig. 2 we plot thescalar field Ψ as a function of time v and radial coordi-nate r in the equatorial plane for spin a = 0.9. The innerand outer boundaries of the shaded region correspondto the curves rmin(v) and rmax(v) and reflect our time-dependent computational domain. As time progressesthe scalar wave packet propagates inwards towards r = 1,becoming increasingly narrower in the process while stay-ing roughly constant in magnitude. As the scalar wave

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FIG. 4: Σ′|r=1 for axisymmetric simulations with spin a = 0.9, 0.95 and 0.99.

packet approaches the inner horizon, the metric at r > 1approaches that of Kerr.

The localization of the scalar wave packet to r = 1results in large radial derivatives of the metric at r = 1.A useful metric component to study is Σ, which is relatedto the volume element via

√−g = Σ2 sin θ. In Fig. 3 we

plot Σ′|r=1 (with ′ ≡ ∂r) in the equatorial plane at severaltimes, again for spin a = 0.9. The ∗ denote the maximumof Ψ at the corresponding time. As is evident from thefigure, there is a dramatic change in Σ′ near the scalarmaxima. In other words, there is a shock in Σ′. Exteriorto the shock Σ′ is well approximated by its Kerr value.The change in Σ′ across the shock grows with time.

In Fig. 4 we plot Σ′|r=1 as a function of v for severalvalues of θ and for a = 0.9, 0.95 and 0.99. Also included

in each plot is eκv where κ = 12

(1

M−√M2−a2 −

1M

)is the

(absolute value of the) surface gravity of the inner horizonof the corresponding Kerr solution. For a = 0.9, 0.95 and0.99 we have κ ≈ 0.386, 0.227 and 0.0821, respectively.Our numerics are consistent with the scaling Σ′ ∼ eκv.

FIG. 5: The Kretschmann scalar K in the equatorial planeat several times for an axisymmetric simulation with a = 0.9.The ∗ denote location of the maximum of the scalar wavepacket at the corresponding time. Exterior to the wave packetK is well described by its Kerr value. At r = 1 K is nearlyconstant but K′ grows with time.

We now turn to the curvature. In Fig. 5 we plot theKretschmann scalar K ≡ RµναβRµναβ as a function ofr in the equatorial plane at several times for the same

simulation shown in Fig. 4. The ∗ denote the location ofthe maximum of Ψ at the corresponding time. Exteriorto the scalar wave packet, K is well approximated byits Kerr value. A prominent feature of Fig. 5 is thatK ′ grows dramatically with time just inside the wavepacket. In Fig. 6 we plot |K ′||r=1 as a function of vat several values of θ for the same simulations shown inFig. 4. Also included in the plots is eκv. Our numericsare consistent with the scaling |K ′||r=1 ∼ eκv. Evidently,the inner horizon becomes a curvature brick wall at latetimes, with a shock in K developing there.

The geometry in the vicinity of the inner horizon canbe studied perturbatively [15, 17, 18]. Many of our resultsfollow from geometric optics. To see this, we introducea bookkeeping parameter ε and solve the Einstein/scalarsystem in the shell r−1 = O(ε) in the limit ε→ 0. Exte-rior to the shell we impose the boundary condition Ψ = 0and demand the metric is that of the Kerr geometry.Since at late times the metric becomes rapidly varyingnear r = 1, inside the shell we assume radial derivativesscale like ∂r ∼ 1/ε. Additionally we assume |Ψ| 1. Itfollows that in the shell the metric is approximately thatof the Kerr geometry. It is straightforward to show thatat leading order the dynamical components of the equa-tions of motion reduce to geometric optics. Allowing noinfalling modes, the dynamical equations further reduceto the first order system

d+Ψ = 0, d+Σ = 0, d+Fa = 0, d+hab = 0, (3)

where d+ = ∂v + Ω∂φ − κ(r − 1)∂r is the directionalderivative along outgoing null geodesics of the Kerr ge-

ometry. Here Ω =√M2−a2+M

2aM is the angular velocity ofthe inner horizon. The remaining metric component Ais non-dynamical and satisfies a second order ODE in rsourced by all the other fields [19]. Note that in addi-tion to Eqs. (3), the metric and scalar field must satisfya system of (nonlinear) initial value constraints.

