Black Hole CollisionsBlack Hole Collisions

Frans PretoriusFrans PretoriusPrinceton UniversityPrinceton University

Black Holes VIBlack Holes VIWhite Point, Nova ScotiaWhite Point, Nova Scotia

May 14, 2007 May 14, 2007

OutlineOutline• Motivation : why simulate black hole collisions?Motivation : why simulate black hole collisions?

– expected to be among the strongest and most promising sources of gravitational waves that could expected to be among the strongest and most promising sources of gravitational waves that could be observed by gravitational wave detectorsbe observed by gravitational wave detectors

– understand the strong-field regime of general relativityunderstand the strong-field regime of general relativity

– in the final stages of coalescence no exact or perturbative analytic techniques known to solve the in the final stages of coalescence no exact or perturbative analytic techniques known to solve the problem problem

• MethodologyMethodology

– brief overview of numerical relativity, the difficulties in discretizing the field equations, and an brief overview of numerical relativity, the difficulties in discretizing the field equations, and an evolution scheme based on generalized harmonic coordinatesevolution scheme based on generalized harmonic coordinates

• ResultsResults

– brief synopsis of recent results in binary black hole merger simulationsbrief synopsis of recent results in binary black hole merger simulations

• most detailed studies carried out for equal mass, quasi-circular inspiralsmost detailed studies carried out for equal mass, quasi-circular inspirals

• much excitement about large recoil velocities for certain cases with spinmuch excitement about large recoil velocities for certain cases with spin

– can understand this as a “frame dragging” effectcan understand this as a “frame dragging” effect

– fine-tuned eccentric orbitsfine-tuned eccentric orbits

• evidence that, to a evidence that, to a certaincertain degree, some of the interesting phenomenology of test particle motion persists in the degree, some of the interesting phenomenology of test particle motion persists in the equal mass merger regime, including unstable equal mass merger regime, including unstable circularcircular orbits and corresponding zoom-whirl like behavior orbits and corresponding zoom-whirl like behavior

• speculative application to the high-energy scattering problemspeculative application to the high-energy scattering problem

Numerical RelativityNumerical Relativity• Numerical relativity is concerned with solving the field equations of general relativityNumerical relativity is concerned with solving the field equations of general relativity

using computers. using computers.

• When written in terms of the spacetime metric, defined by the usual line elementWhen written in terms of the spacetime metric, defined by the usual line element

the field equations form a the field equations form a system of 10 coupled, non-linear, second order partial system of 10 coupled, non-linear, second order partial differential equations, each depending on the 4 spacetime coordinatesdifferential equations, each depending on the 4 spacetime coordinates

– it is this system of equations that we need to solve for the 10 metric elements (plus whatever it is this system of equations that we need to solve for the 10 metric elements (plus whatever matter we want to couple to gravity)matter we want to couple to gravity)

– for many problems this has turned out to be quite an undertaking, due in part to the for many problems this has turned out to be quite an undertaking, due in part to the mathematical complexity of the equations, and also the heavy computational resources mathematical complexity of the equations, and also the heavy computational resources required to solve themrequired to solve them

• The field equations may be complicated, but they are The field equations may be complicated, but they are thethe equations that we believe equations that we believe govern the structure of space and time (barring quantum effects and ignoring govern the structure of space and time (barring quantum effects and ignoring matter). That they can, in principle, be solved in many “real-universe” scenarios is a matter). That they can, in principle, be solved in many “real-universe” scenarios is a remarkable and unique situation in physics. remarkable and unique situation in physics.

πTG 8

dxdxgds 2

Minimal requirements for a formulation of the field equations Minimal requirements for a formulation of the field equations that that mightmight form the basis of a successful numerical integration form the basis of a successful numerical integration

schemescheme• Choose coordinates/system-of-variables that fix the character of the equationsChoose coordinates/system-of-variables that fix the character of the equations

– three common choicesthree common choices• free evolutionfree evolution — system of hyperbolic equations — system of hyperbolic equations• constrained evolutionconstrained evolution — system of hyperbolic and elliptic equations — system of hyperbolic and elliptic equations• characteristiccharacteristic or or null evolutionnull evolution — integration along the lightcones of the spacetime — integration along the lightcones of the spacetime

• For free evolution, need a system of equations that is well behaved off the For free evolution, need a system of equations that is well behaved off the ”constraint ”constraint manifold”manifold”

– analytically, if satisfied at the initial time the constraint equations of GR will be satisfied for all timeanalytically, if satisfied at the initial time the constraint equations of GR will be satisfied for all time

– numerically the constraints can only be satisfied to within the truncation error of the numerical numerically the constraints can only be satisfied to within the truncation error of the numerical scheme, hence we do not want a formulation that is “unstable” when the evolution proceeds scheme, hence we do not want a formulation that is “unstable” when the evolution proceeds slightly off the constraint manifoldslightly off the constraint manifold

• Need well behaved coordinates (or gauges) that do not develop pathologies when the Need well behaved coordinates (or gauges) that do not develop pathologies when the spacetime is evolvedspacetime is evolved

