# Radiation from a *D*-dimensional collision of shock waves: proof of first order formula and angular factorisation at all orders

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## Abstract

In two previous papers [1, 2] we have computed the inelasticity *ϵ* in a head-on collision of two *D*-dimensional Aichelburg-Sexl shock waves, using perturbation theory to calculate the geometry in the future light-cone of the collision. The first order result, obtained as an accurate numerical fit, yielded the remarkably simple formula *ϵ* _{1st order} = 1*/*2 − 1*/D*. Here we show, *analytically*, that this result is *exact* in first order perturbation theory. Moreover, we clarify the relation between perturbation theory and an angular series of the inelasticity’s angular power around the symmetry axis of the collision (*θ* = 0*, π*). To establish these results, firstly, we show that at null infinity the angular dependence factorises order by order in perturbation theory, as a result of a hidden symmetry. Secondly, we show that a consistent truncation of the angular series in powers of sin^{2} *θ* at some order \( \mathcal{O}(n) \) requires knowledge of the metric perturbations up to \( \mathcal{O}\left(n+1\right) \). In particular, this justifies the isotropy assumption used in first-order perturbation theory. We then compute, analytically, all terms that contribute to the inelasticity and depend linearly on the initial conditions (*surface terms*), including second order contributions.

## Keywords

Large Extra Dimensions Black Holes## Notes

**Open Access**

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## References

- [1]C. Herdeiro, M.O.P. Sampaio and C. Rebelo,
*Radiation from a D-dimensional collision of shock waves: first order perturbation theory*,*JHEP***07**(2011) 121 [arXiv:1105.2298] [INSPIRE].ADSCrossRefGoogle Scholar - [2]F.S. Coelho, C. Herdeiro and M.O.P. Sampaio,
*Radiation from a D-dimensional collision of shock waves: a remarkably simple fit formula*,*Phys. Rev. Lett.***108**(2012) 181102 [arXiv:1203.5355] [INSPIRE].ADSCrossRefGoogle Scholar - [3]G. ’t Hooft,
*Graviton dominance in ultrahigh-energy scattering*,*Phys. Lett.***B 198**(1987) 61 [INSPIRE]. - [4]N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali,
*The hierarchy problem and new dimensions at a millimeter*,*Phys. Lett.***B 429**(1998) 263 [hep-ph/9803315] [INSPIRE].ADSCrossRefGoogle Scholar - [5]I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali,
*New dimensions at a millimeter to a Fermi and superstrings at a TeV*,*Phys. Lett.***B 436**(1998) 257 [hep-ph/9804398] [INSPIRE].ADSCrossRefGoogle Scholar - [6]L. Randall and R. Sundrum,
*A large mass hierarchy from a small extra dimension*,*Phys. Rev. Lett.***83**(1999) 3370 [hep-ph/9905221] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar - [7]L. Randall and R. Sundrum,
*An alternative to compactification*,*Phys. Rev. Lett.***83**(1999) 4690 [hep-th/9906064] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar - [8]P.C. Argyres, S. Dimopoulos and J. March-Russell,
*Black holes and submillimeter dimensions*,*Phys. Lett.***B 441**(1998) 96 [hep-th/9808138] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar - [9]T. Banks and W. Fischler,
*A model for high-energy scattering in quantum gravity*, hep-th/9906038 [INSPIRE]. - [10]S.B. Giddings and S.D. Thomas,
