A Black-Hole Excision Scheme for General Relativistic Core-Collapse Supernova Simulations Bailey Sykesand Bernhard Mueller

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A Black-Hole Excision Scheme for General Relativistic Core-Collapse Supernova

Simulations

Bailey Sykes∗and Bernhard Mueller†

School of Physics and Astronomy, Monash University, Clayton, VIC, 3010, Australia

Isabel Cordero-Carri´on

Departamento de Matem´aticas, Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain

Pablo Cerd´a-Dur´an

Departamento de Astronom´ıa y Astrof´ısica, Universitat de Val`encia, 46100 Burjassot, Val`encia, Spain

J´erˆome Novak

Laboratoire Univers et Th´eories, Observatoire de Paris, Universit´e PSL,

CNRS, Universit´e de Paris-Cit´e, 92190 Meudon, France

(Dated: May 24, 2024)

Fallback supernovae and the collapsar scenario for long-gamma ray burst and hypernovae have

received considerable interest as pathways to black-hole formation and extreme transient events.

Consistent simulations of these scenarios require a general relativistic treatment and need to deal

appropriately with the formation of a singularity. Free evolution schemes for the Einstein equa-

tions can handle the formation of black holes by means of excision or puncture schemes. However,

in constrained schemes, which oﬀer distinct advantages in long-term numerical stability in stel-

lar collapse simulations over well above 104light-crossing time scales, the dynamical treatment of

black-hole spacetimes is more challenging. Building on previous work on excision in conformally

ﬂat spacetimes, we here present the implementation of a black-hole excision scheme for supernova

simulations with the CoCoNuT-FMT neutrino transport code. We describe in detail a choice

of boundary conditions that ensures long-time numerical stability, and also address upgrades to

the hydrodynamics solver that are required to stably evolve the relativistic accretion ﬂow onto the

black hole. The scheme is currently limited to a spherically symmetric metric, but the hydrody-

namics can be treated multi-dimensionally. For demonstration, we present a spherically symmetric

simulation of black-hole formation in an 85M⊙star, as well as a two-dimensional simulation of the

fallback explosion of the same progenitor. These extend past 9 s and 0.3 s after black-hole formation,

respectively.

I. INTRODUCTION

Typically the ﬁnal iron core collapse of massive stars

eventually results in the formation of a neutron star and a

supernova explosion [1]. In a substantial fraction of mas-

sive stars, however, iron core collapse will lead to black

hole formation. Observations of supernova progenitors

indicate that red supergiants with initial masses above

15-18M⊙will collapse quietly to black holes [2, 3], and

there is now even direct evidence for the disappearance of

a red supergiant of ∼25M⊙[4]. In some cases, black hole

formation may not proceed quietly, however. It has long

been theorized that the formation of black holes with an

accretion disk during the collapse of rapidly rotating mas-

sive stars may lead to powerful “hypernova” explosions

and long gamma-ray bursts [5, 6] (collapsar scenario).

Furthermore, black holes may be formed, regardless of

progenitor rotation, if an incipient supernova explosion

is not suﬃciently energetic to eject the entire stellar en-

velope and a substantial fraction of the envelope under-

∗bailey.sykes@monash.edu

†bernhard.mueller@monash.edu

goes fallback onto the neutron star during some stage of

the explosion. Recently, a number of multi-dimensional

supernova simulations have followed the long-term evolu-

tion of neutrino-driven explosions in massive progenitors

to study this fallback scenario [7–10].

Simulations of black-hole forming supernovae are tech-

nically challenging, however. In addition to the usual nu-

merical challenges – multi-dimensional ﬂuid ﬂow includ-

ing magnetic ﬁelds and neutrino transport – a general

relativistic treatment of the spacetime is required to con-

sistently follow the evolution of the supernova core up to,

through, and beyond black hole formation. Furthermore,

while a judicious choice of slicing conditions may be suﬃ-

cient to follow the accretion onto the black hole for some

time after its formation [10], special techniques such as

excision [11] or puncture methods [12] are generally re-

quired to avoid problems with the central singularity and

follow the accretion onto the black hole on longer time

scales. As a simpler alternative, one can treat the long-

term evolution after black hole formation in Newtonian

gravity [7, 8], perhaps in combination with a pseudo-

relativistic potential [13, 14], but this approximate ap-

proach is not well suited during the dynamical collapse

phase and may not be accurate enough for treating col-

arXiv:2210.12939v3 [astro-ph.HE] 23 May 2024

lapsar disks and, in particular, fast relativistic outﬂows.