– typically need dynamical coordinate conditions that can adapt to unfolding features of the typically need dynamical coordinate conditions that can adapt to unfolding features of the spacetimespacetime

• Boundary conditions also historically a source of headaches Boundary conditions also historically a source of headaches

– naive BC’s don’t preserve the constraint nor are representative of the physicsnaive BC’s don’t preserve the constraint nor are representative of the physics– fancy BC’s can preserve the constraints, but again miss the physicsfancy BC’s can preserve the constraints, but again miss the physics– solution … compactify to infinity solution … compactify to infinity

• Geometric singularities in black hole spacetimes need to be dealt with Geometric singularities in black hole spacetimes need to be dealt with

Generalized Harmonic Evolution SchemeGeneralized Harmonic Evolution Scheme

• Einstein equations in Einstein equations in generalized harmonicgeneralized harmonic form with form with constraint dampingconstraint damping::

where the generalized harmonic constraints are ,where the generalized harmonic constraints are , is a constant, is a constant, nn is a unit time-like vector is a unit time-like vector

and and are the Christoffel symbols are the Christoffel symbols

• Need Need gauge evolution equationsgauge evolution equations to close the system; use the following with to close the system; use the following with and and nn constants, and constants, and is the so-called lapse function: is the so-called lapse function:

• Matter stress energy supplied by a massless scalar field Matter stress energy supplied by a massless scalar field : : 0

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Computational issues in solving numerical Computational issues in solving numerical solution of the field equationssolution of the field equations

• Each equation contains tens to hundreds of individual terms, requiring Each equation contains tens to hundreds of individual terms, requiring on the order of on the order of several thousand floating point operations several thousand floating point operations per grid per grid point point with any evolution scheme.with any evolution scheme.

• Problems of interest often have Problems of interest often have several orders of magnitude of several orders of magnitude of relevant physical length scalesrelevant physical length scales that need to be well resolved. In an that need to be well resolved. In an equal mass binary black hole merger for example:equal mass binary black hole merger for example:

• radius of each black hole radius of each black hole R~2MR~2M• orbital radius orbital radius ~ 20M (~ 20M (which is also the dominant wavelength of radiation emitted)which is also the dominant wavelength of radiation emitted)• outer boundary outer boundary ~ 200M~ 200M, as the waves must be measured in the weak-field , as the waves must be measured in the weak-field

regime to coincide with what detectors will seeregime to coincide with what detectors will see

– Can solve these problems with a combination of hardware technology — Can solve these problems with a combination of hardware technology — supercomputers — and software algorithms, in particular adaptive mesh supercomputers — and software algorithms, in particular adaptive mesh refinement (AMR)refinement (AMR)

• vast majority of numerical relativity codes today use finite difference techniques vast majority of numerical relativity codes today use finite difference techniques (predominantly 2(predominantly 2ndnd to 4 to 4thth order), notable exception is the Caltech/Cornell pseudo- order), notable exception is the Caltech/Cornell pseudo-spectral code spectral code

• How to deal with the true geometric singularities that exist inside all How to deal with the true geometric singularities that exist inside all black holes?black holes?

• excisionexcision

Brief Synopsis of Recent Progress in the Brief Synopsis of Recent Progress in the Simulation of Binary Coalescence ISimulation of Binary Coalescence I

• Two quite different, stable methods of integrating the Einstein field Two quite different, stable methods of integrating the Einstein field equations for this problemequations for this problem

– generalized harmonic coordinates with constraint dampinggeneralized harmonic coordinates with constraint damping, , F.Pretorius, F.Pretorius, PRL 95, 121101 (2005)PRL 95, 121101 (2005)

• Caltech/Cornell, Caltech/Cornell, L. Lindblom et al., Class.Quant.Grav. 23 (2006) S447-S462L. Lindblom et al., Class.Quant.Grav. 23 (2006) S447-S462

• PITT/AEI/LSU, PITT/AEI/LSU, B. Szilagyi et al., gr-qc/0612150, & L. Lehner et al. B. Szilagyi et al., gr-qc/0612150, & L. Lehner et al.

– BSSN with “moving punctures”,BSSN with “moving punctures”, M. Campanelli, C. O. Lousto, P. Marronetti, Y. M. Campanelli, C. O. Lousto, P. Marronetti, Y. Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Zlochower PRL 96, 111101, (2006); J. G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter PRL 96, 111102, (2006)Meter PRL 96, 111102, (2006)

• Pennstate, Pennstate, F. Herrmann et al., gr-gc/0601026F. Herrmann et al., gr-gc/0601026

• Jena, Jena, J. A. Gonzalez et al., gr-gc/06010154J. A. Gonzalez et al., gr-gc/06010154

• LSU/AEI/UNAM, LSU/AEI/UNAM, J. Thornburg et al., gr-gc/0701038J. Thornburg et al., gr-gc/0701038

• U.Tokyo/UWM, U.Tokyo/UWM, M. Shibata et al, astro-ph/0611522M. Shibata et al, astro-ph/0611522

