Black holes, quantum information, entanglement, and all that IHES, 29.05-01.06 2017 Gravitational Bremsstrahlung from Transplanckian-Energy Collisions: a progress report Gabriele Veneziano
Introduction Why is this topic of any relevance for this workshop? My excuse is that, by looking at these gedanken experiments at E >> M P in a regime where collapse is not expected, we arrive at an S-matrix which is: 1. Unitary (i.e. information preserving), 2. Shows the emergence of the Hawking- temperature scale T H ~ h/GE << M P
Outline � • PART I • An unsolved “textbook” exercise • A classical GR approach • A quantum S-matrix approach, comparison with CGR • GW energy spectrum and a logarithmic divergence • PART II (if time allows) • A claim by DGILS and its reinterpretation by ABV. • A final question
The textbook exercise The problem of computing the GWs emitted by a binary system is (almost) as old as GR. It has become gradually relevant for testing GR, for searches of GWs at bars and interferometers and, more recently, after LIGO’s observation of GWs emitted by the coalescence of BH-BH binaries: EOB (Damour, Buonanno), numerical rel. Most of the time this process is in the NR regime, with the exception of the merging itself when high speeds (v/c ~ 0.3-0.6) are reached.
Much less attention has been devoted in the past to a more academic problem: Consider the collision of two massless or highly relativistic ( γ = E/m >> 1) gravitationally interacting particles in the regime R << b in which they deflect each other’s trajectory by a small angle θ E (with γ -1 << θ E << 1) θ s ≡ θ E = 8 GE ≡ 2 R ; c = 1 b b P roblem: compute the GW spectrum associated with this collision to lowest order in θ E . How can it possibly be an unsolved problem? (A. Gruzinov, private conversation, early 2014)
What we do know 1. The zero frequency limit (ZFL). (see Smarr, prl 1977) We have a solid prediction for dE GW /d ω as ω -> 0. It can be obtained either by a classical or by a quantum argument. Latter uses the well- known soft graviton limit (Weinberg 1965, …) The result (after integrating over angles) dE GW → Gs E log( θ − 2 π θ 2 E ) ; ω → 0 d ω
An interesting problem: Can we get the 1st correction to the ZFL? (Addazi, Bianchi & GV, in progress)
2. Work in the seventies: P. D’Eath; D’Eath and Payne ~ 1978
� S. Kovacs and K.Thorne 1977
� 3. Numerical Relativity (F. Pretorius, U. Sperhake, private comm. ~ 04.14) The calculation in NR is challenging because the deflected particles carry with them two shock waves that travel (almost) as fast as the emitted GWs (and roughly in the same direction) Disentangling the two becomes very tricky for γ ’s >~ 3 and θ E a bit > γ -1 Maybe some hope for the future?
A 1st attempt (2007) by D. Amati, M. Ciafaloni & GV and an “energy crisis” Within ACVs approximations one finds: − | k || b | − ω R 3 R 2 � ⇥ dE gr ; Gs d 2 k d ω = Gs R 2 exp b 2 >> 1 b 2 � => the energy fraction in GWs is O(1) already for b = b* >> R (Gs/h (R/b*) 2 =O(1)). Smells bad (S. Rychkov priv. comm).
Need GR’s answer to: Q1: What’s the cutoff in ω for the GWs emitted in an ultra-relativistic small angle (b >> R) 2-body collision? Related Qs: � Q2 Is the massless limit singular? � Q3: Is the classical limit singular? � My (tentative) answers to Q2 & Q3: No-No!
What’s GR’s answer for θ E > 1/ γ ? (in particular for massless particles) � Recent progress. Classical: Gruzinov & GV (1409.4555), Spirin &Tomaras (1503.02016); � Quantum: Ciafaloni, Colferai & GV (1505.06619), C C Coraldeschi & GV, (1512.00281).
A simple Classical Treatment (A. Gruzinov & GV, 1409.4555) The calculation is done directly in the c.o.m. system for massless particles at small θ s . Obtained via Huygens principle in Fraunhofer approximation.
Schematic illustration of Huygens-Fraunhofer z − = − R log( b − x ) 2 2 R x · b θ b 2 ✓ ◆ log b − b · x x z − = − 2 R x ’ b 2 1 θ s z − = 0 z − = − 2 R log b x · θ 2 z + = 0 b θ s z → + ∞ z = 0
Frequency and angular distribution of GW spectrum: dE GW = GE 2 θ = θ − θ s ; θ s = 2 R b π 4 | c | 2 ; ˜ d ω d 2 ˜ b 2 θ Z d 2 x ζ 2 θ h i e − i ω x · ˜ c ( ω , ˜ e − 2 iR ω Φ ( x ) − 1 θ ) = | ζ | 4 2 ln ( x − b ) 2 Φ ( x ) = 1 + b · x ζ = x + iy b 2 b 2 Z d 2 x ζ 2 � e − iR ω ln ( x − b )2 − e +2 iER ω b · x e − i ω x · θ c ( ω , θ ) = b 2 b 2 | ζ | 4 Re ζ 2 and Im ζ 2 correspond to the two GW polarizations. � � Subtracting the deflected shock wave (cf. P. D’Eath) is crucial!
