Download Towards Gravitational Wave Astronomy

High Speed Black Hole and
Soliton Collisions
Frans Pretorius
Princeton University
Workshop on
Tests of Gravity and Gravitational Physics
May 19-21, 2009
Case Western Reserve University, Cleveland, Ohio
Outline
• Why explore high speed black hole collisions?
– probe the most dynamical, non-linear regime of general relativity
– could be of relevance describing LHC & cosmic ray observations if
there are large extra dimensions
• However, do super-Planck scale particle collision really form black
holes?
– arguments are purely classical, and not overwhelmingly convincing at a first
glance
• Test this hypothesis by studying self-gravitating soliton collisions
using numerical methods
• Results from high-speed black hole collision simulations
– head-on collisions
– speculations about grazing collisions − phenomenology described by zoomwhirl orbits?
Black hole formation at the LHC and in the atmosphere?
•
large extra dimension scenarios [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali,
•
A scale in the TeV range is a “natural” choice to solve the hierarchy
problem
•
Implications of this are that super-TeV particle collisions would probe
the quantum gravity regime
PLB429:263-272; L. Randall & R. Sundrum, PRL.83:3370-3373]
suggest the true
Planck scale can be very different from what then would be an
effective 4-dimensional Planck scale of 1019 GeV calculated from the
fundamental constants measured on our 4-D Brane
–
collisions sufficiently above the Planck scale are expected to be dominated by
the gravitational interaction, and arguments suggest that black hole formation
will be the most likely result of the two-particle scattering event [Banks &
Fishler hep-th/9906038, Dimopoulos & Landsberg PRL 87 161602 (2001), Feng &
Shapere, PRL 88 021303 (2002), …]
•
current experiments rule out a Planck scale of <~ 1TeV
•
The LHC should reach center-of-mass energies of ~ 10 TeV
•
cosmic rays can have even higher energies than this, and so
in both cases black hole formation could be expected
•
these black holes will be small and decay rapidly via Hawking
radiation, which is the most promising route to detection
•
ATLAS experiment at the LHC
One of the water tanks at the Pierre Auger Observatory
if a lot of gravitation radiation is produced during the
collision this could show up as a missing energy signal
at the LHC
But do super-Planck scale particle collisions
form black holes?
• The argument that the ultra-relativistic collision of two particles
should form a black hole is purely classical, and is essentially based
on Thorne’s hoop conjecture
– (4D) if an amount of matter/energy E is compacted to within a sphere
of radius R=2GE/c4 corresponding to the Schwarzschild radius of a black
hole of mass M=E/c2, a black hole will form
– applied to the head-on collision of two “particles” each with rest mass
m, characteristic size W, and center-of-mass frame Lorentz gamma
factors g, this says a black hole will form if the Schwarzschild radius
corresponding to the total energy E=2mg is greater that W
– the quantum physics comes in when we say that the particle’s size is
given by its de Broglie wavelength W = hc/E, from which one gets the
Planck energy Ep=(hc5/G)1/2
Evidence to support this
•
From the classical perspective, evidence to support this would be solutions
to the field equations demonstrating that weakly self-gravitating objects,
when boosted toward each other with large velocities so that the net mass
of the space time (in the center of mass frame) is dominated by the kinetic
energy, generically form a black hole when the interaction occurs within a
region smaller than the Schwarzschild radius of the spacetime
– generic: the outcome would have to be independent of the particular details of
the structure and non-gravitational interactions between the particles, if the
classical picture is to have any bearing on the problem
– interaction: the non-linear interaction of the gravitational kinetic energy of the
boosted particles will be key in determining what happens
• consider the trivial counter-examples to the hoop conjecture applied to a single particle
boosted to ultra-relativistic velocities, or a white hole “explosion”
The infinite boost limit
•
Give an arbitrary, static, charge-free, spherically symmetric soliton of
gravitational rest mass m a Lorentz boost of g. Take the limit g, m0 such
that the energy E= mg is constant. In the limit, the solution is given by the
Aichelburg-Sexl spacetime [GRG 2, 303 (1971)] (originally derived as the infinite
boost limit of a Schwarzschild black hole)
•
the spacetime is a plane-fronted gravitational “shock”-wave, with Minkowski
spacetime on either side of the shock
•
Collide two such spacetimes together … even though the solution to the future of
the collision is not known, trapped surfaces can be found at the moment of
collision [Penrose 1974, Eardley & Giddings PRD 66, 044011 (2002), Yoshino &
Rychkov, PRD D71, 104028 (2005), … ]
The infinite boost limit
•
The presence of trapped surfaces thus gives an argument in favor of the
hypothesis of black hole formation in ultrarelativstic collisions, however
– this is a singular limit, and only in the most trivial sense does the solution
provide a good approximation to the geometry of a finite-g soliton (i.e. it’s a
good approximation when you’re far enough away that the soliton spacetime
is close to Minkowksi)
• going to the infinite boost limit, the algebraic type of the Weyl tensor changes from
type D (two distinct eigenvectors) to type N (1 distinct eigenvector), and the
spacetime ceases to be asymptotically flat
– when two finite-gamma solitions collide, the non-trivial spacetime dynamics
that will (or will not) cause black hole formation will unfold precisely in the
regime where the Aichelberg-Sexl solution is not a good approximation
– simulations of boson star collisions [D. Choi et al; K. Lai, PhD Thesis 2004, C.