• U. Sperhake, gr-qc/0606079U. Sperhake, gr-qc/0606079

Brief Synopsis of Recent Progress in the Brief Synopsis of Recent Progress in the Simulation of Binary Coalescence IISimulation of Binary Coalescence II

• Several groups have now evolved approximations to quasi-circular inspiral, and are getting similar Several groups have now evolved approximations to quasi-circular inspiral, and are getting similar resultsresults

• Waveforms remarkably simple … brief merger regime connects inspiral and ringdown phases well fit Waveforms remarkably simple … brief merger regime connects inspiral and ringdown phases well fit by perturbative modelsby perturbative models

• For the equal mass, non-spinning mergers ~ 3-4% of the total energy of the system is radiated in the For the equal mass, non-spinning mergers ~ 3-4% of the total energy of the system is radiated in the final couple of orbits, collision and ringdown; the final black hole has a spin of ~ 0.7final couple of orbits, collision and ringdown; the final black hole has a spin of ~ 0.7

J. G. Baker, M. Campanelli, F. Pretorius, Y.Zlochower qr/qc J. G. Baker, M. Campanelli, F. Pretorius, Y.Zlochower qr/qc 07010160701016 A. Buonanno, G.B. Cook and F. Pretorius qr/qc A. Buonanno, G.B. Cook and F. Pretorius qr/qc 06101220610122

RepresentatioRepresentations of the ns of the waveform in waveform in terms of the terms of the the Newman the Newman Penrose scalar Penrose scalar , which in , which in the wave zone the wave zone is proportional is proportional to the second to the second time derivative time derivative of the more of the more usual strain usual strain

Brief Synopsis of Recent Progress in the Brief Synopsis of Recent Progress in the Simulation of Binary Coalescence IIISimulation of Binary Coalescence III

• Waveform utility for early gravitational wave detection effortsWaveform utility for early gravitational wave detection efforts

– simplicity of waveforms suggest hybrid (or even completely analytic) templates banks could be simplicity of waveforms suggest hybrid (or even completely analytic) templates banks could be constructedconstructed

– for higher-mass mergers events present waveforms seem to be accurate enough to be used as for higher-mass mergers events present waveforms seem to be accurate enough to be used as templatestemplates

– current approximate and phenomenological templates families for binary black holes (such as BCV) current approximate and phenomenological templates families for binary black holes (such as BCV) are again good for detectionare again good for detection

• get very high overlaps between these families (>0.97) and hybrid PN/Numerical waveformsget very high overlaps between these families (>0.97) and hybrid PN/Numerical waveforms

• parametersparameters of highest overlap templates are often far off of highest overlap templates are often far off

T. Baumgarte et al., qr/qc T. Baumgarte et al., qr/qc 06121000612100

T. Baumgarte et al., qr/qc T. Baumgarte et al., qr/qc 06121000612100

Y. Pan et al. qr/qc Y. Pan et al. qr/qc 0704.1964 0704.1964

Evolution of Cook-Pfeiffer Quasi-circular BBH Evolution of Cook-Pfeiffer Quasi-circular BBH Merger Initial dataMerger Initial data

This animation This animation shows the shows the lapse lapse functionfunction in the in the orbital plane.orbital plane.

The lapse function The lapse function represents the represents the relative time relative time dilation between a dilation between a hypothetical hypothetical observer at the observer at the given location on given location on the grid, and an the grid, and an observer situated observer situated very far from the very far from the systemsystem

The redder the The redder the color, the slower color, the slower local clocks are local clocks are running relative to running relative to clocks at infinityclocks at infinity

If this were in “real-If this were in “real-time” it would time” it would correspond to the correspond to the merger of two merger of two ~5000 solar mass ~5000 solar mass black holesblack holes

Gravitational waves from the Gravitational waves from the simulationsimulation

The real component of The real component of in the orbital plane, in the orbital plane, which is proportional to which is proportional to the second time the second time derivative of the plus-derivative of the plus-polarization component polarization component of the waveof the wave

Recoil velocities in binary mergersRecoil velocities in binary mergers

• ““kick” velocity of remnant black hole due to asymmetric beaming of radiationkick” velocity of remnant black hole due to asymmetric beaming of radiation

– up to 175km/s for non-spinning, unequal mass components [up to 175km/s for non-spinning, unequal mass components [F. Herrmann, D. Shoemaker and P. F. Herrmann, D. Shoemaker and P. Laguna, gr-qc/0601026, J.G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter, M. Coleman Miller, astro-ph/0603204, Laguna, gr-qc/0601026, J.G. Baker, J. Centrella, D. Choi, M. Koppitz, J. van Meter, M. Coleman Miller, astro-ph/0603204, J.A. Gonzalez, U. Sperhake, B. Bruegmann, M. Hannam, S. Husa, gr-qc/0610154J.A. Gonzalez, U. Sperhake, B. Bruegmann, M. Hannam, S. Husa, gr-qc/0610154 ] ]