A rigorous relativistic treatment is also required for ac-

curate predictions of the gravitational wave signal from

black-hole formation in core-collapse supernovae.

For this reason, there have only been few eﬀorts to

consistently simulate collapsars and fallback supernovae

through black hole formation in full general relativity.

Many simulations of fallback supernovae stop when a

black hole is formed, regardless of whether the spacetime

was treated in full general relativity [15] or using ap-

proximate pseudo-Newtonian gravity [16, 17] up to that

point. Chan et al. [7], Chan and M¨uller [8] mapped from

a relativistic simulation (in the conformally ﬂat approx-

imation [18–20]) up to black hole formation to a Newto-

nian moving mesh code to follow the long-term evolution

of fallback supernovae. Two studies have presented ax-

isymmetric (2D) general relativistic simulations beyond

black hole formation based on a puncture method, em-

ploying either multi-group ﬂux-limited diﬀusion or [10]

or a simpler, gray transport scheme [21]. While mod-

els of collapsars and long gamma-ray bursts [e.g. 22–24].

have long moved beyond the use of modiﬁed Newtonian

gravity in early studies [5], they commonly bypass the

pre-collapse phase altogether and insert a black hole at

the beginning.

In order to circumvent the problem of numerical singu-

larities or grid stretching in singularity avoiding slicings,

two classes of methods have been developed to evolve

black hole spacetimes in numerical relativity. Puncture

methods [12, 25] factor out the singular parts of the

spacetime from the regular parts. The singular parts are

dealt with analytically while the regular parts are allowed

to evolve numerically [26, 27]. On the other hand, exci-

sion techniques involve removing a section of the space-

time and imposing boundary conditions on the excised

surface [11, 28]. Applications of puncture and excision

schemes have largely been conﬁned to hyperbolic evolu-

tion schemes.

Since core-collapse supernovae typically need to cover

several 104light-crossing time scales of the compact ob-

ject or more, it has been popular to resort to the con-

formal ﬂatness condition as an elliptic formulation of

the Einstein equations for this problem [29, 30]. While

CFC is exact only in spherical symmetry, it enables sta-

ble long-term evolution by solving the elliptic constraint

equations directly. The CFC approximation is also a

stepping stone towards the fully constrained formalism

(FCF) [31], a non-approximate elliptic-hyperbolic formu-

lation of the Einstein equations in the generalized Dirac

gauge. Because of their stability properties and suitabil-

ity for long-time simulations, adapting such constrained

schemes to black hole spacetimes is a major desidera-

tum. However, the elliptic nature of the constraint equa-

tions poses challenges in formulating excision or puncture

schemes for constrained formulations. Building on ear-

lier work around excision for the black hole initial data

problem [32–34], an excision scheme based on CFC/FCF

was formulated for spherically symmetric spacetimes by

Cordero-Carri´on et al. [35] and applied to several test

cases (static black holes, scalar ﬁeld collapse, collapse of

an isolated neutron star). However, the scheme is yet

to be applied in full core-collapse supernova simulations

with neutrino transport.

In this paper, we implement a modiﬁed version of

the excision scheme of Cordero-Carri´on et al. [35] in the

CoCoNuT-FMT supernova code. We also present re-

sults from simulations of black-hole formation during the

collapse of a massive star in spherical symmetry and in

axisymmetry (with a spherically symmetric metric) to

demonstrate the excision scheme in practice.

