JHEP12(2018)133 Springer December 12, 2018 December 21, 2018 November 26, 2018 : : : Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP12(2018)133 . 3 1811.08827 The Authors. c Classical Theories of Gravity, Models of Quantum Gravity

The geometric description of gravitational memory for strong gravitational , [email protected] Swansea University, Swansea, SA2 8PP, U.K. E-mail: Department of Physics, College of Science, Open Access Article funded by SCOAP Keywords: ArXiv ePrint: wave limit appropriate to nulltimelike congruences. and A null detailed geodesic analysis congruencesof of in impulsive memory Aichelburg-Sexl and is extended type, presented gravitational for and shockwaves astronomy and for to gyratons. quantumscattering, Potential gravity, are especially briefly applications mentioned. infra-red to structure gravitational and wave ultra-high energy waves is developed,gyratons. with particular Memory, which focus mayresidual on be separation shockwaves of of and position neighbouringtational their or ‘detector’ wave spinning velocity-encoded geodesics burst, type, analogues, following and characterisesis the retains the shown passage information how of on memory aspacetime the is and gravi- nature a encoded new of ‘timelike in Penrose the the limit’ wave is Penrose introduced source. to limit complement of the Here, original the it plane original gravitational wave Abstract: Graham M. Shore Memory, Penrose limits and thegravitational geometry shockwaves of and gyratons JHEP12(2018)133 25 16 7 4 23 39 21 – 1 – 24 21 30 19 23 33 9 11 20 15 36 13 27 5 26 6 42 2 34 26 3.3.1 Null congruences3.3.2 and plane waves Timelike congruences 3.3.3 Congruences and memory for the gravitational shockwave 3.2.1 Null congruences 3.2.2 Timelike congruences 3.2.3 Aichelburg-Sexl shockwave 4.1 Gyraton metric 4.2 Geodesics and4.3 orbits Optical tensors4.4 and memory Penrose limit and memory 3.2 Optical tensors and memory 3.3 Penrose limits and memory 2.5 Weak gravitational2.6 waves Gravitational memory 3.1 Geodesics for gravitational bursts and shockwaves 2.1 Geodesic deviation 2.2 Optical tensors 2.3 Penrose limits2.4 for null and Gravitational timelike plane congruences waves C Gyraton metrics 5 Discussion A Planck energy scattering B Symmetries of gravitational shockwaves 4 Gyratons 3 Gravitational shockwaves Contents 1 Introduction 2 Memory, optical tensors and Penrose limits JHEP12(2018)133 ] , i z 2 , 1 (1.1) , and ∆ ˙ i z , j z ]. The important, and very general, ) u 15 ( – j i , defining the null and timelike Penrose 11 ˆ Ω 2 du – 2 – ], so the description of memory for null observers u . Memory resides in the value of ∆ −∞ t Z 10 , 9 = i z ∆ in a given background spacetime is encoded in its Penrose γ defines the expansion, shear and twist optical tensors which ] memory respectively. From a geometric point of view, such ij 4 , ω 3 + ˆ replaced by time ij u σ ]) . + ˆ 26 ˆ θ – 1 2 16 of neighbouring geodesics, which for a null congruence is given in suitable Fermi , = i 8 ]. This limit is a plane wave [ z ij 8 – ˆ Ω 5 We set out this general theory in section Central to this analysis is the observation that the geometry of geodesic deviation In this paper, we develop a geometric formalism for the description of gravitational Gravitational memory refers to the residual separation of ‘detectors’ following the limits and relating ourconsidered approach so to far the conventional inrˆoleof analysis astrophysical gravitational of applications plane weak [ waves gravitational inrelated waves encoding work memory on geodesics is and discussedfound gravitational in in plane some waves, [ including detail. memory Closely effects, may be timelike observers, we defineMoreover, here we a show new that ‘timelike ifthe Penrose the transverse limit’ original geodesic with spacetime equations is theand defining itself same null memory in congruences. property. are the in general fact class the of same pp waves, for both timelike develop the theory ofcongruences geodesic in deviation strong and gravitational memory waves. for both null andaround timelike a geodesic chosen null geodesic limit [ in a general spacetime can be reduced to that in an equivalent gravitational plane wave. For where characterise the congruence. Alightlike similar coordinate expression holds forin timelike the congruences with future the regiondetermined by following integration the of interaction the optical with tensors the through gravitational the wave interaction burst, region. and Here, we is vector ∆ normal coordinates as passage of a gravitational wave burst.or This a may constant take the separation formand velocity, of ‘velocity-encoded’ a or fixed [ both. change inidealised position, We detectors refer are represented to as these neighbouringdescription geodesics of as in memory ‘position-encoded’ a is timelike [ congruence. thereforedeviation. part The To of be the precise, more the general separation geometric of analysis nearby of detectors geodesic is identified as the connecting memory which goes beyond thegravitational conventional weak-field analysis waves, and especially is applicabletions, gravitational to gyratons. strong shockwaves and their spinning generalisa- Gravitational memory is becomingphysics, an not only increasingly because important of topic itsical potential sources, in observation but in gravitational gravitational also wave waves for fromnotably its astronom- soft-graviton importance theorems, in quantum theoretical loop issues effects in and quantum Planck gravity, energy including scattering. 1 Introduction JHEP12(2018)133 ]. ). = 2 51 – (1.3) (1.2) /r uu ) and ] gives r R (1 49 ( O 27 f ] also give + 44 /r . 2 ]. + 1 dφ r 35 2 – r log + , 32 , 2 2 ∼ 18 ) dr dφ r 2 ]. While this simply requires ( ] and dilatonic [ r + f 31 43 + – ), this is the original Aichelburg- – 2 u 29 ), while other black holes such as ( 41 du dφ dr u δ ( ) δ + u ( 2 ) = , and metrics of this type are important J ) in the Aichelburg-Sexl metric while retaining → 2 u r ) typical of a sandwich wave and its im- du ) ( ( ) u 0 Jχ ) f F u ( 2 u ( χ F In this case, however, it is necessary to choose ( F r/r − χ F 1 χ 2 – 3 – χ ) r log ( du ( ) and f ], Kerr-Newman [ u r ( GE + 40 F 4 – χ log − ) 37 r ( ∼ du dv f ) = ) r r ( + ( ], = 2 f ). (Here, ∆ denotes the two-dimensional Laplacian.) With the f 2 u ) is fixed by the Einstein equations through the relation . They are the simplest models in which to study the gravitational ] in the scattering of such shockwaves becomes a model for black ], Kerr [ 28 ( r , J ( ds F 48 36 du dv f ) chosen to be impulsive, ). χ 27 – ) u u ( r ( 45 δ ( = 2 F ). For example, an infinitely-boosted Schwarzschild black hole [ f 2 χ u ∆ ( ds 2 1 δ − = 0. , to gravitational shockwaves. These are described by generalised Aichelburg-Sexl = u 3 uu A natural extension of the Aichelburg-Sexl shockwave metric is to ‘gyratons’ [ Several generalisations are also of interest, giving rise to different potentials For a particle source, Motivated primarily by issues in quantum gravity, our focus in this paper then turns, in of test geodesics as they interact with the shockwave [ Note that this is not achieved by, for example, infinitely boosting a black hole with spin (the Kerr 1 v πGT momentum, related to effect of spin in ultra-high energy scattering. metric), since as mentionedthe above impulsive this profile simply modifies These are a special class of gravitational pp waves with metric, These describe the spacetime generated by a pulse of null matter carrying an angular It would be interestingan if important such rˆolein shockwaves astrophysics.of from gravity, Also extremely the note fast-moving Planck that scale blacktrapped in can holes surfaces certain be have [ higher-dimensional loweredhole to theories production TeV at scales, the in LHC which or case FCC. the formation of distinguishing between an extendedpulsive profile limit a shockwave metric with Reissner-Nordstr¨om[ impulsive shockwaves with modified potentials of the form the solution for a singlescattering null geodesic in in an the interacting Aichelburg-Sexl quantumthe background, field ultra-high geometry theory energy including of loopPenrose the contributions limits, full depends and on congruence. theirunitarity, have importance These been in QFT studied resolving extensively effects, in fundamental the the issues series geometry with of of causality papers and [ the relevant in analyses of Planck energy scattering.by At gravitational such interactions ultra-high and energies, the scattering leading isgenerated eikonal dominated by behoviour ladder of diagrams the representing scatteringidentifying amplitude, multi-graviton the exchange, corresponding can phase be shift reproduced∆ with by the discontinuous lightcone coordinate jump 8 profile function Sexl metric describing thesurface spacetime around an infinitely-boosted source localised on the section metrics of the form [ where the potential JHEP12(2018)133 ruφu R for a null v ) considered here. 1.3 , we study these orbiting 4 ) would give an unphysically u ( δ ∼ in the optical tensors characterising ) u ij ( ω J χ – 4 – , and calculate the leading corrections arising from an s ), so an impulsive profile u ). In appendix B, we describe the symmetries associated with the ( ) to be extended. This is because the curvature component u 0 J u ( χ ( F J χ χ ) for Planck energy scattering to the lightlike coordinate shift ∆ ) and motivating the particular choice of metric ( u ( s, t J ( χ A ) involves 1.3 ) and would be determined by a non-vanishing twist ˆ A key observation is that the geometry of geodesic deviation around a given null In this section, we describe in quite general terms the geometry of geodesic congruences We include three appendices. In appendix A, we review the relation of the scattering u i ( z F geodesic in a curved spacetimewave background limit. is This encoded implies ina the the general corresponding remarkable background Penrose simplification spacetime plane that mayappropriate the be plane wave properties entirely background. described of Welimit by memory also construction studying for describe for congruences here the in a case an generalisation of of timelike the geodesics. Penrose for more foundational questionsalso involving discuss shockwaves the and difference in Planckgeodesics the energy origin acquire scattering. of a position-encoded fixed We memory,encoded separation in memory, after which in neighbouring the which gravitational they wave separate has or passed, focus and with velocity- fixed velocity. for the class ofpp gravitational waves waves — of the Aichelburg-Sexl interest. shockwaveWe and We consider its focus both non-impulsive particularly extension, timelike and onfor geodesics, gyratons. two astrophysical relevant examples for gravitational of the waves, interpretation and in null terms geodesics, of which detectors will also be appropriate Gravitational memory concerns the separation ofsage neighbouring of geodesics a following gravitational the wave burst,limit, pas- either a an shockwave. extended The (sandwich)geometry wave appropriate or, mathematical of in description the geodesic of impulsive congruences,optical memory tensors is in in therefore particular the the Raychoudhuri geodesic equations. deviation characterised by the χ 2 Memory, optical tensors and Penrose limits integer values of theextended CM profile energy shockwave and corresponding planehanced wave symmetries metrics, for impulsive in profiles. particulargyraton Finally, metrics considering in showing appendix potential especially C, how en- we the consider curvature constrains more the general the form of the profiles question is whether the∆ gyraton spin givesthe rise congruence. to a ‘twist memeory’ in whichamplitude the final geodesic in an Aichelburg-Sexl spacetime, illustrating the origin of the poles at complex from ( singular curvature. This alsogeodesic (detector) allows imparting the an spin angulargeodesics momentum. in and the In the associated section metric null and timetensors, timelike congruences to the in relevant act detail, Penrose determining on the limits, optical the and scattering the eventual gravitational memory. A particular the spin profile JHEP12(2018)133 NP , we (2.4) (2.5) (2.1) (2.2) (2.3) γ , this is . This will /dλ      µ γ ) is ensured by dx = 2.3 ) through µ with tangent vector µ ¯ . For lightlike m ), ( γ k , 2 (2.6) , 0 11 0 0 0 0 0 0 0 0 1 0 0 0 1 γ 2. In our previous work on ) , µ , µ      2.2 1 2 2 , m ie is = = 1 µ  , u, v, γ , n µ µ σ , AB 1 ` = k = 0. The FNC basis is just this e , ( ρ A 2 , 2 ν k vanishes, i.e. = 0 1 m µ z = 0, we find √ z µ σ γ = ¯ ρνσ µ z k σ = , µ ) 2 ρ ˙ x µ µ as ν k R m ρ k z ˙ γ x − ν ν = k.Dk = 0 for all along , i, j ρνσ µ 2 D µ = ν µ , m ( n Aµ ρνσ j z R ν e µ vµ − = λ = Ω . The consistency of ( e − µ R – 5 – 2 i µ λ Ω µ − satisfies the geodesic equation. This is in essence k.De ` e which is parallel-propagated along λ − = which will be fundamental to our analysis. Differ- µ µ = ij µ A k = µ δ µ z e = 1, k.Dz k k.Dz 2 µ 2 ν + ) + Ω ¯ z m = ν 2 ν D , n u µ µ Dλ m. D e k.D z = Ω , η uµ ( k µ 1, ) and the tangent vector is given by e ν v λ L − µν e ( B = , i.e. we impose k.D e µ = γ µ + x µ k ν A `.n v = e e µ ` µ AB u e η ], we used this NP notation extensively. = = 22 γ ,