Palenzuela et al Phys.Rev.D75:064005,2007] suggest that gravity becomes
weaker in the interaction as the initial velocity is increased
Use numerical simulations to try to test this
•
Solve the Einstein equations coupled to a matter source that admits soliton
solutions (work with M. Choptuik)
•
Use boson-stars as our soliton model
– the simplest matter model giving boson star solutions is a single massive
complex scalar field f (or can think of it as two interacting real scalar fields)
– boson star solutions take the form
f (r )  Ae
 ( r / d ) 2 iwt
e
and for a range of central amplitudes A, stable eigensolutions exist with
eigenvalues d and w; w is roughly equal to the mass parameter m of the field
– though the constituent fields are oscillatory, the stress-energy tensor is static
•
Solve the coupled Einstein-matters equations using a generalized harmonic
form of the field equations
Results for head-on collision of boosted boson stars
• Very computationally expensive to run high-g simulations,
so need to start with a relatively compact boson star that
will reach hoop-conjecture limits with reasonable g ’s.
– resolution requirements scale like g 3; gauge issues; not
solving the constraint equations for a superposition of
boosted solutions, so solitons need to be far enough apart
for error to be small; etc.
• choose parameters to give a boson star with R/2M ~ 22
– thus, hoop-conjecture suggests a collision of two of these
with g=11 in the center of mass frame will be the marginal
case
Results for head-on collision of boosted
boson stars
• we find black hole formation at g=3 already!
– signs of a Planck scale of O(10 Tev) or smaller should
already have been seen?
Case 1: free-fall collision from rest
Symmetry axis
Both the color and
height of the
surface represent
the magnitude of
the scalar field.
Scale M is the
total rest-mass of
of the boson stars
•
Here, gravity dominates the interaction, causing the boson stars to
coalesce into a single, highly perturbed boson star (this case
eventually collapses to form a black hole)
Case 2 : g = 2
•
Here, though gravity strongly perturbs the boson stars, kinetic energy “wins” and causes
them to pass through each other
–
soliton-like interference pattern can be seen as the boson star matter interacts
–
superposition of initial data, and subsequent truncation, cause some component of the field to
move in the wrong direction; the truncation error part converges away with resolution, the initial
data part lessens the further the initial separation
Case 3 : g = 4
•
Here, the early matter interaction looks similar, but now the gravitational interaction of the kinetic
energy of the solitons causes gravitational collapse and black hole formation
–
NOTE: gauge than previous case: the coordinate spreading of the solitons before collision, and shrinking of
the horizon afterwards, are just coordinate effects; also, different color scale
Case 4 : g ~ 3
•
beginning to see richer interaction dynamics, as expected for threshold solutions
zoom-in of collision region
•
Though note, different vertical and color-scale from previous animation
Back to Particle Collisions
• If these soliton collisions are confirming the expected generic
behavior of high energy particle collisions, this implies we can
use any model of a particle to study the classical gravitational
signatures of super-Planck scale collisions, including black holes!
• Will describe some on-going studies of such scenarios with U.