– typical values for spinning black holes of typical values for spinning black holes of 100’s km/s, but can be as large as 100’s km/s, but can be as large as 4000km/s for equal mass black holes with 4000km/s for equal mass black holes with spins vectors anti-aligned and in the spins vectors anti-aligned and in the orbital plane [orbital plane [F. Herrmann et al., F. Herrmann et al., gr-qc/0701143; M. Koppitz et al., gr-qc/0701163; gr-qc/0701143; M. Koppitz et al., gr-qc/0701163; M. Campanelli et al. gr-qc/0701164 & gr-qc/gr-M. Campanelli et al. gr-qc/0701164 & gr-qc/gr-qc/0702133qc/0702133]]

• uniform sampling over spin vector uniform sampling over spin vector orientations and mass ratios for two orientations and mass ratios for two a=.9 black holes with m1/m2 a=.9 black holes with m1/m2 between 1 and 10 suggested only between 1 and 10 suggested only around 2% of parameter space has around 2% of parameter space has kicks larger than 1000km/s, and 10% kicks larger than 1000km/s, and 10% larger than 500km/s [larger than 500km/s [J. Schnittman & A. J. Schnittman & A. Buonanno, astro-ph/0702641Buonanno, astro-ph/0702641]]

• astrophysical population is most astrophysical population is most likely highly non-uniform, e.g. torques likely highly non-uniform, e.g. torques from accreting gas in supermassive from accreting gas in supermassive merger scenarios tend to align the merger scenarios tend to align the spin and orbital angular momenta, spin and orbital angular momenta, which will result in more modest kick which will result in more modest kick velocities <~200km/s [velocities <~200km/s [T.Bogdanovic et T.Bogdanovic et al, astro-ph/0703054al, astro-ph/0703054]]

J. A. Gonzalez et al., qr/qc J. A. Gonzalez et al., qr/qc 06101540610154

Large recoil velocities in binary mergersLarge recoil velocities in binary mergers

• scenario giving rise to very large kick velocities is, at scenario giving rise to very large kick velocities is, at a first glance, quite bizarre: a first glance, quite bizarre:

– equal masses with equal but opposite spin vectors equal masses with equal but opposite spin vectors in the orbital planein the orbital plane

– kick direction is normal to the orbital plane, but kick direction is normal to the orbital plane, but magnitude depends sinusoidally on the initial magnitude depends sinusoidally on the initial phasephase

– is linearly dependent on the magnitude of the spinis linearly dependent on the magnitude of the spin

– still “small” from a dimensional analysis point of still “small” from a dimensional analysis point of view, but enormous in an astrophysical settingview, but enormous in an astrophysical setting

Frame dragging induced kicksFrame dragging induced kicks

• Can intuitively understand this phenomena Can intuitively understand this phenomena as a as a frame dragging effectframe dragging effect

– think of BH 1 (2) moving in the background think of BH 1 (2) moving in the background space time of BH 2 (1)space time of BH 2 (1)

– relative to BH 1 (2)’s spin vector, BH 2 (1) relative to BH 1 (2)’s spin vector, BH 2 (1) has zero orbital angular momentum … has zero orbital angular momentum … such a “particle” in a BH background is such a “particle” in a BH background is dragged by the rotation of the BH with a dragged by the rotation of the BH with a velocity velocity

v ~ m a sin(v ~ m a sin() / r) / r22

Frame dragging induced kicksFrame dragging induced kicks

• The result of this in the equal mass problem with the The result of this in the equal mass problem with the particular configuration for maximum kicks is that the particular configuration for maximum kicks is that the entire orbital planeentire orbital plane will oscillate sinusoidally in the will oscillate sinusoidally in the direction normal to the orbital plane with the orbital direction normal to the orbital plane with the orbital frequency, and maximum velocity of frequency, and maximum velocity of m a / rm a / r22

• The binary is emitting gravitational radiation, with the The binary is emitting gravitational radiation, with the largest flux normal to the orbital plane … the frame largest flux normal to the orbital plane … the frame dragging induced oscillations will cause a periodic dragging induced oscillations will cause a periodic doppler shift of the radiation along the axis, alternating doppler shift of the radiation along the axis, alternating between red and blue shift in one direction with the between red and blue shift in one direction with the opposite in the otheropposite in the other

• Averaged over an orbit, and without inspiral, the net Averaged over an orbit, and without inspiral, the net momentum radiated is zero. However, the merger event momentum radiated is zero. However, the merger event stops this process, and depending on where it’s stopped stops this process, and depending on where it’s stopped there will be linear momentum radiated normal to the there will be linear momentum radiated normal to the orbital plane … to conserve momentum the final black orbital plane … to conserve momentum the final black hole is given a kick in the opposite direction hole is given a kick in the opposite direction

Frame dragging induced kicksFrame dragging induced kicks

• To estimate the maximum kick:To estimate the maximum kick:

– assume radiation stops once the black hole assume radiation stops once the black hole separation is within separation is within r~2M (M=2m)r~2M (M=2m)