), and using the geodesic equation 8 µν be the ‘connecting vector’ specifying the orthogonal separation to a neighbouring g 2.2 is the covariant derivative. It follows that µ µ z D The next step is to establish a frame adapted to the chosen congruence. That is, we At a point, this is identified with a Newman-Penrose (null) basis ( 2 . Let µ with the usual contractions basis parallel-propagated along Penrose limits [ choose a pseudo-orthonormal frame define Fermi normal coordinateschoose (FNCs) a in frame such the that neighbourhood the of metric in the neighbourhood of the identity, which holds in general giventhe only Raychoudhuri that equation. written in terms of the intrinsic derivative along where the dot denotes a derivative w.r.t. entiating ( which is the Jacobi equationis for affine geodesic parametrised deviation. as In more familiar form, if the geodesic where where we define the tensor Ω Consider a congruence centred on ak chosen (null or timelike) geodesic geodesic. By definition, the Lie derivative of 2.1 Geodesic deviation JHEP12(2018)133 (2.9) (2.8) (2.13) (2.10) (2.11) (2.12) form a γ 1) , 1 , 1 , 1 , − s 3 (2.7) , , ˙ ν t z 2 j ˙ onto the appropriate trans- t , is antisymmetric, while the e 0 µ ). s i µν r ij 0 e = diag ( = 1 ν r ω j ˆ Ω is effectively two-dimensional. . ij e R t, x . δ AB − µν ij , and where the basis vectors satisfy Ω iuju = . ω = γ µ R i r ν + ˆ e ¨ z = 0, so µν u − ) ij e ) become simply g µ σ = ij vν v ) e Ω ˆ 2.4 2 e + ˆ ν ). g µν i ˆ j Ω ij − ( e ˆ – 6 – Ω , r, s ν − = (ˆ ˆ µν θ δ ν v , s e ˆ 1 2 Ω j = e µ u, v, x µν ) is then seen to be equivalent to the Raychoudhuri µ i u µ ˆ = ij Ω ˙ e e r u z ˆ 2.5 e Ω ˙ ij u − ≡ = 0 and ˆ , η rs d Ω du δ ij ν µν uju uν i ˆ g B Ω e + e R ν ≡ µν = 0, defining timelike FNCs ( µ ]). For a null congruence, we have the projection matrix 0 − ˆ Ω A e µ 52 e µν µ = A ˆ g 0 i e , the Jacobi equations ( ¨ z AB γ η − is symmetric and traceless, the twist ˆ k.De , we choose = = ij ˆ Ω. The relation ( γ ] — initially a circle, this distorts as the gravitational wave burst passes σ γ

chosen to be tangent to the geodesic 21 and µν µ µ = tr g k k ˆ θ = = = 0. This defines null FNCs ( µ µ uµ 0 e e A The optical tensors are defined from the projections of Ω In terms of these coordinates, where by the definition of FNCs the Christoffel symbols For timelike Here, the shear ˆ expansion equation for the optical tensors, since for null FNCs it is simply, We define the optical tensors from the decomposition of as and define It is readily checked that see, the special symmetry characterising pptwo-dimensional waves and means the that this optical space tensors remains are effectively identical to theverse null subspace case. (see e.g. [ ence, the transverse spacea spanned cross-section by through the the“Tissot connecting congruence, ring” those vector geodesics [ is at two-dimensional.displaying fixed clearly Taking separation the from effects oftransverse expansion, space shear and and optical twist. tensors are For a in general timelike congruence, three-dimensional although, the as we shall for null, timelike congruences respectively. 2.2 Optical tensors Geodesic deviation, and therefore gravitational memory,tensors is — described expansion, in shear terms of and the twist optical — characterising the congruence. For a null congru- with vanish locally along k.De with JHEP12(2018)133 , ) 3 . (2.18) (2.19) (2.14) (2.20) (2.16) (2.17) (2.15) x . ω ( 1 O +  δ oriented axes), dx × ] γ 52 dx , and the twist ˆ , β 7 × x σ α x γ

]. , 7 , , ν αγβδ ν s s . e . An elegant construction of the ! R e µ γ 1 3 + ω rs r µν ˆ σ (shear with + and e ω oriented shear ˆ + , Ω + ˆ − × rs µ γ + ˆ × × δ r σ ˆ µν θ , e ˆ ) σ 1 2 rs + g σ = σ = , and choosing null FNCs according to the dudx Ω ˆ and ˆ γ ν + ˆ β + ν g ˆ ω s x + ˆ k σ e rs – 7 – α σ µ − ˆ = (ˆ + ˆ k x µν ˆ θ δ γ × ˆ ˆ θ Ω

1 3 + ˆ σ 1 2 µν µ r ˆ Ω

= µν e g αγβu = rs ≡ R ≡ ˆ Ω ij 4 3 rs ˆ Ω ˆ µν , + oriented shear ˆ Ω (expansion), ˆ + ˆ ˆ g θ 2 2 are non-vanishing, so the optical tensors remain effectively ˆ θ 1. In this case, , − du j β = = 1 x dx 2 i α ˆ k x γ dx r, s

ij δ as + αuβu with ij R ˆ Ω rs  ˆ . Illustration of the effect of the optical tensors on the Tissot circle. From left to right, Ω dudv − (twist). Their action on the Tissot ring is indicated schematically in figure =2 ω In the neighbourhood of a null geodesic For a timelike congruence, the analogous projection matrix is Since the transverse space is two-dimensional, we can further simplify this description 2 ds For null congruences, theof geometry the of background geodesic geometry deviationPenrose with is limit the encoded in terms chosen in of geodesic the Fermi Penrose null limit coordinates isconstruction given described in above, [ we can expand the metric as follows [ The timelike Raychoudhuri equations followwith straightforwardly. the Again, defining however, symmetry noteponents that of the pptwo-dimensional waves and considered can here, also we be will visualised find with2.3 a only Tissot the ring. com- Penrose limits for null and timelike congruences defines three-dimensional optical tensors through as where we normalise defining the optical scalars and ˆ Figure 1 the figures show the expansion by writing JHEP12(2018)133 , v 2 − , pre- κ (2.23) (2.24) (2.26) (2.25) (2.21) (2.22) ). The and are . → iuju ) γ 2.8 3 R v x in the plane , ( i u O x , i.e. neglecting ]) entirely in the 2 + → in the plane wave − 35 i  κ u – q x in the original metric, dx 32 ]: , p . . γ , 2 52 18 j s dx s du x dx dx j i as [ r r x x i . γ dx γ dx x

ij γ rs ) around δ

δ which scale as = 0 rpsq ) is that it describes a gravitational + 2 2.8 j + ): iuju R 2 x 2 ds 1 3 R 2 2.21 du dt  j s + − 2.20 x x ) keeping only the curvature terms which p j . i r du dλ γ x x

of ( ) dx  in FNCs with the Brinkmann γ i u

2.25 ) – 8 – i dtdx ( 0 u s iuju s z dx ij ( 0 x j ij R r h r i δ R x − h + γ +

− − 0 2 i ) = truncation 2 x u rps dt dudv ( 2 dudv dλ R ) identified in terms of the curvature tensor of the original d − ij 2 3 u h ( = 2 = = 2 ij + 2 P as 2 P h 2 P 2 s only. γ ds ds dt ds s u dx x r r x dx ]. Keeping only those terms in γ ) here. Note that the curvatures are evaluated on the geodesic

rs i 6 0 δ , s 5 0 + r v, x ( ) is well known: ,[ 2 expressed in Brinkmann coordinates, i.e. R i ), leaves the following 4 x  dt ≡ 1 − 2.21 α − − − x , κ κ 3 = Given this description of geodesic deviation for null congruences, it is now natural to This confirms the claim that the Penrose limit captures precisely the geometry of The geodesic equation for the transverse Brinkmann coordinates The conventional (null) Penrose limit follows from the rescaling − 2 → κ ( ds i Without invoking a scalingsimply argument as make in an the analogousenter original the truncation Penrose Jacobi of limit equation. ( derivation, This we leaves may metric in the neighbourhood of a chosen timelike geodesic simpler and well-studied casewe of demonstrate plane this waves in has the proved context to of be gravitational extremely memory. repeat powerful. the Here, construction forPenrose timelike limit”. congruences, Using defining the what timelike we FNCs defined may above, call we the expand the “timelike original background This is identical towhere the we geodesic identify deviation the equationwave. connecting ( vector geodesic deviation. The ability(such to as analyse quantum physical loop effects corrections controlled in by QFT geodesic in deviation curved spacetime [ metric ( with the profile function spacetime evaluated on We immediately see that thiscisely truncation leaves those only that the curvaturesecond determine components key geodesic property deviation ofplane the through wave Penrose the limit Jacobi metric equation ( ( therefore functions of x O where JHEP12(2018)133 ) 2 , . 2.27 ij (2.28) (2.29) (2.27) = 1 ). ) we only i, j = 0 and can ) is 2.27 . The same 2.29 u ]), and their B 19 2.27 ) with t and given by Ω ( ij γ ) we have ¨ 3 h ). However, this is sufficient 2.22 , = 0. ) for the connecting vector s 2.8 0 i , dx 2.8 30 in ( ) although, to our knowledge, r 2 R , ) ) does not in principle give any ) are dx r = 0 = x 2.27 rs ˙ ( u = 0 δ i O 2.22 2.22 s ˙ x + 3030 ] or Carroll symmetry [ x j 2 2 R x derived from the metric ( )), for the geodesics in the metric ( dt 22  ij = 1 + , r
˙ h t 7 s 2.8 x dt dλ x . r  + 2 x ) ) 2 – 9 – as an affine parameter, simplifying ( t t ˙ = 0 ( u ( s = 1 in ( λ j 2 r rs ˙ u x u h i h = j x x − u − ij j i r , 1 h 2 h x 2 1 2 dλ − d − = 0 + i = ¨ ¨ ¨ v x u 2 P ds . γ