Sperhake, V. Cardose, E. Berti, J.A Gonzalez, T. Hinderer & N.
Yunes
• However, with application to the LHC in mind, such a project
can only be pursued in some approximate manner
– unknown Planck-scale physics
– unknown structure of the extra dimensions
– 4D simulations “barely” feasible … generic 10(11?) dimensional
simulations impossible in foreseeable future
Aside : the Planck Luminosity regime
• The Planck Luminosity Lp is
c5
Lp =  1052 W = 1059 ergs/s
G
– notice that Planck’s constant does not enter, hence this
is a regime that is ostensibly described by classical
general relativity
– Lp may be a limiting luminosity for “reasonable”
processes, and trying to reach it may reveal some of the
more interesting aspects of the theory
• would super-Planck luminosities be associated with violations of
cosmic censorship?
The Planck Luminosity regime
•
Suppose a single black hole is formed during the collision, and a fraction e of
the total energy of the system 2gmc2 (equal mass) is released as gravitational
waves. Estimate that the shortest timescale on which the energy can be
released is the light-crossing time of the remnant black hole ~2R/c = 4G(1e)M/c3. This gives
e
L
Lp
41  e 
•
Implies that a very efficient (>80%) prompt energy release mechanism will
result in super-Planck luminosity, however
– we know in the low-speed regime e is small (fraction of a percent)
– if the limiting solution is a collision of Aichelburg-Sexl shock waves, Penrose found
trapped surfaces at the moment of collision implying e<29%; seems to be consistent
with simulation results (next slides)
– in non-head-on collisions, may be able to make e~100% (though this will not be
prompt, so luminosity might still be sub-Planckian)
g=2.9 collision example
T=0 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=30 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=83 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=115M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=137 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=156 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=202 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=252 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=292 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
T=405 M0
(M=g M0 )
Re[Y4]
g=2.9 collision example
•
roughly 8% of the total energy is emitted in gravitational waves during the collision …
peak luminosity ~0.3% of Lp
–
about a factor of 6 less than the order-of-magnitude estimate (the relevant time scale seems to
be given by the dominant quasi-normal mode frequency of the final black hole, the period of
which is a few times larger that the light-crossing time of the black hole)
•
spectrum is reasonably flat (left) below a cut-off given by the QNM frequencies of the
black hole, a result predicted by the “zero-frequency-limit” (ZFL) approximation
•
extrapolation to infinite boost limit using a curve motivated by the ZFL (right, red line)
suggests result will be ~ 14 %, a bit less than 1/2 the Penrose bound
Non-head on collisions: the black hole scattering problem
•
Consider the off-center collision of two black holes with impact
parameters b
m2,v2
b
m1,v1
•
in general two, distinct end-states possible
• one black hole, after a collision
• two isolated black holes, after a deflection
•
because there are two distinct end-states, there must be some kind of
threshold behavior approaching a critical impact parameter b*
The threshold of immediate merger
•
The following illustrates what could happen as one tunes to threshold,
assuming smooth dependence of the trajectories as a function of b
• non-spinning case (so we have evolution in a plane)
• only showing one of the BH trajectories for clarity
•
solid blue (black) – merger (escape)
•
dashed blue (black) – merger (escape) for values of b closer to threshold
The threshold of immediate merger : geodesics
•
we know the previous argument works for geodesics … in fact, the threshold
solutions are the unstable circular orbits of the black hole, and perturbations of
these give rise to “zoom-whirl” behavior
unbound orbits (green scatter, blue
capture)
bound orbits [the non-capture case is not a
two-leaf orbit … integration just stopped
after the second zoom]
Zoom-whirl orbits
• Zoom-whirl orbits are perturbations of unstable circular orbits that exist
within the ISCO
– In Schwarzschild, radial perturbations of circular orbits in the range
• 4M to 6M lead to elliptic zoom-whirl orbits
• 3M (the “light ring”) to 4M lead to a hyperbolic orbit with one whirl episode
– depend on the sign of the perturbation, the geodesic will fall into the black
hole or not after a whirl phase
• The number of whirls n is related to the magnitude of the perturbation
dr and the instability exponent g of the orbit via
e  dr
n
g
• in relation to the scattering problem, dr is proportional to b-b*
The threshold of immediate merger : equal mass binaries
•
The figures below are from full numerical simulations of the field equations for
equal mass orbits, showing qualitatively the same behavior as the geodesic
problem
– however, the binary in the whirl phase is emitting copious amounts of gravitational
radiation; on the order of 1-1.5% of the total mass of the system per orbit
two cases tuned close to threshold
(only 1 BH trajectory shown)
dominant component of emitted gravitational
waves as measure by NP scalars
Back to particle collisions in extra dimensions
• full simulations of black hole collisions in higher dimensional
settings unfeasible, however, we may be get a decent
approximation to gravitational aspects of the problem, if the
following properties hold for ultra-relativistic collisions
– the setup of the scattering problem (i.e. the LHC and cosmic
rays) “naturally” focuses on the threshold of immediate merger
– the energetics in this regime are dominated by the unstable
light-like circular orbit, and bounded at the two opposite
extremes by the head-on collision result, and 0.