Our paper is structure as follows. In Section II, we re-

view the CFC approximation and describe the boundary

conditions for the excision scheme, including modiﬁca-

tions from the original formulation of Cordero-Carri´on

et al. [35]. Section III provides further details on the nu-

merical implementation of the boundary condition and

the solution of the elliptic equations that are speciﬁc to

CoCoNuT-FMT and more peripheral to the excision

method per se. Section IV describes modiﬁcations to the

hydrodynamics modules of the CoCoNuT-FMT code,

which are required to stably model accretion ﬂow across

the black hole horizon. In Section V, we present the

results of a spherically symmetric (1D) core-collapse su-

pernova simulation of an 85M⊙star. Finally, we show

results from an axisymmetric (2D) simulation of a fall-

back supernovae (using the same progenitor) with the

excision scheme in Section VI.

Throughout this paper we use geometrized units: G=

c= 1. Greek indices run from 0 to 3 while Latin indices

run from 1 to 3.

II. METRIC EQUATIONS AND EXCISION

SCHEME

A. 3+1 Formalism in the conformal ﬂatness

approximation

For the numerical solution of dynamical problems, the

Einstein equations must be formulated as an evolution

problem using spacelike or null foliations. We start from

the 3+1 formalism [36, 37], in which the four-dimensional

spacetime is foliated into a continuous sequence of three-

dimensional spacelike hypersurfaces. This foliation in-

duces a metric with line element,

ds2=−α2dt2+γij (dxi+βidt)(dxj+βjdt),(1)

where αis the lapse function, βiis the shift vector, and

γij is the three-metric on the spacelike hypersurfaces.

In this work, we adopt the approximation that the spa-

tial metric is conformally ﬂat (conformal ﬂatness con-

dition, CFC) [19] and spherically symmetric (although

when assuming spherical symmetry, the former assump-

tion ceases to be an approximation and is just a choice of

isotropic spatial coordinates).1The CFC approximation

for relativistic gravity has been used extensively in var-

ious general relativistic hydrodynamics codes for many

years [29, 38–40].

The CFC approximation reduces the Einstein equa-

tions to a set of elliptic constraint equations. In spherical

polar coordinates, the CFC metric is given by,

gµν =





−α2+βiβiβrβθβφ

β1ϕ40 0

β20ϕ4r20

β30 0 ϕ4r2sin2θ







.(2)

To determine the conformal factor, ϕ, in addition to the

lapse, α, and shift vector, βi, the Hamiltonian constraint

and momentum constraint are supplemented by a gauge

condition, namely maximal slicing, which requires that

the trace of the extrinsic curvature Kij should vanish,

K=γij Kij = 0.(3)

This results in a set of three coupled elliptic equations

that describe the spacetime in the presence of a given

stress-energy tensor, determined by the matter energy,

momentum and stresses at a given time:

∆ϕ=−2πϕ−1E∗+ϕ6Kij Kij

16π,(4)

∆(αϕ)=2παϕ−1E∗+ 2S∗+7ϕ6Kij Kij

16π,(5)

∆βi+1

3∇i∇jβj= 16παϕ−2(S∗)i+ 2ϕ10Kij ∇j

ϕ6,(6)

where E∗,S∗and (S∗)iare the conformally rescaled

energy density, trace of the stress tensor, and momen-

tum density respectively (which are conserved hydrody-

namic variables). The ﬂat-space Laplacian is denoted by

∆ = fij ∇i∇jwhere fij is the ﬂat space three-metric.

At spatial inﬁnity, the metric is ﬂat, meaning ϕ→1,

α→1 and βi→0. Also note that spherical symmetry

causes the angular components of the shift to be zero,

i.e., βθ=βφ= 0, as well as the oﬀ-diagonals of the

extrinsic curvature. In our code, we use a ﬁxed-point it-

eration [41] based on a multipole expansion [42] for the

scalar and vector Poisson equations. The vector Pois-

son equation for the shift is reduced to scalar Poisson

equations following Grandcl´ement et al. [43]. In general,

the source terms on the right hand sides are dominated

by terms containing hydrodynamic quantities; only for

1The CFC approximation is intimately related to an exact mixed

hyperbolic/elliptic formulation of the Einstein equations in the

generalized Dirac gauge known as the fully constrained formalism

(FCF) [31].

strong gravitational ﬁelds do the strongly non-linear cur-

vature terms (e.g. Kij Kij ) become signiﬁcant.