– energy of a gravitational wave ~ frequency-energy of a gravitational wave ~ frequency-squared … doppler shift by maximum frame squared … doppler shift by maximum frame dragging velocity at dragging velocity at rr

– let the net energy radiated over the last ½ orbit let the net energy radiated over the last ½ orbit be be MM, then:, then:

vvrecoilrecoil ~ ~ (a/M)/4(a/M)/4

– with typical with typical ~ 1-2%, ~ 1-2%, get get vvrecoilrecoil ~ (a/M) 1000 km/s ~ (a/M) 1000 km/s

Frame-dragging induced kicksFrame-dragging induced kicks• this seems to be consistent with what’s happening in this seems to be consistent with what’s happening in

simulationssimulations– however, none of this is gauge invariant, so take this in the same however, none of this is gauge invariant, so take this in the same

spirit as the order-of-magnitude calculation spirit as the order-of-magnitude calculation

equal mass merger, “scattering” initial conditions with impact parameter equal mass merger, “scattering” initial conditions with impact parameter ~14m, initial velocities ~0.12 in +- y direction, initial spins a/m=0.5 anti-~14m, initial velocities ~0.12 in +- y direction, initial spins a/m=0.5 anti-aligned in +- x direction aligned in +- x direction

Fine-tuned eccentric orbitsFine-tuned eccentric orbits

• Use scalar field collapse to construct binariesUse scalar field collapse to construct binaries

– initial, equal mass scalar field pulses separated a coordinate (proper) initial, equal mass scalar field pulses separated a coordinate (proper) distance distance 8.9M8.9M ( (10.8M10.8M ) on the x-axis, one boosted with ) on the x-axis, one boosted with boost parameterboost parameter kk in the in the +y+y direction, the other with direction, the other with kk in the in the -y-y direction direction

• note, resultant black hole velocities are related to, but not equal to note, resultant black hole velocities are related to, but not equal to kk

• To find interesting orbital dynamics, tune the parameter To find interesting orbital dynamics, tune the parameter kk to get to get as many orbits as possibleas many orbits as possible

– in the limit as in the limit as kk goes to 0, get head-on collisions goes to 0, get head-on collisions– in the large in the large kk limit, black holes are deflected but fly apart limit, black holes are deflected but fly apart

• Generically these black hole binaries will have some eccentricity Generically these black hole binaries will have some eccentricity (not easy to define given how close they are initially), and so (not easy to define given how close they are initially), and so arguably of less astrophysical significancearguably of less astrophysical significance

– want to explore the non-linear interaction of BH’s in full general want to explore the non-linear interaction of BH’s in full general relativity relativity

Scalar field Scalar field rr, compactified (code) , compactified (code) coordinatescoordinates

)2/tan(),2/tan(),2/tan( zzyyxx

Sample OrbitSample Orbit

Lapse function Lapse function , orbital plane, orbital plane

Lapse and Gravitational WavesLapse and Gravitational Waves6/8h resolution, v=0.21909 merger example6/8h resolution, v=0.21909 merger example

Real component of the Newman-Penrose Real component of the Newman-Penrose scalar scalar 44( times rM), orbital plane( times rM), orbital plane

The threshold of “immediate” mergerThe threshold of “immediate” merger

• Tuning the parameter Tuning the parameter kk between merger between merger and deflection, one approaches a very and deflection, one approaches a very interesting dynamical regimeinteresting dynamical regime

• the black holes move into a state of the black holes move into a state of near-circular evolution before either near-circular evolution before either merging (merging (k<k*k<k*), or moving apart ), or moving apart again (again (k<k*k<k*)---at least temporarily)---at least temporarily

• for for kk near near k*k* , there is exponential , there is exponential sensitivity of the resultant evolution sensitivity of the resultant evolution to the initial conditions. In fact, the to the initial conditions. In fact, the number of orbits number of orbits nn spent in this spent in this phase scales approximately likephase scales approximately like

where for this particular set of initial where for this particular set of initial conditions conditions

• In all cases, to within numerical error In all cases, to within numerical error the spin parameter of the final black the spin parameter of the final black hole for the cases that merge is hole for the cases that merge is a~0.7a~0.7

*kken

The threshold of “immediate” mergerThe threshold of “immediate” merger• The binary is still radiating significant amounts of energy (on the order of 1-1.5% The binary is still radiating significant amounts of energy (on the order of 1-1.5% per per

orbitorbit), yet the black holes do not spiral in), yet the black holes do not spiral in

• Technical comment: the coordinates seem to be very well adapted to the physics of the Technical comment: the coordinates seem to be very well adapted to the physics of the situation, as the situation, as the coordinatecoordinate motion plugged verbatim into the quadrupole formula for two motion plugged verbatim into the quadrupole formula for two point masses gives a very good approximated to the actual numerical waveform point masses gives a very good approximated to the actual numerical waveform measured in the far-field regime of the simulationmeasured in the far-field regime of the simulation

• taking these coordinates at face-value, they imply the binaries are orbiting well taking these coordinates at face-value, they imply the binaries are orbiting well within the inner-most stable circular orbit of the equivalent Kerr spacetime of the within the inner-most stable circular orbit of the equivalent Kerr spacetime of the final black holefinal black hole