– the instability exponents of near-equal mass collisions are
similar to those of geodesics about a black hole with the same
mass and angular momentum parameters of the final collision
product of the two particles
Back to Particle Collisions in extra dimensions
• This implies we can study the problem by exploring the
following ingredients
– The head on-collision case, for the baseline of energy radiated
• due to the symmetries of the problem this will be a lower dimensional
problem (2+1D in most scenarios)
– An estimate of the luminosity of energy emission during the whirl
phase of the zoom-whirl orbits
• PN-like techniques, calibrated by ultra-relativistic 4D simulations
– A study of the instability exponents of the unstable geodesics about
the relevant black hole solution
4D example
•
Look at the equal mass, non-spinning case
– unequal mass collisions will be important for LHC/cosmic ray examples
– in the infinite boost limit charge and spin are irrelevant , though both may play
some role at LHC energies
•
Energy emitted in the ultra-relativistic, head-on case is ~ 14% [Sperhake et.
al. Phys.Rev.Lett.101:161101, 2008]
•
What’s the relevant black hole geodesic analogue?
– it seems “natural” that in this limit the final spin of the black hole at threshold is
a=1. This is consistent with geodesic estimates of the initial energy and angular
momentum at threshold, and of the energy/angular momentum radiated per orbit
from quadrupole physics:
d ( J / E 2 ) 1 dE  1
J 
=
2 2 

dn
E dn  Ew
E 
– for the scattering problem with the same impact parameter as a threshold
geodesic on an extremal Kerr background, the initial J/E2=1. The Boyer-Lindquist
value of Ew is ½ for a geodesic on the light ring of an extremal Kerr BH, so in that
regime d(J/ E2)=0
Sample energy radiated vs. impact parameter curves (normalized)
•
Quadrupole formula says dE/dn ~ p/40 in this
limit; however, early results from full simulations
suggest this is a significant underestimate!
•
Furthermore, one limitation of this analysis is that
in extremal Kerr black holes have no unstable
circular orbits, i.e. the instability exponent g
diverges!
•
This is similar to one example of a higher
dimensional black holes, namely the Myers-Perry
black holes, where g becomes quite large
regardless of the spin [C. Merrick, Princeton JP
thesis ‘08; V. Cardoso et al. arXiv:0812.1806]
•
This may imply an even simpler scenario … when the
impact parameter is within a factor of 2 of the
threshold, almost all energy is lost to gravitational
wave emission
–
Good (?) for LHC detection prospects, bad for
cosmic rays
Conclusions
• Black hole collisions are a fascinating probe of strong-field general
relativity
– remarkable that the cut-and-paste Penrose construction of the infinite boost limit
seems to capture the leading order behavior of what is a highly non-linear
interaction in the finite boost case
• Soliton collision simulations suggest a critical boost of around 1/3
less than hoop-conjecture estimates lead to black hole formation
– Rich dynamics will unfold near the threshold, which existing studies of
gravitational collapse suggest would be a Type-II critical solution, where for
some fixed impact parameter one would get
M BH ( ) = ( Enet  EGW )    c  ,

  c ,   0
– since the matter seems to becoming irrelevant, the critical solution should be a
vacuum solution − is it the Abrahams & Evans Brill wave critical collapse
spacetime [A.Abrahams and C.Evans, PRL 70, 2980 (1993)] ? Even with non-zero b?