0 s 0 r R as the affine parameter (so ˙ and can at best parametrise such that λ − 2 ) = r u x ) = ( t ( O rs = h ¨ t The full set of geodesic equations for the metric ( Of course, introducing the Penrose limit metric ( The geodesic equation for the coordinates To complete the demonstration that the two-dimensional optical tensors are the same for the null and . However, it does allow us to exploit the whole body of knowledge on the geometry in the original spacetime. This confirms that the timelike Penrose limit metric ( 3 ij r to establish the equivalence of the optical tensors defined in the neighbourhood of This allows us to immediately take timelike cases, we need thesimply further take observation that whereashave in the null case ( background can begeometry reduced of to geodesic the congruencesonly simpler in a plane case brief waves of is summaryimportant the well physical of understood Penrose example some in and limit key their we plane own results. present right. wave. here Of The course, gravitational plane waves are an benefits should also arise formetrics the of timelike this Penrose form limit have ( not been2.4 so widely studied in Gravitational the plane generalThe waves relativity discussion literature. above shows that memory for null observers in a general curved spacetime controlled by geodesic deviation.(expressed as In an particular, extended the Heisenbergclassification, enhanced algebra brings symmetries [ of considerablecongruences plane insight and, waves into by extension, the intoshockwaves nature the and form of their of plane the gravitational memory. wave geodesic Penrose The solutions symmetries limits of and are described in appendix here, we find thatare only non-zero, the since two-dimensional for transverse these components backgrounds we have information that is not already presentΩ in the original derivation ofof the gravitational optical plane tensors from waves, and to expose a large measure of universality in phenomena which is identical to the timelikez geodesic deviation equation ( fully captures the geometry of geodesic deviation. Moreover, for the pp waves of interest with In general, therefore, we define the timelike Penrose limit as a metric of the form JHEP12(2018)133 ], . . In ra 53 δ , ) (2.30) (2.34) (2.35) (2.31) (2.32) (2.33) B u 22 ) for the ( ) r i ( 0) for null f 2.31 ). We therefore ) = η < u 2, as ( , 2.2 a i ) in appendix E = 1 solutions. The choice = 0 ( ) a (the dot now signifying B.11 r i ( η ), ij g ) u 1 ( . a − i are readily seen to satisfy b , and ) and ( E b . ˙ ) ij EE r dX i X ( ab a f a ab
j B.10 , = (  x X ) (referred to as “BJR” coordinates E ) = 0 ) j a ) ij E dX i ij u u T h ) 2) defined with canonical ‘parallel’ and T ( ( ab , ˙ a u Ω E
j ab ( = ), these Ω = Ω T − E ab Ω − ij = 1 ) x u, V, X Ω ) 1 2 , ˙ C , 1 u 2 E r 2.31 (Ω a ( − ( − T j − + ab T – 10 – i X ) ˆ E ) give the change of variables from Brinkmann label the geodesic and ) ω h r E Ω are used to contract the transverse Rosen indices. ˙ i ( ηu  EE + (Ω u defined as Ω g a ( − ab a ij + ) = = = = 2 = ) 2.30 i ij ˙ du dV i Ω u E V E here precisely matches that given in ( ˙ ( ab x and V,X T a i ij ) W = = E = 2 r ¨ i i E ( v 2 f x with Ω = ( ] that the Wronskian associated with a particular choice of ds ab 22 ab ) ). C E 2.5 Ω is the twist in Rosen coordinates. It follows that for a congruence to T ) to Rosen coordinates ( i E ˆ ωE T ) for specified boundary conditions. This is discussed in detail in [ = ( u E u, v, x = 0, the expressions ( ( ab a i = η Ω E ]), in terms of which the plane wave metric takes the form, ab ), ˆ 21 ω – ). It follows immediately that Now, it is shown in [ The nature of the congruences is determined by the particular solutions of ( With The solutions are then written in terms of a zweibein 2.30 19 where exhibit twist, the Wronskian of the zweibein must not vanish. However, noting that the zweibein is the former reference focusingthe on complete set geodesics of exhibiting‘spray’ solutions twist. boundary conditions It respectively, as isgeneral, given convenient the in to zweibein ( consider isof a zweibein linear corresponding combination ofspacetime to region these an before initially an parallel encounter with congruence, a as shockwave, is appropriate therefore in the flat The metric components zweibein to be compared with ( coordinates ( in [ so we see that theuse definition the of Ω same notation for economy. Using ( In ( d/du where the integration(timelike) constants geodesics. The zweibein satisfies the key ‘oscillator equation’, JHEP12(2018)133 (2.37) (2.36) ) = 0, so 0 , it follows u ( u ab , W ! 1 2 dX dX . It follows that while a ij 2 , ! , ) = 0, that h δ ) 0 uu u tr u ) ( = 1 ( R 1 2 u ) + ) into the individual Raychoud- ( r h i ( − + ˙ × f h ij 2 − ij , 2.33 h i r ω 1 δ − tr ˆ ) , = . − u ) = ( 0 2 ij viz + u × iuju σ , a, b iuju ( h h ) b C C r tr ˆ i ( – 11 – f − , 1 + − dX ], this description of geodesic congruences in the a ij ij 2 ˆ ˆ θ ˆ σ ω

35 1 2 ˆ ˆ θ θ dX – 
− − − ) 2 u 32 , = = = ( dX ) is transverse and traceless. The ‘weak’ condition means ab ij ij u h ˆ 22 θ ˆ ˆ σ ω ( , 1 + ab d d d 18 du du du h dX ) only. ab h δ . (  O ). Similar considerations apply to timelike congruences, following the and the Weyl tensor is + + 2.3 ij h 2.36 -independent and can therefore be evaluated at any value of tr dudV dudV u − = 2 = 2 = ) and the boundary conditions 2 uu ds R 2.35 A weak gravitational wave is described by the metric, As discussed in refs. [ The implications for gravitational memory are that since we start with detectors form- Indeed, this is already apparent from expanding ( that is, the perturbation that we may work to we briefly review geodesicviewpoint deviation of the and general memory formalism for developed weak in this gravitational section. waves from the shockwave backgrounds and discuss in detail how they determine gravitational2.5 memory. Weak gravitational waves A very natural class of gravitationalthe waves from weak an gravitational observational waves, point viewed of as view a are small of course perturbation around flat spacetime. Here, by expansion and shear alone. plane wave spacetime can be developedMorette in many matrix ways, which notably enters in calculating theHere, the loop-corrected Van we Vleck- propagators focus needed for on QFT the applications. plane waves which arise as Penrose limits of various gravitational ing a twist-free congruenceerned in flat by spacetime, the and appropriateduring since Penrose and their after limit subsequent its spacetime, evolution encountertational is the with twist gov- the congruence memory, impulsive showing will gravitational that wave. remain the This twist-free evolution rules of out the gravi- Tissot ring is always determined where non-vanishing expansion and shear canof be non-vanishing induced curvature as suchlast the as equation congruence a of encounters shockwave, ( a thediscussion region twist in remains section zero by virtue of the from ( an initially parallel congruence can never develop a non-vanishinghuri twist. equations for expansion, shear and twist, Wronskian is JHEP12(2018)133 ij
1 and − ) = 0 )), are of (2.42) (2.38) (2.39) (2.43) (2.44) (2.45) (2.40) (2.41) ij having . i γ ˙ z EE ij ,Ω −∞ 2.14 h a ( i i = z and ˙ E ij . , uju i . ! ) for ) R 2 + = 0 × . h ¨ h ( ¨ h ˆ 2.40 ω ) − . . 2 O ) h j ), ( 2 ( + + × h dx O . This gives (see ( ( ¨ ¨ h h i ij O 2.39 . i +

dx , j constant, and integrate to give ij 1 2 z + ! × , δ h i ˙ h ) i + ) 1 2 z + u 00 = h i ! h , we find ( × 2 u j j a h ( = − + i z z ij ) as a plane wave in Rosen coordi- h h du 1 ) E × × and recognising that 1 2 ) j uju 0 u h ˆ σ i 4 ( x u 1 2 i − 2.37 ( R + ), x ) , with 1 h 00 uju 2 ) × i uju h ) in Brinkmann form. Defining Ω u h i ( du R ( – 12 – + ) 2.43 R 1 + 0 O ij − 0 h u , , h ¨ h × 2.37 1 2 −∞ h

+ du Z + −∞ ! 2 1 ˙ ) = h ( 0 ) in ( i u 1 2 + u 1 + h = ( z × ˙ −∞ h ( i du ˙ h Z ) corresponding to the time coordinate along the geodesic ¨ z = −

− O t dudv ab ( u ) − + 0 −∞ + j u C σ 0 Z ( i + × i i = 2 j ˙ ˙ h h R z − ) = ) = 0, we find ) = z h u 2 u in terms of the zweibein

( ≡ = ( i a ˙ ds 1 2 ) z −∞ ab i ( ) = u
˙ E ( h u , we clearly have, to i E = ( z i ij T = 0 ˆ
z ∆ 1 ij E ˆ θ − Ω ) in the weak-field limit. That is, starting from = ¨ EE ), with ) is the transverse connecting vector, ab 2.4 u ( C = i 2.34 z ij h Gravitational memory is usually discussed in this context by integrating the Jacobi With the metric in this form, we can simply transcribe everything we have described We immediately recognise the metric ( ), we can set For definiteness, we write the equations for a null congruence. The timelike case is essentially the same, . In particular, we can read off the optical tensors from Ω h 4 ( ij with the curvature replaced by If we now considerand the spacetime relevant with case of an initially parallel congruence with ˙ equation ( where O and we see immediately thatexhibit for shear, but these no weak expansion. gravitationalzero This waves, trace. the is a geodesic direct congruences consequence of the perturbation for plane waves inh general, substituting the specific forms ( The Brinkmann plane wave metric is therefore This allows us to re-express the metricand ( nates ( Writing JHEP12(2018)133 γ − ) f ) this u (2.48) (2.50) (2.46) (2.47) (2.49) ) = 0. ( i ij 2.47 h = 0. i ≡ z ), ( memory, the u < u k ( ij i L h z 2.46 , i.e. γ . memory. u −∞

) . ) = 0 but ∆ z i f j position-encoded u z ( h possible to express the curvature ij
= (Ω , ˙ j h ) ) from a reference null geodesic ] for a selection. z is the tangent vector to the geodesic i u ij , ( ij  ,
) is non-zero. From ( 15 i µ j z  2 j – −∞ z k 2 z z ( always Ω ) ) velocity-encoded 11 ij + Ω ˙ 2.45 , u u h with solution, ( ( i z + j j j ˙ ˙ i Ω Ω + Ω 5 i − z ij   h h ) ), it is 0 0 , where Ω ) u i while ( ( – 13 – u k = du du d 2.5 = Ω ( j du ij f ij h u u i D i 2 ˙ −∞ −∞ h z Z Z = 2 1 2 1 2 du = dz du d = u > u ij ij ) = h ) = ) = u u −∞ ( ( ( i i i z z ˙ z ∆ ∆ ˙ − ) ), these simplify to give u u ( ( i 6= 0, we have in addition a ˙ , we start with the most general form of the Jacobi equation for z ij ) ¨ h . As we have seen in ( , we see that in order to find a purely f 1 2 γ f u