Kerr equatorial geodesic Kerr equatorial geodesic analogue analogue

work with work with D. Khurana, D. Khurana,

gr-qc/0702084 gr-qc/0702084

• We can play the same fine-tuning game with equatorial geodesics on a black hole backgroundWe can play the same fine-tuning game with equatorial geodesics on a black hole background

– here, we tune between capture or escape of the geodesichere, we tune between capture or escape of the geodesic

– regardlessregardless of the initial conditions, at threshold one tunes to one of the unstable circular orbits of the initial conditions, at threshold one tunes to one of the unstable circular orbits of the of the Kerr geometry (Kerr geometry (for equatorial geodesics)for equatorial geodesics)

• I.e. I.e. anyany smooth, one parameter family of geodesics that has the property that at one extreme of the smooth, one parameter family of geodesics that has the property that at one extreme of the parameter the geodesic falls into the black hole, while at the other extreme it escapes, exhibits this behaviorparameter the geodesic falls into the black hole, while at the other extreme it escapes, exhibits this behavior

un-bound orbit exampleun-bound orbit example bound orbit example; the threshold orbit in this bound orbit example; the threshold orbit in this case is sometimes referred to as a homoclinic case is sometimes referred to as a homoclinic

orbitorbit

Kerr geodesic analogue cont. Kerr geodesic analogue cont.

• Can quantify the unstable behavior by calculating the Lyapunov Can quantify the unstable behavior by calculating the Lyapunov exponentexponentof the orbit (or in this case calling it an “instability” of the orbit (or in this case calling it an “instability” exponent may be more accurate)exponent may be more accurate)

– easy way: measure easy way: measure n(k-k*)n(k-k*) , and find the slope , and find the slope of of nn versus versus –ln|k-k*|–ln|k-k*| … … ..

– easier(?) way: do a perturbation theory calculation (following easier(?) way: do a perturbation theory calculation (following N. Cornish and J. Levin, CQG 20, 1649 (2003)N. Cornish and J. Levin, CQG 20, 1649 (2003)) …) …

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Comparison of Comparison of • The dashed black lines are the The dashed black lines are the

previous formula evaluated for previous formula evaluated for various ranges of a & rvarious ranges of a & r

• The colored dots are from The colored dots are from calculations finding the capture-calculations finding the capture-threshold using numerical threshold using numerical integration of geodesicsintegration of geodesics

• Technical note: analytic Technical note: analytic expression derived using expression derived using Boyer-Lindquist Boyer-Lindquist coordinates, numerical coordinates, numerical integration done in Kerr-integration done in Kerr-Schild coordinates, though Schild coordinates, though neither expect nor see neither expect nor see significant differencessignificant differences

• The red-dashed ellipse is the The red-dashed ellipse is the single “dot” from the full single “dot” from the full numerical experiment numerical experiment performed. The size of the performed. The size of the ellipse is an indication of the ellipse is an indication of the numerical uncertainty, though numerical uncertainty, though the trend suggests that r is the trend suggests that r is slowly decreasing with the slowly decreasing with the approach to threshold, so the approach to threshold, so the ellipse may move a bit to the ellipse may move a bit to the left if one could tune closer to left if one could tune closer to thresholdthreshold

• interestingly, the final spin interestingly, the final spin parameter of the black hole parameter of the black hole that forms in the merger that forms in the merger case is ~0.7case is ~0.7

How far can this go in the non-linear How far can this go in the non-linear case?case?

• System is losing energy, and quite rapidly, so there must be a limit to the System is losing energy, and quite rapidly, so there must be a limit to the number of orbits we can getnumber of orbits we can get

• Hawking’s area theoremHawking’s area theorem: assume cosmic censorship and “reasonable” : assume cosmic censorship and “reasonable” forms of matter, then net area of all black holes in the universe can forms of matter, then net area of all black holes in the universe can notnot decrease with timedecrease with time

– the area of a single, isolated black hole is: the area of a single, isolated black hole is:

– initially, we have two non-rotating initially, we have two non-rotating (J=0)(J=0) black holes, each with mass black holes, each with mass M/2M/2::

– maximum energy that can be extracted from the system is if the final black hole maximum energy that can be extracted from the system is if the final black hole is also non-rotating:is also non-rotating:

in otherwords, the maximum energy that can be lost is a factor in otherwords, the maximum energy that can be lost is a factor 1-1/√2 ~ 29%1-1/√2 ~ 29%

– If the trend in the simulations continues, and the final If the trend in the simulations continues, and the final J~0.7MJ~0.7M22, we still get close , we still get close to to 24%24% energy that could be radiated energy that could be radiated

• the simulations show around 1-1.5% energy is lost per whirl, so we may get close to 15-the simulations show around 1-1.5% energy is lost per whirl, so we may get close to 15-30 orbits at the threshold of this fine-tuning process!30 orbits at the threshold of this fine-tuning process!

