High Speed Black Hole and Soliton Collisions

1. 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

2. 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 zoom-whirl orbits?

3. Black hole formation at the LHC and in the atmosphere? • large extra dimension scenarios [N. Arkani-Hamed , S. Dimopoulos & G.R. Dvali, 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 1019GeV calculated from the fundamental constants measured on our 4-D Brane • 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 • 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 & Fishlerhep-th/9906038, Dimopoulos & Landsberg PRL 87 161602 (2001), Feng & Shapere, PRL 88 021303 (2002), …] One of the water tanks at the Pierre Auger Observatory • 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 • if a lot of gravitation radiation is produced during the collision this could show up as a missing energy signal at the LHC ATLAS experiment at the LHC

4. 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 purelyclassical, 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

5. 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”

6. 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), … ]

7. 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-gsoliton(i.e. it’s a good approximation when you’re far enough away that the solitonspacetime 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

8. Use numerical simulations to try to test this • Solve the Einstein equations coupled to a matter source that admits solitonsolutions (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 and for a range of central amplitudes A,stable eigensolutions exist with eigenvaluesd 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

9. 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

10. Results for head-on collision of boosted boson stars • we find black hole formation at g=3already! • signs of a Planck scale of O(10 Tev) or smaller should already have been seen?

11. Case 1: free-fall collision from rest • 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) 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

12. 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

13. 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

14. Case 4 : g ~ 3 • beginning to see richer interaction dynamics, as expected for threshold solutions

15. zoom-in of collision region • Though note, different vertical and color-scale from previous animation

16. 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

17. Aside : the Planck Luminosity regime • The Planck Luminosity Lp is • 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?

18. 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(1-e)M/c3. This gives • 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)

19. g=2.9 collision example T=0 M0 (M=gM0 ) Re[Y4]

20. g=2.9 collision example T=30 M0 (M=gM0 ) Re[Y4]

21. g=2.9 collision example T=83 M0 (M=gM0 ) Re[Y4]

22. g=2.9 collision example T=115M0 (M=gM0 ) Re[Y4]

23. g=2.9 collision example T=137 M0 (M=gM0 ) Re[Y4]

24. g=2.9 collision example T=156 M0 (M=gM0 ) Re[Y4]

25. g=2.9 collision example T=202 M0 (M=gM0 ) Re[Y4]

26. g=2.9 collision example T=252 M0 (M=gM0 ) Re[Y4]

27. g=2.9 collision example T=292 M0 (M=gM0 ) Re[Y4]

28. g=2.9 collision example T=405 M0 (M=gM0 ) Re[Y4]

29. 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

30. Non-head on collisions: the black hole scattering problem • Consider the off-center collision of two black holes with impact parameters b • in general two, distinctend-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* m2,v2 b m1,v1

31. 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

32. 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]

33. 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 gof the orbit via • in relation to the scattering problem, dr is proportional to b-b*

34. 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

35. 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

36. 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

37. 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: • 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

38. Sample energy radiated vs. impact parameter curves (normalized) • Quadrupole formula says dE/dn ~ p/40in 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

39. 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 • 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?