FRAGMENTATION OF GENERAL RELATIVISTIC QUASI-TOROIDAL POLYTROPES
Burkhard Zink,1, 2 Nikolaos Stergioulas,3 Ian Hawke,4 Christian D. Ott,5 Erik Schnetter,1, 6 and Ewald Müller7
1Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA
2Horace Hearne Jr. Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA
3Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
4School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
5Department of Astronomy and Steward Observatory, The University of Arizona, Tucson, AZ, USA
6Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
7Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germany
How do black holes form from relativistic stars? This ques-
tion is of great fundamental and practical importance in grav-
itational physics and general relativistic astrophysics. On the
fundamental level, black holes are genuinely relativistic ob-
jects, and thus the study of their production involves ques-
tions about horizon dynamics, global structure of spacetimes,
and the nature of the singularities predicted as a consequence
of the occurrence of trapped surfaces. On the level of as-
trophysical applications, systems involving black holes are
possible engines for highly energetic phenomena like AGNs
or gamma-ray bursts, and also likely a comparatively strong
source of gravitational radiation.
The most simple model of black hole formation from, say,
cold neutron stars, is a fluid in spherically symmetric poly-
tropic equilibrium moving on a sequence of increasing mass
due to accretion [1]. This assumes that (i) the stellar structure
and dynamics are represented reasonably by the ideal fluid
equation of state and the polytropic stratification, (ii) accre-
tion processes are slow compared to the dynamical timescales
of the star, and (iii) rotation is negligible. Our focus has been
to study the effects of relaxing the third assumption.
In spherical symmetry, the sequence of equilibrium poly-
tropes has a maximum in the mass function M(ρ
), where ρ
denotes the central rest-mass density of the polytrope. This
maximum is connected to a change in the stability of the fun-
damental mode of oscillation [1], and thus collapse sets in
via a dynamical instability to radial deformations. During the
subsequent evolution, a trapped tube forms at the center which
traverses the stellar material entirely [2].
How much of this behaviour is preserved when rotation is
taken into account? Rotation is known to change the equi-
librium structure of the star, and, in consequence, its modes
of oscillation and set of unstable perturbations. The collapse
might also lead to the formation of a massive disk around the
new-born black hole, and finally only systems without spher-
ical symmetry can be a source of gravitational radiation.
Numerical simulations have been used to study the collapse
and black hole formation of general relativistic rotating poly-
tropic stars [3]. For the uniformly and moderately differen-
tially rotating models investigated in those studies, the dynam-
ical process is described by the instability of a quasi-radial
mode and subsequent collapse of the star up to the formation
of an accreting Kerr black hole at the star’s center.
Will strong differential rotation modify this picture? Even
before our study, there was evidence that this should be the
case. (i) Strong differential rotation can deform the high-
FIG. 1: Development of the fragmentation instability in a model of a
strongly differentially rotating supermassive star. The darker shades
of grey indicate higher density. The closed white line in the last plot
is a trapped surface.
density regions of a star into a toroidal shape, thus chang-
ing the equilibrium structure considerably. (ii) It admits stars
of high normalized rotational energy T/|W | [1] which are
stable to axisymmetric perturbations. (iii) It admits non-
axisymmetric instabilities, for example by the occurrence of
corotation points[4], at low values of T/|W |[5]. (iv) A bar-
mode instability of the type found in Maclaurin spheroids[6]
would likely express itself by the formation of two orbiting
fragments if the initial high-density region has toroidal shape.
This last property has motivated us to ask this question:
Can a bar deformation transform a strongly differentially ro-
tating star into a binary black hole merger with a massive
accretion disk? If so, this process might occur in supermas-
sive stars if the timescales associated with angular momentum
transport are too large to enforce uniform rotation.
We have investigated black hole formation in strongly dif-
ferentially rotating, quasi-toroidal models of supermassive
stars [7, 8], and found that a non-axisymmetric instability can
lead to the off-center formation of a trapped surface (see fig-
ure). An extensive parameter space study of this fragmen-
tation instability [8] reveals that many quasi-toroidal stars of
this kind are dynamically unstable in this manner, even for low
values of T/|W |, and we have found evidence that the coro-
tation mechanism observed by Watts et al. [4] might be active
in these models. Since, on a sequence of increasing T/|W |,
one of the low order m = 1 modes becomes dynamically un-
http://arxiv.org/abs/0704.0431v1
stable before m = 2 and higher order modes, one would not
expect a binary black hole system to form in many situations
(although this may depend on the rotation law and details of
the pre-collapse evolution as well). Rather, the off-center pro-
duction of a single black hole with a massive accretion disk
appears more likely.