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28 MAi

Can we go even further?Can we go even further?• The preceding back-of-the-envelope calculation assumed the The preceding back-of-the-envelope calculation assumed the

energy in the system was dominated by the rest mass of the energy in the system was dominated by the rest mass of the black holesblack holes

• What about the black hole scattering problem?What about the black hole scattering problem?

– give the black holes sizeable boosts, such that the net energy of give the black holes sizeable boosts, such that the net energy of the system is dominated by the kinetic energy of the black holesthe system is dominated by the kinetic energy of the black holes

– set up initial conditions to have a one-parameter family of solutions set up initial conditions to have a one-parameter family of solutions that smoothly interpolate between coalescence and scatter that smoothly interpolate between coalescence and scatter

• ““natural” choice is the impact parameternatural” choice is the impact parameter

– it is it is plausibleplausible that at threshold, that at threshold, allall of the kinetic energy is of the kinetic energy is converted to gravitational radiation (think of what happens to a converted to gravitational radiation (think of what happens to a “failed” merger, and what the resultant orbit “failed” merger, and what the resultant orbit mustmust look like in the look like in the limit)limit)

• this can be an this can be an arbitrarily large fraction of the total energyarbitrarily large fraction of the total energy of the system of the system (scale the rest mass to zero as the boost goes to 1) (scale the rest mass to zero as the boost goes to 1)

An application to the LHC?An application to the LHC?• The Large Hadron Collider (LHC) is a particle accelerator currently The Large Hadron Collider (LHC) is a particle accelerator currently

under construction near Lake Geneva, Switzerlandunder construction near Lake Geneva, Switzerland

– it will be able to collide beams of protons with center of mass it will be able to collide beams of protons with center of mass energies up to energies up to 14 TeV14 TeV

• In recent years the idea of In recent years the idea of large extra dimensionslarge extra dimensions have become have become popular [popular [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, Phys.Lett.B429:263-272; L. Randall & N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, Phys.Lett.B429:263-272; L. Randall & R. Sundrum [Phys.Rev.Lett.83:3370-3373R. Sundrum [Phys.Rev.Lett.83:3370-3373]]

– we (ordinary particles) live on a 4-dimensional we (ordinary particles) live on a 4-dimensional branebrane of a of a higher dimensional spacetimehigher dimensional spacetime

• ““large” extra dimensions are sub-mm in size, but large large” extra dimensions are sub-mm in size, but large compared to the 4D compared to the 4D Planck lengthPlanck length of of 1010-33-33 cm cm

– gravity propagates in all dimensionsgravity propagates in all dimensions

• The 4D The 4D Planck Energy,Planck Energy, where we expect quantum where we expect quantum gravity effects to become important, is gravity effects to become important, is 10101919 GeV GeV; ; however the presence of extra dimensions can change however the presence of extra dimensions can change the “true” Planck energy the “true” Planck energy

• A Planck scale in the TeV range is preferred as this A Planck scale in the TeV range is preferred as this solves the hierarchy problemsolves the hierarchy problem

• current experiments rule out Planck energies <~ current experiments rule out Planck energies <~ 1TeV1TeV

• Collisions of particles with super-Planck energies in Collisions of particles with super-Planck energies in these scenarios would cause black holes to be these scenarios would cause black holes to be produced at the LHC!produced at the LHC!

• can “detect” black holes by observing energy loss can “detect” black holes by observing energy loss (from gravitational radiation or newly formed black (from gravitational radiation or newly formed black holes escaping the detector) and/or measuring the holes escaping the detector) and/or measuring the particles that should be produced as the black holes particles that should be produced as the black holes decay via Hawking radiationdecay via Hawking radiation

The black hole scattering problem The black hole scattering problem

• Consider the high speed collision of two black holes with impact parameter Consider the high speed collision of two black holes with impact parameter bb

– good approximation to the collision of two partons if energy is beyond the Planck regimegood approximation to the collision of two partons if energy is beyond the Planck regime

– for sufficiently high velocities charge and spin of the parton will be irrelevant (though both will for sufficiently high velocities charge and spin of the parton will be irrelevant (though both will probably be important at LHC energies) probably be important at LHC energies)

• threshold of immediate merger threshold of immediate merger mustmust exist exist

• if similar scaling behavior is seen as with geodesics and full simulations of the equal mass/low if similar scaling behavior is seen as with geodesics and full simulations of the equal mass/low velocity regime in general, can use the geodesic analogue to obtain an approximate idea of the cross velocity regime in general, can use the geodesic analogue to obtain an approximate idea of the cross section and energy loss to radiation vs. impact parameter … Ingredients:section and energy loss to radiation vs. impact parameter … Ingredients:

– map geodesic motion on a Kerr back ground with map geodesic motion on a Kerr back ground with (M,a)(M,a) to the scattering problem with total initial to the scattering problem with total initial energy energy E=ME=M and angular momentum and angular momentum aa of the black hole that’s formed near threshold of the black hole that’s formed near threshold