Postponing momentarily the properties of this GW spectrum let me jump to: A quantum treatment of same problem (Ciafaloni, Colferai & GV, 1505.06619), CC&Coraldeschi & GV, 1512.00281)
In CC(C)V (1505.06619 & 1512.00281) the same problem has been addressed at the quantum level improving on the earlier (ACV07) treatment. One observation is that the usual soft-graviton recipe (emission from external legs) has to be amended since the internal exchanged gravitons are almost on shell (“fractionation” of the exchanged transverse momentum). � Emission from such internal lines is important for not-so-soft gravitons (hence for the energy loss). Q: is it taken care of by NL correction? (ABV)
Another point is that, for gravitons with ω > R -1 , there are decoherence effects. � At fixed graviton helicity and momentum production amplitudes depend in a precise way upon the incidence angle, which changes along the fast-particle trajectory. � This decoherence causes a break from the flat spectrum at ω ~ R -1 .
If this effect is kept into account when summing over diagrams in which the graviton can be emitted by any rung in the ladder diagram, the result for Z Z c( ω , θ ) is: Z 1 Z d 2 z d ξ h s ( z )e i ω b z · ( θ − ξ Θ s ( b )) θ 0 ✓ E ◆ 1 ⌘ � Φ R ( z ) b � ω � � � � � ˆ � ˆ h s ( z ) ⌘ ω log � � log E z b � z � � � � π 2 z ⇤ 2 π 2 z ⇤ 2 � � � ⇣ ⌘ ˆ � ˆ Φ R ( z ) ! Φ ( z ) ⌘ b · z + log , b � z � � � S imilar -but not identical- to the classical result of G+V.
However, as argued by CCCV, one should also take into account the difference between the eikonal phase of the final 3-particle state and that of an elastic 2-particle state. � When this is done, the classical result of G+V is exactly recovered in the limit h ω /E << 1!
We have analyzed (mostly numerically) the properties of the spectrum in the classical limit. To be illustrated below in a few pictures. But first some more qualitative remarks. For b -1 < ω < R -1 the E-spectrum is almost flat in ω dE GW ∼ − 8 G π θ 2 s log( ω R ) � d ω Below ω = b -1 it freezes reproducing the ZFL dE GW → 4 G s E 2 log( θ − 2 θ 2 s ) d ω π
Above ω = R -1 the energy spectrum becomes scale-invariant � dE GW E ∼ θ 2 s d ω ω � This gives a log ω * in the “efficiency” for a cutoff at ω * Using ω * ~ R -1 θ s -2 (where our approximations break down and the “Dyson bound” dE/dt < 1/G is saturated) we find (to leading-log accuracy) the suggestive result: E GW = 1 2 π θ 2 s log( θ − 2 s ) √ s For ω > ω * G+V argued for an ω -2 spectrum (TBC): it turns out (extrapolating to θ s ~1) to be that of a time- integrated BH evaporation!
ω - θ distribution log-log plot (azimuth integrated) θ Z θ s / ω R dE GW d θ 1 s log 1 = θ 2 θ = θ 2 s Gs d ω ω R θ s 1 ω R = θ s θ Weinberg Lipatov θ s ( θ − θ s ) = θ s √ ω R W ZFL L ω *? θ s 1 θ s-2 ω R dE GW 1 = θ 2 s log( θ − 1 s ) Gs d ω Z θ s / √ ω R dE GW d 2 ( θ − θ s ) = θ 2 1 s = Gs d ω ω R
SOME NUMERICAL RESULTS ON THE SPECTRA
θ s = 10 -3 M. Ciafaloni, D. Colferai & GV, 1505.06619
Frequency spectrum 1/( ω R)
Angular (polar and azimuthal) distribution ω R = 10 -3 ω R = 0.125 M. Ciafaloni, D. Colferai, F. Coraldeschi & GV, 1512.00281
Angular (polar and azimuthal) distribution ω R = 8.0 ω R = 1.0
We now want to understand what, if any, provides a large-frequency cutoff and extend the reasoning towards the large-angle/collapse regime. First steps (involving some educated guesses) already made by Ciafaloni & Colferai (1612.06923). � The emerging picture is quite appealing: transverse � momenta are limited by 1/b while longitudinal ones (and energies) are controlled by the larger scale 1/R (with some leakage at higher frequencies) If that behavior persists as b -> b c ~ R, the GW/graviton distribution becomes more and more “isotropic” with <n> ~ Gs/h and (again!) characteristic energy O(h/R ~T H ).
END PART I (check time!)
A claim by Dvali, Gomez, Iserman, Luest, Stieberger (1409.7405) NB: D=4, no attempt to project on fixed impact parameter. � �
Expected phase diagram from collapse criteria b weak gravity DGILS14 l s strong string gravity gravity Critical line? Collapse ACV87+GV04 l P E = M P E= E th ~ M s /g s2 >> M P l P l s R(E)
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