The general solution to each equation in (3) is an ar-bitrary function of u ≡ eκv(r − 1), θ and χ ≡ ϕ − Ωv.Curves with u, θ, χ all constant are simply outgoing nullgeodesics in the shell. These geodesics circle the inner

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FIG. 6: |K′||r=1 evaluated at several polar angles for axisymmetric simulations with spin a = 0.9, 0.95 and 0.99.

FIG. 7: |K′||r=1 in the equatorial plane for a non-axisymmtricsimulation with spin a = 0.95. The left plot is evaluated atϕ = 0 whereas the right is at ϕ = Ωv.

horizon at angular frequency Ω, which is due to framedragging, and eventually terminate on the inner horizonas v → ∞. The value of the fields on these geodesics isconstant. Since the r dependence comes in the combina-tion eκv(r − 1), it follows that e−κv plays the role of ourbookkeeping parameter ε.

The above analysis implies that as v → ∞, the scalarwave packet must approach r = 1, just as seen in Fig. 2.Additionally, since Σ and K only depend on v, r, ϕ viathe combinations eκv(r − 1) and ϕ− Ωv, it follows that

Σ′|r=1 = eκvH(θ, ϕ− Ωv), K ′|r=1 = eκvQ(θ, ϕ− Ωv),(4)

for some functions H and Q. The scaling relations (4)match those shown in Figs. 4 and 5 for our axisymmetricsimulations.

The scaling relations (4) also demonstrate rotation in-variance in ϕ can be broken: a small non-axisymmetricperturbation in initial data results in violations of ax-isymmetry in Σ′ and K ′ which are exponentially ampli-fied. To demonstrate this, in Fig. 7 we plot K ′|r=1 atθ = π/2 as a function of time for a non-axisymmetricsimulation with a = 0.95. The left figure is evaluatedat ϕ = 0 while the right figure is evaluated at ϕ = Ωv.At ϕ = 0 we see that K ′ grows exponentially with sinu-soidal oscillations superimposed. In the rotating frame,

where ϕ = Ωv, the sinusoidal oscillations are not present,just as (4) requires. Evidently, the curvature brick wallat r = 1 retains angular structure contained in the ini-tial data. Oscillating features of the curvature were alsoreported in [21].

Shocks that form near the inner horizon of Reissner-Nordstrom black holes result in the affine distance fromthe inner horizon to the central singularity decreasinglike e−κv [9–11, 16]. While in our analysis we have notevolved geometry up to the singularity, reasonable ex-trapolations indicate a similar conclusion. Since the sin-gularity has zero area, it must lie at a point r∗ whereΣ|r∗ = 0. Since our radial coordinate is the affine param-eter of infalling null radial geodesics, the affine distancebetween the inner horizon and singularity is just 1− r∗.We estimate 1−r∗ ∼ Σ/Σ′|r=1. Employing (4) we obtain

1− r∗ ∼ e−κv/H(θ, ϕ− Ωv), (5)

meaning that at late times the central singularity is ex-ponentially close to the inner horizon. This conclusion isbolstered by the observation of the curvature brick wallat the inner horizon, where K ′ grows exponentially large.

What does (5) mean for the end state of the instabil-ity? Consider the experience of an infalling observer atlate times. Until they’re exponentially close to r = 1,they will measure the local curvature to be given by itsKerr value, just as Fig. 5 suggests. Near r = 1 theywill encounter the curvature brick wall, where K subse-quently increases from its Kerr value to ∞ over a propertime ∆τ ∼ e−κv. At late enough times v, ∆τ must beshorter than the Planck time. It is reasonable to expectquantum gravity effects to dominate the physics on thesescales. Because of this, a reasonable conclusion is that atlate enough times the classical geometry of the black holeeffectively ends at r = 1, where there is a singular surfacewhose dynamics are described by quantum gravity.