2.1 u ( − ij ) should vanish for ˙ ≤ h = iuju ) can be immediately integrated R u . This is satisfied by Ω 2.44 − uju ij i ≤ 2.48 = R i 6= 0. If u ij ) h i As in section These moments of the curvature can be related to specific astrophysical sources of u Explicitly, we have the self-consistent solution, ( 5 . Then ( , say ij where we impose theThis initial is simply condition the of defining equation ancharacterisation for in initially the terms connecting parallel vector of congruence, following the from ˙ the vanishing alternative of its Lie derivative along for some Ω γ an exactly analogous way.therefore satisfies, The connecting vector with in the form potentially strong gravitationalnatural wave realisation bursts, in the especially language shockwaves. of Penrose limits This developedgeodesic finds here. deviation, a and very immediatelyordinates. adopt Again, the we description present in results terms for of a Fermi null normal congruence, co- the timelike case following in gravitational waves, e.g. flybys,are core-collapse discussed supernovae, at black length hole in mergers, the etc. literature; see2.6 These e.g. [ Gravitational memory We now generalise this conventional description of gravitational memory to the case of u integral in ( requirement in terms ofh the metric amplitudes is that Considering this in the context of a gravitational wave burst confined to a finite region of Writing JHEP12(2018)133 = 0 a . Any (2.51) (2.54) (2.52) (2.53) j = 0. u ) would E u ) u = 0 and so ( 2.54 j u i , we can inte- h ij −
). These regions 1 i a i − ¨ ). In effect, the orig- ]), these metrics de- E . The corresponding ˙ EE 54 ab ,
u < u 2.30 a in this form necessarily ( E a ) would be = X i T ij
E ij E ) i 2.53 , descriptions of flat spacetime. 6= , = . ) a f ) regions are simply flat spacetime. ab ) in the Penrose limit of the original X u < u , u ) would be localised at ) ( u ( a f C u ( a f θ ( i i X δ a x i ) E u > u ( a u a ( − in the form Ω i a u > u + ) for the curvature, so a physical realisation u > u ) i ( E ij u f a a non-equivalent would necessarily be negative for some values i ( E i ), where 0 δ ˙ δ ). E – 14 – u ∼ ( ∼ iuju 2.34 u > u a ) = ( i R ) a u i ) = ( E u and final iuju i ( . Of course, expressing Ω u E z ( a a i R i i z X E ∆ ˙ and vanishes for a zweibein which is at most linear in . The velocity-encoded memory is then, f ) = u < u ab u u ) ( ¨ i E ≤ z ) with such a zweibein is therefore diffeomorphic to flat spacetime, T , which is just the defining oscillator equation u ∆ E ij
≤ 2.34 1 i = ( − u ¨ EE aubu R = ) directly, giving ). ij u h ( 2.50 a i Now consider how these forms of the zweibein give rise to memory. For a purely The subtle point here is that we are considering gravitational wave bursts and shock- As explained above, these expressions are precisely those following from the geodesic To describe gravitational memory, we need to compare the relative positions and ve- Now, provided only that we can write Ω E The corresponding velocity change proportionalwould to not affect the futurerequire memory a region. too-singular However, dependence taken literally,would the have form to ( be smoothed. Even so, shockwave considered below, where theand full slide’ spacetime construction, may dividing be flat described spacetime by along a the Penrose shockwave ‘cut position-encoded localised memory, on an idealised form which realises ( being related by the Brinkmann-Rosen coordinateinal transformation spacetime ( links inequivalent andmemory distinguishable copies being of the flat physical spacetime,in signature. gravitational Bondi-Sachs In gravitational the wavescribe backgrounds language inequivalent (for adopted gravitational a vacua. in All review, discussions this see of is [ memory especially evident in the Aichelburg-Sexl Nevertheless, gravitational memory requires must therefore be described byThis two different, is best seencurvature in is the Rosen metricmetric of ( the form ( determined entirely by the zweibein waves which in the initial locities of neighbouring geodesicsburst before confined and to after an encounter with a gravitational wave and the position-encoded memory is equations for the Brinkmann transverse coordinate spacetime. Memory for a generalappropriate background gravitational spacetime plane is wave. therefore entirely encoded in an grate ( for some integrationimplies constants for JHEP12(2018)133 ) ρ 3.4 (3.1) (3.2) (3.3) (3.4) (3.5) (2.55) ) with u ( ρδ = with energy- ). The most u 6 uu ]. This solution ( , /r , T ). δ ) u 23 ], together with a ( , = 1 r, φ F 27 ) = χ φ rφ 18 , u ) , Γ ( 8 r 2 ( F 0 ] for which χ dφ rf 2 56 , r 1 2 ), which is characteristic of an u 55 − + . , ( , , δ 2 ) ) 18 = 2 and is a pp wave. The Christoffel u u , ( dr ∼ (  ) 0. F V 0 u r + ∂ r χ u θ ( ≥ φuφu 2 ) iuju a F  i r R ( du a 0 ) iuju f χ ). Solving the Einstein equations using ( r , log f u + u R ( 1 2 ∆ − ( a F i 1 2 δ ) δ = = χ GE – 15 – − ) x = tr 4 ( r v ur r φφ ( 2 ∼ − ,R = . Γ uu Γ f ) 0 ) r R u Eδ uu u + ( ) = ( R r F a ( i χ ) = f ) . E u 2 r dudv ( , ( δ 00 ) ) = 0, corresponding to the profile , r f u = 2 ( ( ) ), so we have velocity-encoded memory [ 1 2 πGρ r u 2 ρ F u 4 u ( ( − χ − ds = θ 0 F ) = χ r ∼ ( uu ) = ) 0 r ) T r ( f ( u f ruru 1 2 ( f a R 1 2 i − ˙ E = = v uu r uu Γ Γ The original Aichelburg-Sexl shockwave describes an impulsive gravitational wave lo- We therefore consider the metric, In the following sections, we will see how all this is realised in two important examples Next, consider zweibeins of the form , and to be compatible with the null energy condition for the source we would still need It is also interesting to consider a homogeneous beam source [ 6 u for some short-distance cut-off scale constant, for which with ∆ denoting the two-dimensional Laplacian in polar coordinatescalised ( on theimportant lightcone case is themomentum tensor shockwave formed bygives an infinitely boosted particle, and symbols are while the non-vanishing curvature components are smoothed (sandwich wave) extension in which the impulsive limit is relaxed. This has a manifest symmetry with Killing vector 3 Gravitational shockwaves This brings us tooptical the tensors main and Penrose topicthis limits, of section, for this important we paper, classes consider the of in characterisation gravitational of detail wave bursts. memory, the via In Aichelburg-Sexl the shockwave [ impulsive gravitational shockwave. Clearly,extended the wave burst, same where comments we apply would to find both a position smoothedof and or gravitational velocity memory. wave burstsextensions) — and its the spinning Aichelburg-Sexl generalisation, shockwave the (including gyraton. non-impulsive to impose the non-trivial constraint In this case, corresponds to a localised source with curvature of JHEP12(2018)133 > uu (3.9) (3.6) (3.7) (3.8) (3.10) T ). In this u ( ) we have δ , 3.6 equation to be ) ) = u ¨ φ ( u θ ( ) F b ( χ 0 f = 0 after encountering 1 2 , r ) ), depends on the impact b = , directly from the metric as u ( ( ˙ 0 r v η , = 0 axis. In what follows, we ) allows the uθ . = 0 . 2 b/f γ = 0 and the geodesics are curves r ) γ r 2 = 2 b 3.1 ` ) ˙ ( − 0 2 u , ˙ ( f φ 2 ., F 1 8 = r χ = 0 ) − u + 2 r ˙ ) ( ) provided the null energy condition φ 2 0 u , r r f ( ) , 3.5 θ = const − u ) ) ( ) = 0. Then ` b ) ) + ˙ u ) + – 16 – θ ( u ), ( ( u u 2 ( u f = ( ( ) ˙ , ( φ b 1 2 ˙ F F uθ 3.4 0 F φ ( ) 0 χ χ 2 χ )) b ) ) − r f ) ( = 0 0 r r 0 for a timelike geodesic. r 1 8 ( ˙ ( f ( 3.6 0 φ ]) both classically and including the special issues arising ηu ˙ f r f 1 2 f − r 2 2 1 1 2 + 58 η < ) + , + u = 0 for the chosen geodesic v + + V − ( b 57 2 ˙ δ ¨ φ ¨ ¨ v r φ ) = = b ( v r f 1 2 = 0. The focal point, at at the instant of collision. This raises immediate questions concerning − u η v ] (see also [ 0, which follows from ( = 56 ˙ , < v ) in b ) ( b 0 35 ( , 0 f ) constant, taking is the conserved angular momentum about the is the impact parameter for the chosen geodesic f 1 2 18 u = 0 for a null geodesic, ` b ( − φ η For Now focus on the Aichelburg-Sexl (AS) shockwave with profile The cylindrical symmetry of the original metric ( = v parameter. The most striking∆ feature of the solution,causality however, in these is shockwave the spacetimes. ‘back Thispapers in has [ been time’ thoroughly jump explored infrom our previous vacuum polarisation in QFT in these spacetimes. where 0 is respected,the the shockwave geodesics at converge towards the source at and so with case we can solve the geodesic equations exactly. For the first integrals of ( where choose the natural initialon condition the (before test particle the described incidence by of ( the gravitational wave burst with integrated, implying and we can immediately write the integrated expression for ˙ The geodesic equations for a general profile are 3.1 Geodesics for gravitational bursts and shockwaves JHEP12(2018)133 . ) 7 u 3 ( . v # ), with (3.11) 2 u ( v !! ˜ but are at u 8 GE πL ). The notable 8 r , b 1 3.10

1  − 2   L for the extended profile. The erf 2 L − ,

5 − ) in the form # u − u 2 u "  ( and  ) from this expression for v θ 2 α !! ) for the particle source in terms of the exp A ˜ u  r − ( = 0. For definiteness, we consider ! f  ˜ u u ) = 1 and has the impulsive limit GE πL can be adjusted to smooth the profile. 2 tanh L 8 ). Numerical values of the parameters ) are different, as illustrated in figure α − u GE r u πL + u, L ( 8 ( ) for three values of the impact parameter ) with b 1 ). u F F for null geodesics and with impulsive and u 

Θ( r ( 3.6 χ  χ 1 b 2 1 3.6 5 r L ). Notice that outside the gravitational wave θ − du