Prior to black hole formation, the extended CFC

(XCFC) formalism of [20] is used as it is robust against

the uniqueness issue which sometimes plagues the orig-

inal CFC approach (Equations (4), (5) and (6)). We

prefer not to use the XCFC after black hole formation

as it introduces an auxiliary vector Xi, which is diﬃcult

to deﬁne a boundary condition for. We observe no issues

with the uniqueness of the solution in the simulation re-

sults.

B. Boundary conditions for the metric variables

To extend our supernova code, CoCoNuT-FMT,

to simulations beyond black hole formation, we follow

Cordero-Carri´on et al. [35] and perform excision of a

spherical region inside the PNS core by imposing suit-

able boundary conditions on the metric variables at the

excision surface. This excision surface lies strictly within

the apparent horizon once it forms. The boundary con-

ditions for ϕ,αand βirespectively are, in the original

formulation,

∂tϕ=βk∇kϕ+ϕ

6∇kβk.(7)

α=ϕ6(Lβ)ij sisj

2ˆ

Aij sisj

,(8)

βisi= constant,(9)

where siis an outward-directed spacelike

unit vector normal to the excision surface,

(Lβ)ij := Diβj+Djβi−2

3fij Dkβkand ˆ

Aij is the con-

formally rescaled extrinsic curvature, i.e., ˆ

Aij =ϕ10Kij .

Unless otherwise noted, boundary conditions are implied

to apply on the excision surface only. We evolve ˆ

Aij

according to the time evolution equation as given by

Cordero-Carri´on et al. [44]. Under the assumption of

spherical symmetry only the component ˆ

Arr has an

eﬀect on the boundary conditions and it suﬃces to

integrate,

∂ˆ

Arr

∂t =βr∂rˆ

Arr +5

3ˆ

Arr 2

rβr+∂rβr−2ˆ

Arr ∂rβr

+ 2Nϕ−6(ˆ

Arr )2−8πNϕ6ϕ4Srr −S

3

+16

3N(∂rϕ)2+16

3ϕ∂rϕ∂rN

−2

3ϕ∆(Nϕ) + N ϕ∆ϕ+ 2∂rϕ(ϕ∂rN+N∂rϕ),

(10)

where Srr is the rr component of the stress energy

tensor, and Sis its trace.

The set of equations (7) - (9) represent one possible

gauge choice for the elliptic system. Equation (7) comes

from the kinematic relations of Bonazzola et al. [31].

Equation (8) is the result of imposing conformal ﬂatness,

hence this relation (or some equivalent equation) is a re-

quired feature in the boundary conditions; see Equation

(2.14) in Cordero-Carri´on et al. [35]. Equation (9) is a

choice of gauge. It is possible to make an alternative

gauge choice and, indeed, we ﬁnd a modiﬁed gauge con-

dition oﬀers better numerical stability when combined

with our hydrodynamics code. Speciﬁcally, we ﬁnd the

original boundary conditions to be sensitive to errors in

the evolution of ˆ

Aij , and that they can become unstable

if ˆ

Aij →0. Furthermore, while these boundary condi-

tions do preserve the inward direction of the characteris-

tics of the metric equations inside the apparent horizon,

we ﬁnd that null-characteristics are often produced out-

side the excision boundary, prompting the boundary to

move outward over a duration of hundreds of milliseconds

until it encompasses most of the grid (i.e. the apparent

horizon becomes thousands of kilometers in radius). This

behaviour is not desirable from a numerical point of view.

Furthermore, we do not ﬁnd good long-term conservation

of the ADM mass with the original scheme if there is on-

going accretion onto the black hole over time scales of

hundreds of milliseconds (as opposed to the test cases in

[35], which quickly settled to a vacuum black hole space

time).