In the present Letter we only considered perturbationsin the interior of the black hole and did not allow infallingradiation. For Reissner-Nordstrom black holes the addi-tion of infalling radiation results in a weak null curva-

5

ture singularity developing on the inner horizon via theso-called ‘mass inflation’ scenario [5, 6]. This additionalstructure exponentially approaches the inner horizon, be-coming a Plankian feature of the curvature brick wall atlate enough times, just as our observed violations of ro-tation invariance do. It is reasonable to expect a similarphenomena to occur with rotating black holes [22]. Weleave the study of effects of infalling modes for futurework.

Acknowledgments.—This work was supported by theBlack Hole Initiative at Harvard University, which isfunded by a grant from the John Templeton Founda-tion. EC is also supported by grant 312032894 from theDeutsche Forschungsgemeinschaft. We thank Peter Gal-ison for useful discussions.

∗ Electronic address: pchesler@g.harvard.edu† Electronic address: erik@strangebeautiful.com‡ Electronic address: rnarayan@cfa.harvard.edu

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1

Supplemental Materials: Numerical evolution of shocks in the interior of Kerr blackholes

To discretize the equations of motion we make a linear change of coordinates from r to z ∈ (−1, 1) via

r = a(v)z + b(v), (1)

where

a(v) = 12 (rmax(v)− rmin(v)), (2a)

b(v) = 12 (rmax(v) + rmin(v)). (2b)

Following [19], we expand the z dependence of all functions in a pseudo-spectral basis of Chebyshev polynomials. Weemploy domain decomposition in z direction with 30 equally spaced domains, each containing 8 points.

For the (θ, ϕ) dependence we employ a basis of scalar, vector and tensor harmonics. These are eigenfunctions ofthe covariant Laplacian −∇2 on the unit sphere. The scalar eigenfunctions are just spherical harmonics y`m. Thereare two vector harmonics, Vs`mi with s = 1, 2, and three symmetric tensor harmonics, T s`mij , s = 1, 2, 3. Explicitrepresentations of these functions are easily found and read [23]

V1`mi = 1√

`(`+1)∇iy`m, (3a)

V2`mi = 1√

`(`+1)ε ji ∇jy

`m, (3b)

T 1`mij =

hij√2y`m, (3c)

T 2`mij = 1√

`(`+1)(`(`+1)/2−1)ε k(i ∇j)∇ky

`m, (3d)

T 3`mij = 1√

`(`+1)(`(`+1)/2−1)[∇i∇j + `(`+1)

2 hij ]y`m, (3e)

where ε ji has non-zero components ε ϕθ = csc θ and ε θϕ = − sin θ, and hij = diag(1, sin2 θ) is the metric on the unitsphere. The scalar, vector and tensor harmonics are orthonormal and complete.

We expand the metric and scalar field as follows,

g00(v, z, θ, ϕ) =∑`m

α`m(v, z)y`m(θ, ϕ), (4a)

g0i(v, z, θ, ϕ) =∑s`m

βs`m(v, z)Vs`mi (θ, ϕ), (4b)

gij(v, z, θ, ϕ) =∑s`m

γs`m(v, z)T s`mij (θ, ϕ), (4c)

Ψ(v, z, θ, ϕ) =∑`m

χ`m(v, z)y`m(θ, ϕ). (4d)

Derivatives in θ, ϕ can then be taken by differentiating the scalar, vector and tensor harmonics.

In order to efficiently transform between real space and mode space, we employ a Gauss-Legendre grid in θ with`max + 1 points. Likewise, we employ a Fourier grid in the ϕ direction with 2`max + 1 points. These choices allow thetransformation between mode space and real space to be done with a combination of Gaussian quadrature and FastFourier Transforms.

We truncate the expansions (4) at maximum angular momentum `max = 100. For axisymmetric simulations we alsotruncate at azimuthal quantum number mmax = 0. For non-axisymmetric simulations we truncate at mmax = 20.