Since the normalized angular momentum J/M2 of the ini-
tial model is greater than unity, there is another interesting
consequence of this formation process: the resulting black
hole, unless it is ejected from its shell, may very well be
rapidly rotating, spun up by accretion of the material remain-
ing outside the initial location of the trapped surface. Investi-
gating the late time behaviour of this accretion process, esti-
mating possible kick velocities of the resulting black hole, and
finding the mass of the final accretion disk is, however, beyond
our present-day capabilities and subject of future study.
[1] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs and
Neutron Stars (Wiley 1983).
[2] S. Shapiro and S. Teukolsky, Astrophys. J. 235, 199 (1980).
[3] M. Shibata, T. Baumgarte and S. Shapiro, Phys. Rev. D 61,
044012 (2000). L. Baiotti, I. Hawke, P. Montero, F. Löffler, L.
Rezzolla, N. Stergioulas, J. A. Font and E. Seidel, Phys. Rev. D
71, 024035 (2005), and references therein.
[4] A. Watts, N. Andersson and D. Jones, Astrophys. J. L37 (2005).
[5] J. Centrella, K. New, L. Lowe and J. Brown, Astrophys. J. 550
(2001).
[6] S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale UP
1969).
[7] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E.
Müller, Phys. Rev. Letters 96, 161101 (2006).
[8] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E.
Müller, astro-ph/0611601 (2006).
http://arxiv.org/abs/astro-ph/0611601

  We investigate the role of rotational instabilities in the context of black
hole formation in relativistic stars. In addition to the standard scenario - an
axially symmetric dynamical instability forming a horizon at the star's center
- the recently found low-$T/|W|$ instabilities are shown to lead to
fragmentation and off-center horizon formation in differentially rotating
stars. This process might be an alternative pathway to produce SMBHs from
supermassive stars with inefficient angular momentum transport.

FRAGMENTATION OF GENERAL RELATIVISTIC QUASI-TOROIDAL POLYTROPES
Burkhard Zink,1, 2 Nikolaos Stergioulas,3 Ian Hawke,4 Christian D. Ott,5 Erik Schnetter,1, 6 and Ewald Müller7
1Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA
2Horace Hearne Jr. Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA
3Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
4School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
5Department of Astronomy and Steward Observatory, The University of Arizona, Tucson, AZ, USA
6Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
7Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching bei München, Germany
How do black holes form from relativistic stars? This ques-
tion is of great fundamental and practical importance in grav-
itational physics and general relativistic astrophysics. On the
fundamental level, black holes are genuinely relativistic ob-
jects, and thus the study of their production involves ques-
tions about horizon dynamics, global structure of spacetimes,
and the nature of the singularities predicted as a consequence
of the occurrence of trapped surfaces. On the level of as-
trophysical applications, systems involving black holes are
possible engines for highly energetic phenomena like AGNs
or gamma-ray bursts, and also likely a comparatively strong
source of gravitational radiation.
The most simple model of black hole formation from, say,
cold neutron stars, is a fluid in spherically symmetric poly-
tropic equilibrium moving on a sequence of increasing mass
due to accretion [1]. This assumes that (i) the stellar structure
and dynamics are represented reasonably by the ideal fluid
equation of state and the polytropic stratification, (ii) accre-
tion processes are slow compared to the dynamical timescales
of the star, and (iii) rotation is negligible. Our focus has been
to study the effects of relaxing the third assumption.
In spherical symmetry, the sequence of equilibrium poly-
tropes has a maximum in the mass function M(ρ
), where ρ
denotes the central rest-mass density of the polytrope. This
maximum is connected to a change in the stability of the fun-
damental mode of oscillation [1], and thus collapse sets in
via a dynamical instability to radial deformations. During the
subsequent evolution, a trapped tube forms at the center which
traverses the stellar material entirely [2].