*High-energy colliders as black hole factories: the end of short distance physics*,*Phys. Rev.***D 65**(2002) 056010 [hep-ph/0106219] [INSPIRE].ADSGoogle Scholar - [11]S. Dimopoulos and G.L. Landsberg,
*Black holes at the LHC*,*Phys. Rev. Lett.***87**(2001) 161602 [hep-ph/0106295] [INSPIRE].ADSCrossRefGoogle Scholar - [12]J.L. Feng and A.D. Shapere,
*Black hole production by cosmic rays*,*Phys. Rev. Lett.***88**(2002) 021303 [hep-ph/0109106] [INSPIRE].ADSCrossRefGoogle Scholar - [13]L. Anchordoqui and H. Goldberg,
*Experimental signature for black hole production in neutrino air showers*,*Phys. Rev.***D 65**(2002) 047502 [hep-ph/0109242] [INSPIRE].ADSMathSciNetGoogle Scholar - [14]R. Emparan, M. Masip and R. Rattazzi,
*Cosmic rays as probes of large extra dimensions and TeV gravity*,*Phys. Rev.***D 65**(2002) 064023 [hep-ph/0109287] [INSPIRE].ADSGoogle Scholar - [15]CMS collaboration,
*Search for microscopic black holes in pp collisions at*\( \sqrt{s}=7 \)*TeV*,*JHEP***04**(2012) 061 [arXiv:1202.6396] [INSPIRE].ADSGoogle Scholar - [16]ATLAS collaboration,
*Search for TeV-scale gravity signatures in final states with leptons and jets with the ATLAS detector at*\( \sqrt{s}=7 \)*TeV*,*Phys. Lett.***B 716**(2012) 122 [arXiv:1204.4646] [INSPIRE].ADSGoogle Scholar - [17]CMS collaboration,
*Search for microscopic black holes in pp collisions at*\( \sqrt{s}=8 \)*TeV*,*JHEP***07**(2013) 178 [arXiv:1303.5338] [INSPIRE].ADSGoogle Scholar - [18]ATLAS collaboration,
*Search for quantum black hole production in high-invariant-mass lepton + jet final states using pp collisions at*\( \sqrt{s}=8 \)*TeV and the ATLAS detector*,*Phys. Rev. Lett.***112**(2014) 091804 [arXiv:1311.2006] [INSPIRE].ADSCrossRefGoogle Scholar - [19]V. Cardoso et al.,
*NR/HEP: roadmap for the future*,*Class. Quant. Grav.***29**(2012) 244001 [arXiv:1201.5118] [INSPIRE].ADSCrossRefGoogle Scholar - [20]V. Cardoso, L. Gualtieri, C. Herdeiro and U. Sperhake,
*Exploring new physics frontiers through numerical relativity*, arXiv:1409.0014 [INSPIRE]. - [21]P.C. Aichelburg and R.U. Sexl,
*On the gravitational field of a massless particle*,*Gen. Rel. Grav.***2**(1971) 303 [INSPIRE].ADSCrossRefGoogle Scholar - [22]H. Yoshino and Y. Nambu,
*High-energy headon collisions of particles and hoop conjecture*,*Phys. Rev.***D 66**(2002) 065004 [gr-qc/0204060] [INSPIRE].ADSMathSciNetGoogle Scholar - [23]H. Yoshino and Y. Nambu,
*Black hole formation in the grazing collision of high-energy particles*,*Phys. Rev.***D 67**(2003) 024009 [gr-qc/0209003] [INSPIRE].ADSGoogle Scholar - [24]H. Yoshino and V.S. Rychkov,
*Improved analysis of black hole formation in high-energy particle collisions*,*Phys. Rev.***D 71**(2005) 104028 [*Erratum ibid.***D 77**(2008) 089905] [hep-th/0503171] [INSPIRE]. - [25]L. Álvarez-Gaumé, C. Gomez, A. Sabio Vera, A. Tavanfar and M.A. Vazquez-Mozo,
*Critical formation of trapped surfaces in the collision of gravitational shock waves*,*JHEP***02**(2009) 009 [arXiv:0811.3969] [INSPIRE].CrossRefGoogle Scholar - [26]J.L. Albacete, Y.V. Kovchegov and A. Taliotis,
*Asymmetric collision of two shock waves in AdS*_{5},*JHEP***05**(2009) 060 [arXiv:0902.3046] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar - [27]K.A. Khan and R. Penrose,
*Scattering of two impulsive gravitational plane waves*,*Nature***229**(1971) 185 [INSPIRE].ADSCrossRefGoogle Scholar - [28]Y. Nutku and M. Halil,