+  1 – 17 – erf R u − u, L ), we solve the geodesic equations piecewise in the three 1 , centred on L

 u erf α − = 0. The radial geodesics are also insensitive to the u, L  " ≡ r L GE ) ) and integrating then gives ) = Θ( 8 exp u tanh but different b ( 3.6 r b  ) = Θ( u, L F b u L χ 1 ( ) = + 2 F ) are chosen purely to demonstrate the key general properties u ( u χ J ∼ r ) is shown in figure 1) ˜ ) u ( ) and − ) = Θ( v ). In this case, analytic solutions may still be found b u u, L u u ( ( ( δ Θ( F δ (log χ ) = ), is normalised so that L u we show the radial coordinate GE u ) = 2. Substituting into ( ( ( 4 2, we can solve the radial equation in ( shows the 3-dim behaviour of a circle of geodesics with the same impact 2 δ , so are the same for null and timelike geodesics. Note that the asymptotic F in the lightcone coordinate L/ + η 4 χ u, L + L V and subsequently u across the interaction region is easily found (see appendix term reproducing the Aichelburg-Sexl shift. gives the duration of the burst and the parameter Θ( 0 b = v L ) = L < u < L/ u , u → The behaviour of Figure In figure We now present illustrations of these solutions for both null and timelike geodesics, and It is also interesting to relax the impulsive limit and consider an extended pulse of ( 2 With the extended profile For numerical results, we frequently use a smoothed version L v 8 7 for both L/ GE These analytic forms reproduceshift the ∆ numerical plots shownthe in log figures − inverse error function to give with ˜ where regions before, during and after the wave burst and match at the boundaries. In the interaction region is clearly seen. extended profiles. A similar behaviournot is seen constant for before timelike geodesics, the but arrival of of course with the gravitational wave, as given in ( parameter slope of the geodesics forThe equal final geodesic remembers the nature of the gravitationalparameter wave but burst. different initial angles to the shockwave axis. The focusing of the geodesics of the geodesics. b burst, the metric simply describesmay flat continue unperturbed spacetime so through the trajectories are straight lines, and lim first sight lessgeodesics illuminating found than from in numerical solutions the of impulsive ( limit,for so both below the we impulsive( show and plots extended profiles of the duration which, like JHEP12(2018)133 . 2 ) of geodesics u ( but with different r ) plots in figure b u ( r = 10. L = 0 axis. The focusing effect for geodesics r ) indicated (with a different vertical scale) by u, L – 18 – ) = Θ( u ( ). The parameters match those of the F χ as the corresponding test particle is struck by an impulsive u, L b = 0. The right-hand plot shows the same result for an extended u . The left-hand plot shows the behaviour of the radial coordinate . The red curve shows the trajectory of a geodesic in the background of a gravitational . The final transverse velocity of the test particle has a memory of the gravitational wave around the gravitational wave source on the φ Figure 4 wave burst with extended profileThe Θ( shaded surface is mappedangles out by geodesics with thewith same impact the parameter same impact parameter is evident. Figure 3 profile, and is greater for the extended wave burst. gravitational wave burst with profile the grey curve. In this and all subsequent plots, we have taken Figure 2 with different impactgravitational parameters shockwave at JHEP12(2018)133 . b ]. 56 (3.13) (3.12) , 35 , ), a detailed 18 3.2 relative motion ). b , ), we have is directly related ( f = 0. v      3.1 1 2 ˙ ) φ − u ( F = ) in the impulsive shock- χ u ˙ for the congruence centred v φ ) ( , ˙ r 1 2 r v µ r ( k      f ν across the shockwave allows us φφ D 0 0 + v ˙ , now shows v = v ˙ r      µν rr rr Ω = ) for ˙ − ν 3.7 k 0 0 Ω µν g ˙ r rr – 19 – ) immediately gives ∆ rr = Ω 0 00 0 0 0 0 Ω ˙ r Ω µ ˙ ) of a test particle exhibits a jump as it encounters the r − 3.10 u ( is a vector field, the tangent vector to the individual      v µ k = ), which depends on the CM energy and impact parameter , k µν Ω s, b      ˙ ˙ ˙ r 1 v φ . Here, γ      = , we discuss briefly how this shift ∆ µ A k . The lightcone coordinate Taking the covariant derivatives (using the Christoffel symbols given in ( This summarises the properties of a single geodesic in a gravitational shockwave back- In appendix calculation using in particular the expression ( and we can immediately use angular momentum conservation to set 3.2 Optical tensorsTo and find the memory optical tensors,on we start a by chosen calculating Ω geodesic geodesics in the congruence. With the generalised shockwave metric ( For the Aichelburg-Sexl shockwave, ( ground. To describe gravitationalof memory, neighbouring however, geodesics. we We needchosen therefore geodesic to need and to find determine describe the the the optical full tensors. congruence centred on a tories, and especially their implications for ‘time machines’, mayto be compute found in the [ scattering, scattering which amplitude is in mediated byto the graviton the eikonal exchange. scattering phase In limit Θ( this for picture, ultra-high ∆ energy particle shockwave case. feature is the discontinuous jumpwave case, in not the least since lightcone the coordinate immediately jump is raises backwards issues in of the corresponding causality time and coordinate. observability. Further This discussion of these trajec- Figure 5 gravitational wave burst, forprofiles. both impulsive Note (left-hand the figure) discontinuous and (and extended backwards (right-hand in figure) time) jump in the impulsive gravitational JHEP12(2018)133 ). 2.16 (3.19) (3.17) (3.18) (3.16) (3.14) (3.15) . according , where by      i µν ) and ( /r 0 0 0 g 1 2.10 = 0 is simply the u, v, x      µ = = µ A 2 , k.Dk ). ). A = e . 3.2 2.12 uµ ! , e 1 ˙ ˙ r , r . , as described in ( r − ,      r k.De ∂ ˙ ij r φφ ) 0 1 0 ! 1 0 0 − = Ω g , = 2 φφ 1 = 0, which requires the null geodesic Ω ˆ

r λ 0 jν which determine the optical tensors are Ω      λ k g e 2 2 1 k  + r ij k = λ φφ µν λ rr ˆ Ω φφ = (ˆ rr Γ µ Ω Γ = 1 Ω ij rr iµ − 2 0 − 1 e ˆ r Ω Ω φ – 20 – uν r k e k = − ). These all follow as purely algebraic conditions r φ

uµ ˆ ∂ ∂ ij Ω = Ω rr e , e ˆ 2.6 Ω = Ω µν = = ). It follows directly that      ij g  = tr is symmetric and therefore unsurprisingly the congruence 1 ˆ Ω = 0 immediately, we have 1 2 = 0. The first identity, 0 0 0 ˆ rr θ φφ − ˙ ij 3.15 φ will be the same for both null and timelike congruences, for Ω as in ( Ω ) as = ˆ Ω Aµ ij      ij in ( ˆ Ω ˆ σ AB 2.11 = η iµ = 0. Explicitly, the expansion is k.De e vµ ij ω . v . Imposing , with µ ) and k AB , e , that is, = η ) for ˙ γ 3.13 µ      are readily verified using the Christoffel symbols ( = u ˙ ˙ 3.7 r 0 1 v e in ( iµ e Bν      µν e ). The required transverse components = Aµ We see immediately that With the FNC basis established, we can construct the projection matrix ˆ It is readily checked that they satisfy the appropriate orthonormalityWe also condition need to verify that this set of basis vectors is parallel transported along the The null FNC basis vectors can be chosen in this case as and 2.9 e uµ e µν vµ leaving the shear as and the optical tensors are read off from thehas decomposition vanishing twist, ( ˆ then given directly from ( with Ω null geodesic defining geodesic equation itselfe so is satisfied by definition. The remainingto identities ( for g except for the lightconecondition ( identity dinates and take the transverse projection While the final results for definiteness we consider the null case first. definition 3.2.1 Null congruences Before evaluating these expressions more explicitly, we first introduce Fermi Normal Coor- where JHEP12(2018)133 . = and      Bν (3.25) (3.22) (3.23) (3.24) (3.20) (3.21) e 33 /r 0 0 0 ˆ Ω 1 Aµ e      ), this spec- µν g = u, r µ 2 . b , e      ˙ r , such that 0 1 0 − , i . 3 , . , 2      , 1 ))  = , , the normalisation being fixed = 0 )    η µ u, r 1 2 A , ( = 3 φφ u, r b ) , − ∂r ( 0 Ω u A , e √ 2 ( − 1 e ∂b r / µ r 0 µ uθ ( e − ) now shows that the components k ) 1 rr u b 1 − ( 0 , r, s =  3.20 f      1 u sν 0 0 0 00 Ω 0 1 2 µ 1 – 21 – e 0 ) = 0 0 0 e − + u =    µν ( b θ Ω ˙ ) r      = b = r η rµ ( 0 ∂ r 2 e f ) is unchanged, so we can project the three-dimensional = 0 is ) to define a vector field describing the whole congruence rs 1) and − = b 1 2 , ˆ Ω = ; 1 p Aµ u rr , = 3.14 ( 1 Ω . In this case, a suitable choice which also satisfies the par- rs ˙ = r r , η ˆ Ω 1 µ ), ( k.De 3 ), i.e. given a geodesic passing through the point ( − = 2 ) for 3.13 u, r , e 2 ( k b      3.10 ˙ ˙ r 1 0 v = diag(      η AB 2 η 1 − as given in ( ) such that √ µν with = 3.7 To achieve this, we invert the solution For this metric, the three-dimensional transverse space in the timelike case becomes Ω vanish, leaving µ i 0 3 AB e ˆ and so, implicitly to define ifies the corresponding impact parameter. Then, we may write 3.2.3 Aichelburg-Sexl shockwave We can now evaluate the optical tensorsSexl explicitly metric for in the null the congruence impulsive, ingeodesic shockwave the limit. solution Aichelburg- ( The only subtletyrather is than that a we single have to geodesic use specified the by the fixed impact parameter becomes clear below whenspective we of present the the Penrose limits. descriptionidentical The of to optical geodesic those tensors in deviation for the the fromof timelike null the a congruence case, are per- Tissot and therefore ring. can be visualised as before in terms of deformations effectively two-dimensional. The deeper reason for this, which is a property of pp waves, transverse components in the same way from A simple calculation with theΩ basis vectors ( For a timelike congruence, weη need an FNCby basis ( allel transport condition 3.2.2 Timelike congruences JHEP12(2018)133 (3.30) (3.29) (3.26) (3.27) (3.28) as , b ) u ( , we find for θ  2.2 u ) . In this case the b 2 ( b 0 indicating focusing), f ˆ θ ) after we have derived = 0. The focal point is at b . ( 7 . πGρ r , + 00 ) 4 which specify the optical σ f . 1 u 6 , − ( , − ) ) ij + ). θ u u  ˆ Ω ) (  ( )  b θ θ ) with impact parameter u ) = ) ( b and so ˆ ( 3.24 0 1 1 u b r , ( ( ( γ b f − − 0 ) 2 uθ . f (shear) as in section . 2 f 1 u  ), (  ) u uθ ( πGρ ) b 2 u 2 u ) + 8 ( u ) ) − ) b u ) ( σ 00 b b − ( 2 u θ ) + ) θ 3.23 ( u ( 9 b f ) 00 b b 0 ( 2 00 ( ( ) and = GE 1 2 f ) θ f f 00 00 u 1 2 2 (4 1 2 ( 1 2 f f GE /b πGρu ) GE   4 b πGρ 1 + − (4 ρ δ ( 8 0 1 + 4  −   – 22 – 1 + 1 + GEb f − b 2(4 ) u u 1  = 4   4 ) ) u b b b − ) ) − ( ) ( ( b b b 0 0 ) = b θ uu ) now gives ( = b f f ( ) b = = T 0 ( 00 b ˆ θ 1 2 1 2 u, r 00 ( f ˆ ∂r θ f ( + (expansion) and ˆ f 00 2 1 2 1 3.23 ˆ σ ˆ f θ ∂b 1 + 1 + 1 2 = =   1 1