– find find andand b* b* using geodesic motionusing geodesic motion

– assume a constant fraction assume a constant fraction of the remaining energy of the system is radiated per orbit near of the remaining energy of the system is radiated per orbit near threshold (estimate using quadrupole formulathreshold (estimate using quadrupole formula))

– Integrate near-threshold scaling relation to find Integrate near-threshold scaling relation to find E(b)E(b) with the above parameters and the following with the above parameters and the following “boundary” conditions: “boundary” conditions: E(0), E(b*)E(0), E(b*) and and E(infinity)

• E(b*)E(b*) must be ~ 1 in kinetic energy dominated regime must be ~ 1 in kinetic energy dominated regime• E(infinity)=0E(infinity)=0• E(0)E(0) … need some other input, either perturbative calculations, or full numerical simulations. … need some other input, either perturbative calculations, or full numerical simulations.

The black hole scattering problem The black hole scattering problem

• What value of the Kerr spin parameter to use?What value of the Kerr spin parameter to use?

– in the ultra-relativistic limit the geodesic asymptotes to the light-ring at in the ultra-relativistic limit the geodesic asymptotes to the light-ring at thresholdthreshold

– it also seems “natural” that in this limit the final spin of the black hole at it also seems “natural” that in this limit the final spin of the black hole at threshold is threshold is a=1a=1. This is consistent with simple estimates of energy/angular . This is consistent with simple estimates of energy/angular momentum radiated momentum radiated

• quadrupole physics gives the following for the relative rates at which energy vs. quadrupole physics gives the following for the relative rates at which energy vs. angular momentum is radiated in a circular orbit with orbital frequency angular momentum is radiated in a circular orbit with orbital frequency ::

• for the scattering problem with the same impact parameter as a threshold geodesic on for the scattering problem with the same impact parameter as a threshold geodesic on an extremal Kerr background, the initial an extremal Kerr background, the initial J/EJ/E22=1=1. The Boyer-Lindquist value of . The Boyer-Lindquist value of EE is ½ for is ½ for a geodesic on the light ring of an extremal Kerr BH, in that regime a geodesic on the light ring of an extremal Kerr BH, in that regime d(J/ Ed(J/ E22)=0)=0

• But now we have a bit of a dilemma, as the extremal Kerr background has But now we have a bit of a dilemma, as the extremal Kerr background has no no unstableunstable circular geodesics, and hence circular geodesics, and hence tends to infinity in this limit tends to infinity in this limit

– will use will use aa close to but not exactly 1 to find out what close to but not exactly 1 to find out what E(b)E(b) might look like might look like

2

2

211)/(

E

J

Edn

dE

Edn

EJd

Sample energy radiated vs. impact parameter curves Sample energy radiated vs. impact parameter curves (normalized) (normalized)

• An estimate of E(0) from An estimate of E(0) from Cardoso et Cardoso et al., Class.Quant.Grav. 22 (2005) L61-R84al., Class.Quant.Grav. 22 (2005) L61-R84

• Cross section for black hole Cross section for black hole formation (b<~1) would thus be formation (b<~1) would thus be 22EE22

• In higher dimensions for equatorial In higher dimensions for equatorial geodesics of Myers-Perry black holes geodesics of Myers-Perry black holes becomes quite small regardless of becomes quite small regardless of the spin (the spin (C. MerrickC. Merrick))

– probably related to the fact that there probably related to the fact that there are no are no stablestable circular geodesics for circular geodesics for d>4d>4

– implies implies E(b)E(b) is well approximated by is well approximated by the function the function

• dE/dn ~ dE/dn ~ /40/40 in this limit, so expect all the in this limit, so expect all the energy to be radiated away in around a dozen energy to be radiated away in around a dozen orbits. orbits.

bbEbE *0~)(

ConclusionsConclusions

• the next few of decades are going to be a very exciting time for the next few of decades are going to be a very exciting time for gravitational physicsgravitational physics

– numerical simulations are finally beginning to reveal the fascinating numerical simulations are finally beginning to reveal the fascinating landscape of binary coalescence with Einstein’s theory of general relativity landscape of binary coalescence with Einstein’s theory of general relativity

• most of parameter space still left to exploremost of parameter space still left to explore

– the “extreme” regions, though perhaps not astrophysically relevant, will be the most the “extreme” regions, though perhaps not astrophysically relevant, will be the most challenging to simulate, and may reveal some of the more interesting aspects of the challenging to simulate, and may reveal some of the more interesting aspects of the theorytheory

– gravitational wave detectors should allow us to see the universe in gravitational wave detectors should allow us to see the universe in gravitational radiation for the first timegravitational radiation for the first time

• even if we only see what we expect to see we can learn a lot about the universe, even if we only see what we expect to see we can learn a lot about the universe, though history tells us that each time a new window into the universe has been though history tells us that each time a new window into the universe has been opened, surprising things have been discoveredopened, surprising things have been discovered

• if we don’t see anything, something is “broken” … unless it’s the detectors even if we don’t see anything, something is “broken” … unless it’s the detectors even that will be a remarkable discovery that will be a remarkable discovery