How much of this behaviour is preserved when rotation is
taken into account? Rotation is known to change the equi-
librium structure of the star, and, in consequence, its modes
of oscillation and set of unstable perturbations. The collapse
might also lead to the formation of a massive disk around the
new-born black hole, and finally only systems without spher-
ical symmetry can be a source of gravitational radiation.
Numerical simulations have been used to study the collapse
and black hole formation of general relativistic rotating poly-
tropic stars [3]. For the uniformly and moderately differen-
tially rotating models investigated in those studies, the dynam-
ical process is described by the instability of a quasi-radial
mode and subsequent collapse of the star up to the formation
of an accreting Kerr black hole at the star’s center.
Will strong differential rotation modify this picture? Even
before our study, there was evidence that this should be the
case. (i) Strong differential rotation can deform the high-
FIG. 1: Development of the fragmentation instability in a model of a
strongly differentially rotating supermassive star. The darker shades
of grey indicate higher density. The closed white line in the last plot
is a trapped surface.
density regions of a star into a toroidal shape, thus chang-
ing the equilibrium structure considerably. (ii) It admits stars
of high normalized rotational energy T/|W | [1] which are
stable to axisymmetric perturbations. (iii) It admits non-
axisymmetric instabilities, for example by the occurrence of
corotation points[4], at low values of T/|W |[5]. (iv) A bar-
mode instability of the type found in Maclaurin spheroids[6]
would likely express itself by the formation of two orbiting
fragments if the initial high-density region has toroidal shape.
This last property has motivated us to ask this question:
Can a bar deformation transform a strongly differentially ro-
tating star into a binary black hole merger with a massive
accretion disk? If so, this process might occur in supermas-
sive stars if the timescales associated with angular momentum
transport are too large to enforce uniform rotation.
We have investigated black hole formation in strongly dif-
ferentially rotating, quasi-toroidal models of supermassive
stars [7, 8], and found that a non-axisymmetric instability can
lead to the off-center formation of a trapped surface (see fig-
ure). An extensive parameter space study of this fragmen-
tation instability [8] reveals that many quasi-toroidal stars of
this kind are dynamically unstable in this manner, even for low
values of T/|W |, and we have found evidence that the coro-
tation mechanism observed by Watts et al. [4] might be active
in these models. Since, on a sequence of increasing T/|W |,
one of the low order m = 1 modes becomes dynamically un-
http://arxiv.org/abs/0704.0431v1
stable before m = 2 and higher order modes, one would not
expect a binary black hole system to form in many situations
(although this may depend on the rotation law and details of
the pre-collapse evolution as well). Rather, the off-center pro-
duction of a single black hole with a massive accretion disk
appears more likely.
Since the normalized angular momentum J/M2 of the ini-
tial model is greater than unity, there is another interesting
consequence of this formation process: the resulting black
hole, unless it is ejected from its shell, may very well be
rapidly rotating, spun up by accretion of the material remain-
ing outside the initial location of the trapped surface. Investi-
gating the late time behaviour of this accretion process, esti-
mating possible kick velocities of the resulting black hole, and
finding the mass of the final accretion disk is, however, beyond
our present-day capabilities and subject of future study.
[1] S. Shapiro and S. Teukolsky, Black Holes, White Dwarfs and
Neutron Stars (Wiley 1983).
[2] S. Shapiro and S. Teukolsky, Astrophys. J. 235, 199 (1980).
[3] M. Shibata, T. Baumgarte and S. Shapiro, Phys. Rev. D 61,
044012 (2000). L. Baiotti, I. Hawke, P. Montero, F. Löffler, L.
Rezzolla, N. Stergioulas, J. A. Font and E. Seidel, Phys. Rev. D
71, 024035 (2005), and references therein.
[4] A. Watts, N. Andersson and D. Jones, Astrophys. J. L37 (2005).
[5] J. Centrella, K. New, L. Lowe and J. Brown, Astrophys. J. 550
(2001).
[6] S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale UP
1969).
[7] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E.
Müller, Phys. Rev. Letters 96, 161101 (2006).
[8] B. Zink, N. Stergioulas, I. Hawke, C. D. Ott, E. Schnetter and E.
Müller, astro-ph/0611601 (2006).
http://arxiv.org/abs/astro-ph/0611601