= 11 22 1 = u u ˆ ˆ Ω Ω ) ) rr b b ( ( Ω ], for which 00 00 f f 1 2 1 2 56 ) that , r ( f 55 1 + 1 + , ), we therefore determine the projections component is found directly from ( 1 2 1 4 where the congruence focuses in one direction while diverging in the orthog- ˙ r 35 , r 3.17 = = and the expansion is ˆ θ = + GE 18 , ˆ σ 4 8 πGρ / φφ 4 2 / b It is interesting to consider other shockwave sources, for example a uniform density We illustrate these properties of the congruence in figures Defining the optical scalars From ( For the beam shockwave, we have 9 = = 1 geodesics independent of their impact parameter. u u onal direction. beam [ congruence has only expansion and no shear, and there is a single focal point for all the these results in the Penrose limit formalism. Note immediately the singular feature at Evaluating for the particle shockwave, we find (the negative sign for a general profile The Ω tensors for the null congruence centred on the geodesic and so we find Taking the partial derivative of ( JHEP12(2018)133 → ) r (3.35) (3.31) (3.32) (3.33) (3.34) ( f ], where is simply 8 ) defining ij h 3.6 , j ), we can replace . below. u ) ( dx 6 i u δ ( ) F dx → u ) is readily found using the FNC χ ij ( ) δ γ F , ,

u 3.1 ( χ ) + , jσ r ! F γ 2 ( e r

0 χ ) 22 iρ 0 f du r = 0 e h j ( 1 2 00 γ j x

i x f 11 x ) 0 1 2 = ) h u ) and we immediately find ). The only non-vanishing components of the u ( ρuσu ( γ j i

– 23 – =

R ij 3.3 h h 3.15 − γ

− = + φuφu . = i ) is the solution of the geodesic equation ( b ¨ R x u ij ij ruru ( 2 h h 1 r r R dudv given in ( . This is already known from our previous work [ − − = = 2.3 = 2 φuφu 22 11 , and solutions are plotted in figure 2 h h R ds 2.4 ]. in the original spacetime, the plane wave profile function and 35 γ , -dependent. That is, the memory associated with an impulsive shockwave ruru 18 , u are the basis vectors in ( R 8 . That is, the Penrose limit metric is iµ e ρµσν R The geodesics for the transverse coordinates are then, Interpreting in terms of gravitational memory, we see immediately that after the pas- ) for constant impact parameter b ( . For the Aichelburg-Sexl shockwave only, where f as described in section In these expressions, the function γ with with where the curvature are With the identification ofalong Brinkmann the geodesic coordinates forthe the projection plane onto the wave FNC metrictensor basis with of FNCs the relevant components of the shockwave curvature our earlier papers onbackgrounds causality [ and quantum field theoretic effects3.3.1 in QED in Null shockwave congruencesWe and consider first plane the FNC waves construction of the Penrose limit corresponding to null geodesics. 3.3 Penrose limitsThe and Penrose memory limit of the generalisedmethod shockwave described metric in ( section it was derived using anthe alternative plane method wave involving metric. the Indeed construction most of of the the Rosen results of form this of subsection are already known from sage of the shockwaveTissot the ring, is relative positionis of of the neighbouring purely geodesics, velocity-encodeding type. visualised test After by particles the encounter the (recall withwith that the fixed shockwave, the velocities, neighbour- null in and a timelike pattern optical exhibiting tensors both are expansion identical) (focusing) fly and apart shear. JHEP12(2018)133 ij ˆ Ω (3.38) (3.39) (3.40) (3.41) (3.36) (3.37) . Evalu- ij ) component 1 v − ˙ EE ( that the solutions ≡ . ij 2.4 ν 0 e , s . sσ , e ) ) ) are easily found and, with dx µ u u r u 0 ( ( ( e . θ θ , δ dx rρ ) ) 1 1 rs e u u − − δ γ ( ( .

. ) =   + u u u is 3. This follows from the symmetries ij u θ u θ ( ) 2 ) h ) s ) b b F ρµσν b ( b = 0 ( b dt ( 0 ( χ b 00 R 0 ) or a 00 = f f s j f f r − 1 2 2 1 x E 1 2 1 2 r 2 ij in the Penrose limit plane wave and the j x i ) ). = h – 24 – ij t 1 + 1 + ) with (   − γ 3.1 rs

+ (Ω) ) ) a 0 h b i b 3.34 s ) in the plane wave are written in terms of a zweibein ) = 1 + ) = 1 + ( ij ( b 0 ¨ 0 u E 00 u u ˙ − r Ω ( ( ( f f i R 2 1 = 0 if either (1 1 2 1 2 x . What we have called the ‘timelike Penrose limit’ has the 1 2 − rs E E − = = h 2.3 = 11 22 2 Ω Ω ) = ) given by ( t ds ), i.e. u ]: ( ( 8 rs ij h h 3.20 ), ) is the projection of the curvature tensor of the shockwave onto the timelike t ( rs 2.27 ) established directly in the full shockwave spacetime. h ) diagonal with [ ) which solves the oscillator equation u u This confirms the identification of Ω The optical tensors are therefore identical, and we can directly verify the Raychoudhuri To see this explicitly in the impulsive limit, recall from section Our key assertion is that the geodesics in this plane wave metric are the same as those ( ( 3.28 a a i i in the original shockwave spacetime ( FNC basis vectors ( We now see immediately that of the curvature tensor together with the fact that there is no non-vanishing For timelike geodesics, corresponding toogous massive formalism test from section particles/detectors, wemetric use ( the anal- where This discussion therefore confirmstional how memory for is null entirely geodesics, encoded geodesic in deviation the and3.3.2 corresponding gravita- Penrose limit plane Timelike wave. congruences of ( equations, written here as ating this, we find The optical tensors in this plane wave are found from the tensor Ω The solutions with boundary conditions appropriate forE a congruence of initially parallel geodesics, we have of the congruence in the tubularb neighbourhood of the null geodesic with impact parameter of the geodesic equations forE the JHEP12(2018)133 ) rs u ( h ) = r u ( (3.42) (3.43) F χ . where the geodesics 3.2 ) in the appropriate u ) to those in the null . 3 3.35 γ . This clearly shows the

) = 0 by definition) before ) involve the solutions u 7 ( being a defining property of i that after the passage of the x φuφu v 3.35 ∂ R 7 , 2 1 r in the original metric.    (with − ) and governed by the optical scalars b and , see footnote 22 i 0 γ 6 = h x 3.35 -translation symmetry of the shockwave 22 v 11 h ) (specifying the null congruence for definite- u ) explicitly for the impulsive Aichelburg-Sexl 0 00 0 0 0 ( i – 25 – x .    3.35 ) changes sign. For the impulsive shockwave, this , h Gµ implied by ( γ 4 = .