In theory, the excision radius could be kept constant,

which may produce a more realistic metric evolution for

the original boundary conditions, however this would ne-

cessitate being able to handle the extremely relativistic

ﬂows near the centre of the grid. CoCoNuT is unable to

accurately model hydrodynamics under these conditions

and hence, the excision surface must be moved outward

where possible, for any set of boundary conditions.

To resolve these issues, we use new boundary condi-

tions for the lapse and radial component of the shift vec-

tor on the excision surface. Respectively, these are,

ϕ2βr= const. (11)

˜γik∇kβj+ ˜γkj ∇kβi−2

3˜γij ∇kβk= 2αϕ−6ˆ

Aij .(12)

We also change the boundary condition for the conformal

factor to explicitly guarantee conservation of the ADM

mass, or rather an appropriate change of the ADM mass

according to the energy ﬂux through the outer boundary

of the grid. The boundary condition for ϕthus changes

from a boundary condition on the excision surface to a

condition on the outer boundary,

ϕ′(R) = −MADM

2R2,(13)

where MADM is the ADM mass (see V A), ϕ′is the radial

derivative of ϕand R= 1010 cm is the radius of the

outer boundary of the grid. An inner boundary is not

required as the solution for ϕis fully determined by the

outer boundary condition and the sources on the grid.

Equation (10) is still used to evolve ˆ

Aij in each timestep.

The motivation behind each of these changes is explained

in more detail in the following two sections.

1. Reformulated boundary condition: Lapse and shift

The constraint from the hyperbolic sector imposed

in Equation (8) can be trivially rearranged to produce

Equation (12), which acts as a boundary condition for

βr. Some form of this condition must be present in any

form of the boundary conditions. This leaves the freedom

to manually set the lapse function, instead of the radial

component of the shift vector. It is possible, and perhaps

is most simple, to impose a constant value of the lapse on

the excision boundary, α(RAH) = const.; set such that α

is smooth and continuous in time over the transition from

non-excised to excised regimes. While this is the obvious

choice, we ﬁnd that this gauge can fail to preserve the

inward-pointing nature of outward-directed light rays in-

side the apparent horizon. This means that the apparent

horizon can vanish if the metric functions, and speciﬁ-

cally the boundary conditions, adjust such that,

ϕ2−βr>0 (14)

inside the excised region. Here the LHS is just the coor-

dinate velocity of radial outward-directed light rays. It is

helpful then to consider an inner boundary condition for

the lapse function where the “outgoing” null characteris-

tic still point inward at the excision surface. We ﬁnd that

this can be achieved by maintaining the ratio α/(ϕ2βr),

as per Equation (11), where its value is set, once again,

by continuity over the transition to the excised metric.

Tests with this boundary condition for the lapse func-

tion demonstrated that it prevents instabilities in the

metric equations which can otherwise occur if the light

cone at the excision surface is not properly pointing in-

ward any more due to the adjustment of the metric as

material accretes onto the black hole. It is also simple to

implement, and does not involve any derivatives, which

makes it eﬃcient to compute and less prone to numerical

accuracy problems under the restriction of ﬁnite spatial

resolution.

The shift vector is now constrained by a Robin bound-

ary condition on the excision surface. We ﬁnd no partic-

ular issues with imposing this boundary condition.

2. Reformulated boundary condition: Conformal factor

We now turn our attention to the boundary condition

for the conformal factor, ϕ. In the original scheme, ϕon

the excision boundary is governed by the time evolution

equation (7). This approach produces a stable simulation

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ABlack-HoleExcisionSchemeforGeneralRelativisticCore-CollapseSupernovaSimulationsBaileySykes∗andBernhardMueller†SchoolofPhysicsandAstronomy,MonashUniversity,Clayton,VIC,3010,AustraliaIsabelCordero-Carri´onDepartamentodeMatem´aticas,UniversitatdeVal`encia,46100Burjassot,Val`encia,SpainPabloCerd´a-Dur´...

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A Black-Hole Excision Scheme for General Relativistic Core-Collapse Supernova Simulations Bailey Sykesand Bernhard Mueller

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