*Colliding impulsive gravitational waves*,*Phys. Rev. Lett.***39**(1977) 1379 [INSPIRE].ADSCrossRefGoogle Scholar - [29]P.D. D’Eath and P.N. Payne,
*Gravitational radiation in high speed black hole collisions. I. Perturbation treatment of the axisymmetric speed of light collision*,*Phys. Rev.***D 46**(1992) 658 [INSPIRE].ADSMathSciNetGoogle Scholar - [30]P.D. D’Eath and P.N. Payne,
*Gravitational radiation in high speed black hole collisions. II. Reduction to two independent variables and calculation of the second order news function*,*Phys. Rev.***D 46**(1992) 675 [INSPIRE].MathSciNetGoogle Scholar - [31]P.D. D’Eath and P.N. Payne,
*Gravitational radiation in high speed black hole collisions. III. Results and conclusions*,*Phys. Rev.***D 46**(1992) 694 [INSPIRE].ADSMathSciNetGoogle Scholar - [32]P.D. D’Eath,
*High speed black hole encounters and gravitational radiation*,*Phys. Rev.***D 18**(1978) 990 [INSPIRE].ADSMathSciNetGoogle Scholar - [33]U. Sperhake, V. Cardoso, F. Pretorius, E. Berti and J.A. Gonzalez,
*The high-energy collision of two black holes*,*Phys. Rev. Lett.***101**(2008) 161101 [arXiv:0806.1738] [INSPIRE].ADSCrossRefGoogle Scholar - [34]W.E. East and F. Pretorius,
*Ultrarelativistic black hole formation*,*Phys. Rev. Lett.***110**(2013) 101101 [arXiv:1210.0443] [INSPIRE].ADSCrossRefGoogle Scholar - [35]L. Rezzolla and K. Takami,
*Black-hole production from ultrarelativistic collisions*,*Class. Quant. Grav.***30**(2013) 012001 [arXiv:1209.6138] [INSPIRE].ADSCrossRefGoogle Scholar - [36]F.S. Coelho, C. Herdeiro, C. Rebelo and M. Sampaio,
*Radiation from a D-dimensional collision of shock waves: higher-order setup and perturbation theory validity*,*Phys. Rev.***D 87**(2013) 084034 [arXiv:1206.5839] [INSPIRE].ADSGoogle Scholar - [37]L. Smarr,
*Gravitational radiation from distant encounters and from head-on collisions of black holes: the zero frequency limit*,*Phys. Rev.***D 15**(1977) 2069 [INSPIRE].ADSGoogle Scholar - [38]M.O.P. Sampaio,
*Radiation from a D-dimensional collision of shock waves: numerical methods*,*Int. J. Mod. Phys.***A 28**(2013) 1340019 [arXiv:1306.0903] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar - [39]F.G. Friedlander,
*The wave equation on a curved space-time*, Cambridge University Press, Cambridge U.K. (1975).MATHGoogle Scholar - [40]K. Tanabe, S. Kinoshita and T. Shiromizu,
*Asymptotic flatness at null infinity in arbitrary dimensions*,*Phys. Rev.***D 84**(2011) 044055 [arXiv:1104.0303] [INSPIRE].ADSGoogle Scholar - [41]K. Tanabe, T. Shiromizu and S. Kinoshita,
*Angular momentum at null infinity in higher dimensions*,*Phys. Rev.***D 85**(2012) 124058 [arXiv:1203.0452] [INSPIRE].ADSGoogle Scholar - [42]F.S. Coelho, C. Herdeiro, C. Rebelo and M.O.P. Sampaio,
*Radiation from a D-dimensional collision of shock waves: an insight allowed by the D parameter*, arXiv:1301.1073 [INSPIRE]. - [43]F.S. Coelho, C. Herdeiro, C. Rebelo and M.O.P. Sampaio,
*Radiation from a D-dimensional collision of shock waves: a summary of the first order results*,*Springer Proc. Math. Stat.***60**(2014) 189 [INSPIRE].CrossRefMathSciNetGoogle Scholar - [44]F.S. Coelho, C. Herdeiro, C. Rebelo and M.O.P. Sampaio,
*Radiation from a D-dimensional collision of shock waves: numerics and a charged case*,*Springer Proc. Math. Stat.***60**(2014) 193 [INSPIRE].CrossRefMathSciNetGoogle Scholar