1 / 3.29 2 x b rs 2.6 ruru h = R ), and the generalisation with an extended profile, with impact parameter u − u ( γ = δ , we now see that the geodesics for the transverse coordinates ) are identical (for small 11 h ) = 2.3 3.40 u ( F ) in the Penrose limit geodesic equation ( χ r ( , we show the behaviour of f ). 6 3.30 direction focus on to the original geodesic . In turn, this can be traced to the 1 direction and divergence in ). Analytic solutions have been given above for the impulsive limit, whereas in x 2 ρµσν In terms of memory, it is evident from figures In figure The behaviour of the Tissot ring is illustrated in figure We illustrate this here by solving ( Following section Then, evaluating as before, we find ) in the metric ( x R u u, L ( i gravitational wave burst, there iseventually a diverging with velocity-encoded straight-line memory trajectories. with Inthere neighbouring the is case geodesics of in the addition extended-profile wave, after a the shift interaction in with position theexpectations in gravitational discussed the wave in burst. geodesics section This compared is immediately in before accord and with the general combination of shear and (initiallysquashes due negative) to expansion the described + oriented above.in shear, The the and contracts initial up circle to thediverging point as in the denominatordegenerate in line is ( reached at ness) in the impulsivethe and extended cases.given This in ( demonstrates the initial convergence in shockwave, with Θ( the extended (sandwich wave)the case functions a numerical solutioncharacterising is the geodesic used. This is necessary since 3.3.3 Congruences andAs memory we for have the seen, gravitationalgravitational the shockwave behaviour memory, of is nearby describedPenrose geodesics limit by in metric. the the congruence, geodesic and equations therefore ( the x Penrose limit. This confirms thatcongruences the in optical the tensors shockwave are metric, the as same is for already the null implicit and in timelike section with metric, the existence ofpp the waves. corresponding It Killing isis therefore vector effectively a two-dimensional. general feature of pp waves, including the shockwave, that in JHEP12(2018)133 (4.1) ) of a ) as in u 2 ( 2 , x x 1 x = 0 is r , ) and 2 u ( dφ 1 2 x r ]. The motivations for + . 51 2 – J dr 49 + dudφ ) u causes the Tissot circle to deform to an ( J + σ Jχ 2 together with a discussion of more general − C 2 ), while the right-hand figure describes the case of u – 26 – ( = 0) through the encounter with a gravitational wave du axis has been rescaled by a factor 5 compared to the δ ) and shear ˆ i u u x ˆ θ ( ) = F u χ ( ) ) and exhibit an initial focusing followed by divergence, while F (with r ). u ( χ ( γ 1 f x u, L + ) show a divergence. The left-hand figure refers to the Penrose limit u ( 2 x ) are profile functions, which in this case may be chosen indepen- ) = Θ( dudv u u ( ( J F = 2 χ χ 2 ds ) and u ( . Illustration of the evolution of the Tissot ring of geodesics through the passage of . These plots show the behaviour of the transverse coordinates F . The combination of expansion χ 6 The metric in the vacuum region outside the spinning source centred at where dently. The angular momentum of the source is proportional to 4.1 Gyraton metric The special form ofAichelburg-Sexl the shockwave to gyraton accommodate a metric spinningthis we source choice consider [ are here described isgyraton briefly the metrics. in simplest appendix extension of the ellipse, degenerate to a line,gravitational then wave form burst. an expanding (Noteprevious ellipse figures that in for the the clarity.) far memory region following the 4 Gyratons Figure 7 an extended plane-wave burst.figure The red curve shows a single geodesic described by ( the blue curves denoting plane wave with impulsivean profile extended profile Figure 6 geodesic in the neighbourhoodburst. of The green curves denote JHEP12(2018)133 , (4.7) (4.5) (4.6) (4.2) (4.3) (4.4) ) u ( J , χ , ). viz. , r r J 1 ) u 4.8 = = ( = 0 ) and there is a ), extending our 0 J ˙ φ rφ φ rφ ), rather than the A χ r v 4.1 u Γ ) ˙ r J ( Γ u ( F = J χ χ , ) ) is of relatively little in- J r , ruφu u u 2 η , ( ( ) δ u F + ( χ r 0 J , ) ) ˙ ) = 2 r χ ) (see appendix u ( ) = u 2 0 u ,R ( ( J u ( r ) f F ( 2 J J , u 1 2 ) = 0 J − χ χ ( χ . We therefore choose ) as the affine parameter. It is usually u χ ) 2 F ( 2 r = = r u ( J χ F 0 ) 2.4 u, L χ ur φ uu f ) may be chosen independently is actually r ) − v ( ) and Γ u 0 r -translation symmetry of the metric, which 2 Γ ( ( u r v ( J ) + equation directly from the metric, f rf u χ F – 27 – 1 2 v ∆ ( ) = Θ( χ ) + ˙ 0 J 1 2 , u − r , u ( χ ) , ( − ) J u − = F u ( χ ) and ( = 0 J χ = , u J = 0 ) χ ) = 0 as in the Aichelburg-Sexl case. ( = 0, to choose χ ) r uu r 2 φuφu F φφ r ( ( ˙ u 2 φ 2 R = 0 u χ ( Γ f f r r J ˙ J φ + ˙ χ r − − 2 2 v r 2 ) ) r 2 ˙ J u , ), together with the smoothed form for some of the numerical ,R ) typical of a sandwich wave produces an orbital motion in the ), which raises issues with the null energy condition. u ( ) ( ) u u − F ( ) + u ( u 0 F ( ( ) J χ u 0 J 3.11 χ ( F u ) F χ χ ) ( 0 J r χ χ r ( ) 0 F χ ) ( 0 ∼ r r χ f f 0) for a null (timelike) geodesic and we have already used ( 2 ) ( ( ) J 0 r 1 2 1 2 00 u r f < ( ( f 2 1 − + F − f ) as in ( 1 2 ¨ χ 1 2 ¨ ¨ φ v r − − = 0 ( u, L = = = η v uu r uu The geodesics are The fact that the profiles Now as we show below, the impulsive choice The non-vanishing Christoffel symbols of this gyraton metric are Γ Γ ruru R implies the geodesic equationsimpler to ¨ use the integrated form of the where where we have immediately exploited the We now study the geodesics,earlier null analysis and for timelike, for the the spinless gyraton gravitational metric shockwave. ( with Θ( plots. For the most part,impulsive limit. we also use the same form4.2 for the profile Geodesics and orbits hand, an extended geodesics. It isfor of a particular single interest geodesic, to thein corresponding verify accordance congruence explicitly with does how, the not despite general acquire a theory this non-vanishing of orbital twist, section motion a consequence of thegeneral cylindrical case, symmetry the we Einstein have equationsconstraint assumed link for the metric. In the more terest as the effect on a test particle is merely to give it a sideways kick. On the other and in the vacuum region where ∆ while the curvature components are JHEP12(2018)133 ), we (4.8) (4.9) (4.12) (4.11) (4.10) u, L ) = Θ( . and we find a . u ) ( 2 J r χ  u, L ∼ ) = 1. For a central 2 L ) r ) Θ( ( − 2) (4.13) u, L r f ( u , ) will precess (in contrast , 2  eff ) ˜ δ V  u ˜ ( u 2 4.11 − b J 2, giving J ≡ χ   where Θ( < u < L/ 2 2 2 profiles to be extended. Choosing L . 2 2 after the passage of the gyraton, u r ) J , ) + 1 2 L/ u arctan ( u < u < L/ u, L both − 2. This describes a straight line trajectory. ) J J (  2 ) = 0 ) + L/ Θ( + δ u > L/ u Jχ L/ u, r ( , ˜ u +  (  − u, r F 2 ) u = 2  ( 2 / eff r χ , 2 1 r 2 ˙ J ( V φ = b eff ) J  2 we need – 28 –  1 2 r V u 2 eff 1 2 r ∂ ( ˜ ) gives u 2 ˜ ˜ u ∂u 4 V b f ∂ + 2 10 J b − ∂r J 1 2 4.9  0 η E, r ) = ), and considering a particle source for the gyraton = ≡ r +  − u . Here, ˜ ( ¨ r 1 + + 0 `  ) = u, L  v log 0 ) r ) = v b ( = v u, r eff GE u, r ) = ) = ) = arctan ( ˜ V 4 ( u u u ( ( ( ) = 0 and just keep the angular momentum term with eff  r v eff r + φ V ( V 2 f ˙ r only. So the orbit corresponding to ( + ) to be Θ( 1 2 2 u ˙ ) = and initial r ( /r J 1 2 b χ u, r  ( 2 and into the ‘memory’ region or 1 . d eff 2 du L/ r ) and gives a bound orbit in the region of V r − ) and ( u f ( = const F u < χ E To be more explicit, integrating ( Now, we can readily see that in order to find solutions for which the test particle The azimuthal equation is also immediately integrable, giving the angular momentum For example, if we take ) as u 10 ( for impact parameter with can solve the geodesic equations exactly in the region So away from theregion initial and final kickswe have from the straight line trajectories for the initial to that with astable, homogeneous closed beam orbit). source for the gyraton, where This becomes a typicalvided central by force problem withpotential, a Bertrand’s logarithmic theorem attractive states potential thatportional pro- every to bound orbit is periodic for potentials pro- gyraton and a secondboth kick as it passes, shockwave, we then have with independently of whether the geodesic is null or timelike. exhibits orbital motion rather than simply receiving a kick at first encounter with the ` Substituting back into the radial geodesic gives a characteristic orbit equation JHEP12(2018)133 = u = 0. r but different initial b (for a slightly smoothed 8 = 10. The right-hand plot shows the ) through the encounter with a gyraton L u ( r – 29 – shows the form of the geodesic as it evolves with 9 ), shown here with u, L . u )), clearly showing the precessing orbit and the final kick at ) = Θ( u ( J u, L χ ) = u ( F χ . The red curve shows a single geodesic orbiting around the gyraton centred on . The left-hand plot shows the behaviour of to the gyraton axis. φ All this clearly illustrates the difference between the geodesics in the Aichelburg-Sexl For the logarithmic potential characterising the particle-source gyraton, we do not 2 into the memory region. Figure approximation to Θ( L/ the lightcone coordinate shockwave and the gyraton. While the initial and final trajectories are of course straight angles have analytic expressions fornumerical solutions. the A geodesic typical orbiting orbits, solution is so shown in we figure illustrate the key features with Figure 9 The shaded envelope is the set of geodesics with the same impact parameter Figure 8 with profile test particle following a precessingin orbit the around memory the region gyraton after axis the before passage emerging of as the a gyraton. straight line JHEP12(2018)133 ) 4.8 ) we u (4.15) (4.14) ( v ) reflecting ) and ( u ( v 4.7 ,      ) u ( in ultra-high energy F v χ ). From ( ) ) , for different values of r u η , ( 4.6 5 ( f J + 2 + ) ˙ Jχ r 1 ˙ u φ ). Note also that depending on ( ) − u J u ) are non-vanishing. ˙ ( φ ( χ r u 2 J ( 2 r 2 J and covers a single orbit. r J Jχ χ J 1 2 , with the bumpy change in , since − J r ˙ v ) + ) and u ) through the encounter with a gyraton. The left- (      u u ( F ( = – 30 – F χ v χ ) ν ) as the gyraton passes may have either sign. See r k in the gyraton metric. Naturally, for a trajectory u ( ( in terms of ˙ f µν . v J g 1 2 ˙ φ ) u = − ( 2 µ J ˙ and r χ 1 2 v 2 J r ) reflects the oscillations in − u is the tangent vector field corresponding to the geodesics in a ( = = v ˙ ˙ µ v φ k , k      ˙ ˙ ˙ , we show the analogue of the jump in the lightcone coordinate r 1 v φ , where 10      µ for a brief discussion of the relevance of the jump ∆ k = . Plots of the lightcone coordinate ν is given by the first integrals of the geodesic equations ( A µ D µ k k = With the gyraton metric, we have With this description of the behaviour of an individual geodesic in the gyraton back- In figure µν where we can immediately express ˙ 4.3 Optical tensorsThe and first memory step inΩ calculating the optical tensorscongruence for based the on gyraton the background solutions is described to above. evaluate gravitational scattering. ground, we now move on to analysein the particular congruence to in the see neighbourhoodmemory whether of with such this a twist. rotation geodesic, is inherited in the optical tensors in the form of found for the Aichelburg-Sexlthe or angular extended momentum shockwave parameter covering in several orbits, figure the metric parameters, theappendix jump in the number of orbits. The right-hand plot is forlines bigger with the testmomentum particle of being the deflected gyraton by metricin its induces the encounter an region with orbital where the motion the gyraton, for gyraton the the profiles angular test particle geodesics Figure 10 hand plot is for relatively small angular momentum JHEP12(2018)133 ,      (4.21) (4.19) (4.20) (4.16) (4.17) (4.18) ˙ φ parallel r = 0 and − uµ not ˜ e /r /r 0 0 ) 1 u = 0 is a rotated ( k.D J υ ) for the spinless 2 Jχ ˙ r , , and the result holds r 3.20 r k.De      , . = ), we may now evaluate η , . While = µ µ ! 1 + φφ 4.2 µ k ˜ e µ µ 2 2 Ω 1 2 ˙ ) ˜ e ˙ φ ˜ ˜ e e φ , , u − ( = 0 and J φφ      µ = χ , 1 Ω rφ φφ ˙ ! µ 2 0 is (compare ( 2 φ 2      r ˙ Ω Ω J φ ˜ e φ , ˙ φ r + r 1 2 k.De 0 1 0 ˙ r − − sin rr ur φr − cos φr − ) = may be expressed in the form      ˙ Ω u φ , ( 0 Ω 0 Ω ˙ φr φ = µν φ F φ rφ at a point µ χ + – 31 – with tangent vector Ω 1 ru uu ) 0 0 00 0 0 Ω ˙ ˜ ) e = Ω sin cos φ r Ω Ω γ − ( , k.D ˙ rφ rφ r 2.6 f

µ      1 2 Ω Ω 2 , rr ˜ e r = Ω ˙ = + φ      + ˙ 2 − ˙ 1 ). µν ˙ r = r 0 0 0 ! = 1 2 − Ω , µ rr µ µ 1 ˙ 1 2 − r , 4.14 ˜ e ru Ω r ) for the spinless shockwave. However, this basis is e e      r ∂ = 0 = =

u φ k.D . These give the same result for the optical tensors in the effectively = ˙ = Ω = 3.20 k k . At this point, we have not yet had to specify ˙ vµ as in ( rr 2 ur ˜ e uu 3.2 r Ω Ω Ω /r ) = ˙ u , ( r J k      ˙ ˙ ˙ Jχ r 1 v φ = 1, = = 0, a short calculation shows that in fact      v ˙ φ k = vµ It is relatively straightforward to see that an appropriate basis which satisfies the Next we need to find a basis for Fermi normal coordinates. We show this explicitly for Given the Christoffel symbols for the gyraton metric in ( ˜ e . After some calculation, we find that Ω uµ µν ˜ e It follows that the correct choiceset of defined FNC by basis with to be compared withtransported ( along the chosenk.D geodesic shockwave), the null congruence, with FNCsdescribed for in the section timelike congruencetwo-dimensional being transverse constructed space. similarly as required orthonormality conditions ( where for both null and timelike congruences. with Ω while In particular, this gives JHEP12(2018)133 , ) ) b u ( not = φ 4.25 r (4.22) (4.24) (4.28) (4.26) (4.27) (4.23) (4.25) . ! φ φ , the angle ij sin cos O is symmetric, it can φ ij vanishes. Even in the φ ˆ Ω φ . φ , ij sin ω cos − . , we define the optical tensors T ) sin 2 ) cos 2 u O u ). ( . ( 2.2 ! u J J ) ( ! ). Note that in ij χ , χ u ω ( /r r φφ J r J r/r , rφ , ) ˙ J jν r 4.21 Ω + ˆ u e Ω r 1 )-dependent rotation of the FNC basis. χ + + ˙ ( 2 jµ − r 1 1 r u J j r ij ˜ ˙ e r ( µν φ for the ordinary shockwave to incorporate φ σ r j − ) are the solutions of the geodesic equations φ i , so we simply find Ω ∂ Jχ ij u + ˆ ij O ( iµ ˆ φr Ω = ˆ ) rr sin 2 cos 2 Ω φ e coincide with the basis defining the FNC coor- ij however means that in this case we have non- Ω u ) = – 32 – Ω = ij r ( 1   ˙ ˆ u ˙ ˙ θ δ r r r J ˆ = Ω ( r /r 1 2 ˙ r r not 1 1 χ ) φ iµ ∂

). We find, r J e ) and u ij = − − ( u ˆ O Ω − = tr is symmetric, so the twist ˆ ( ˙ ˙ J r r orientations. Of course, since ij 4.22 r , and in the ˆ θ r r ˆ ij Ω = ∂ ∂ Jχ ), ˙ × φr ˆ Ω u   ˆ − Ω ! ( ij 1 2 1 2 r ˆ φ Ω = = φ = = , which we take as the null geodesic with initial conditions sin rφ rφ + and + × γ cos ˆ Ω ˆ ˆ − σ σ 2 is the orthogonal matrix in ( both , φ φ sin cos = 1 ) in full we therefore have

, i, j 4.24 = ij O ij ˆ The expansion is given by the trace of We see immediately that This is a very natural generalisation of Now, following the construction described in section Ω = 0. The optical tensors — expansion, shear and twist — are then read off from ( dinates. Explicitly, since the rotation ofterms the proportional FNC to basis Ω vanishing plays shear no in rˆole. The presencebe here diagonalised of to theorientation find off-diagonal only a — rotating however, this basis does in which the shear is non-vanishing in a single gyraton background, the fact thatimply an individual a geodesic relative orbits rotation around of the neighbouring source geodesics does in the congruence. φ with the usual definitions, To interpret this, recall that for the chosen geodesic the spin inherentthe in off-diagonal the terms gyratonWriting spacetime. ( This is evident first in the appearance of with the basis vectors defined in ( where is a solution of the geodesic equation, from the projection that is, JHEP12(2018)133 . T in T O rs ˜ h O h ˜ (4.33) (4.30) (4.31) (4.32) (4.34) (4.29) Ω O ) along O u ( ) and the = φ h 4.3 ˆ Ω = φφ . These are not Ω ) and u rφ . ( r Ω ) ) implies 3 φ 3.2.3 1 r ( , T j 4.33 + O , the Penrose limit metric is dx , rφ T i    11 T Ω , O ) ) dx γ O rr u ( u ! Ω ij (  F δ r 1 ˙ 0 J φ . The fact that the gyraton is also a pp χ , χ + ) + , we should have i φuφu r ruφu 2 ˙ 2 ij 2 J ( ˙ r φ ˜ r 3.3 Ω 0 φ 2 R R ˆ ˙ Ω f r 2 r 1 − du 1 , r 1 2  r 1 h j + x φφ ij − − i ) Ω ˆ ˙ x Ω r ˜ u Ω ) jσ 2 r ) ( 1 e u r φuru d u – 33 – ruru ( F d identical to that considered here for the null congruence. du ( ˙ du R iρ 0 J φ ∂ χ R − e r ij 1 ij χ )  = h h − r rr 2 ( J ˙ r O ij

φ Ω + 00 h ρuσu − f  =  1 2 O R ˙ r + ij − − r 1 ˆ dudv ). Explicitly, Ω ,     = = ˙ − ) φ ) d rφ ˙ ˙ u du r r = 2 ij φ ( 4.22 Ω r ( h 2 r 2 0 J ∂ r -components of the curvature tensor vanish, so the three-dimensional 1 ) component by component. Equation ( O χ v ). Recall that for the null geodesic ), (  ds  − 2 = J ¨ r + φ 4.32 d ij ). To verify this, note first that ¨ 4.22 du 4.21 φ − − h = = = = ), ( and carry through an analysis analogous to section 4.25 12 γ ˜ h 4.19 is the antisymmetric symbol and we use the temporary notation as in ( degenerates to a two-dimensional defined in ( ij ij  2.3 ˆ Ω O Now according to the general theory in section To evaluate further we would need to find explicit solutions for The timelike case follows in the same way as in section 11 wave again means thatsection the where We can then verify ( with, for example, with with with profile function, The Penrose limit isFNC now basis readily ( found giventhe the plane gyraton wave, curvature tensors ( the geodesic known in analytic formthese for results in a terms logarithmicnumerically. of central the potential. Penrose limit, Instead, then we study4.4 first the re-express behaviour of Penrose the limit congruences and memory JHEP12(2018)133 . 2.4 (4.35) . of shear, = 5. The 2 0 ) (in green) × u /r u ( σ 2 1 r x log and ˆ 5 and − + σ GE = ) shown covers a single 4 2 u shows the behaviour of − , x memory. 1 11 x ) = r . Here, under the influence of ( f through the passage of the gyraton, 12 γ = 10. Parameters are chosen such that ) plane. , 2 L , x = 0 1 x j x velocity-encoded and both orientations ˆ ) ˙ ) with u ˆ θ ( j – 34 – i u, L h − i ¨ x ) = Θ( u ( around the gyraton axis. The right-hand plot shows the ), and the particle source J ) in the final step. The other components follow similarly χ γ 4.6 u, L ) = u ). We have solved these equations numerically for the extended ( F makes one orbit of the gyraton axis between χ ) = Θ( 4.31 γ -dependent expansion u ( u shift from before to after the encounter with the gyraton. Subsequently J χ ) = u ( . The left-hand plot shows the behaviour of the transverse coordinates position defined in ( F ) (in blue) of a geodesic in the neighbourhood of χ ij u ( h 2 The evolution of the Tissot circle is shown in figure The results are illustrated in the following figures. Figure x and the geodesic congruences in the Penrose plane wave limit. These are found by ij ˆ shockwave spacetimes. 5 Discussion In this paper, weformalism have which developed encompasses the strong geometric gravitational description waves, of and gravitational have applied memory our in results a to the Tissot circleEventually, is in deformed the in farellipse, a whose memory orientation complicated is region, way governed byappearances, the during diagonalising this the Tissot the change shear in matrix. ring passage orientationthe is Despite settles of not superficial interplay due of the to to the gyraton. any become two twist directions of an of the expanding congruence, shear, simply confirming to the general analysis in section orbit of the originalhow geodesic the transverse positionthere of is a the geodesic,the i.e. geodesic the follows connecting a straight vector, line, evolves. exhibiting Clearly, non-vanishing and wuth profiles the transverse coordinates for athrough. member We have of chosen the parameters geodesic so that congruence the as evolution the of gyraton ( passes and we confirm the linkΩ between the derivatives ofsolving the the optical plane tensors wave found geodesic directly equations, from and shown here with profile the reference geodesic right-hand plot shows this motion in the transverse ( using the geodesic equation ( Figure 11 JHEP12(2018)133 – 13 ]). Of 64 , 63 ] (see also [ ] and neutron star inspi- 62 60 , . In this case, the Tissot circle 59 11 – 35 – . × σ and ˆ + σ . The evolution of the Tissot circle through the passage of the gyraton. The red curve ]. As well as the observed oscillatory signal, these and other astrophysical sources 61 Gravitational wave astronomy has been revolutionised with the recent LIGO and Virgo The geometric formalism was applied to two examples of strong gravitational waves A key observation is that memory is encoded in the Penrose limit of the original grav- ] and more certainly with satellite detectors such as eLISA [ 15 course, these observed signals areour weak-field gravitational analysis waves, but of it gravitationalphysics. may be shockwaves As hoped discussed may that earlier, also theseof shockwaves eventually extremely would find highly-boosted be produced, black applications for holes. in example, by astro- fly-bys used to characterise the eventual gravitational memory. observations of gravitational waves from blackrals hole [ mergers [ may also produce a gravitational memory effect, potentially observable at LIGO/Virgo [ of particular interest —spinning generalisations, gravitational gyratons. shockwaves Analytic of andthe numerical the evolution methods Aichelburg-Sexl were of used type null togravitational and and illustrate wave timelike their burst, geodesic and congruences the through their optical encounter tensors with — the expansion, shear and twist — were tion and memory inrelevant plane for wave gravitational spacetimes, wave whichdefined observations include a in the new astronomy. weak-field ‘timelike approximations showed Penrose For that timelike limit’ if spacetime, congruences, the which we originalgeodesic is spacetime equations less is determining well-studied. in memory thenull However, are wide Penrose we the limit. class same of as pp those waves, then for the the transverse plane waves in the itational wave spacetime. Foranalysis null enhances congruences, the the range Penrose limit of is applications a of plane existing wave studies so our involving geodesic devia- Figure 12 shows a single geodesicevolves in to the congruence an asinterplay expanding of described ellipse the in two figure in shear scalars the ˆ far memory region, with orientation determined by the JHEP12(2018)133 ] for 35 , 18 . These two flat = 0 represents a v u = 0 plane and with ) is at most linear in ] (see also [ u u ( a 31 i – E 29 ] how quantum loop contributions to 35 – = 0 to form the global AS metric in the Penrose 32 , u 18 One of our principal motivations in studying – 36 – in which the zweibein 12 ab ) in the lightcone coordinate for a null geodesic in the E v T E ]). Much of the research in this area has focused on the represents the shift by which the future and past Minkowski spacetimes = ( v 54 ab ], we may say that the shockwave localised at C and displays both position and velocity-encoded memory. This 54 12 Finally, we have shown in previous work [ One theoretical area of intense current interest is the relation of gravitational memory In Brinkmann coordinates, ∆ 12 . In the language of [ of ladder graviton-exchange diagrams, the phasemay shift be determining the calculated scattering from amplitude Aichelburg-Sexl the shift shockwave ∆ background. are displaced when they are glued back together along A Planck energy scattering One of theultra-high most energy interesting scattering. applications At CMis of energies dominated the of by order gravitational the theQFT shockwave gravitational Planck loop geometry interactions. mass, effects), in particle is the As scattering eikonal in shown limit where in the [ interaction may be approximated by a sum I am grateful to Timgravitational Hollowood shockwaves, from for our which earlier thesupported collaboration current in on work part quantum has by field STFC developed. theory grant This and ST/P00055X/1. research was Tissot circle in figure establishes the essential geometric frameworkeffects for future in investigations quantum of field gravitational theory. spin Acknowledgments photon propagation, and to Planckof energy geodesic scattering, deviation are that governedanalysis determines by of the gravitational gravitational same memory memory. geometry in Here,energy the scattering we Aichelburg-Sexl to shockwaves have include relevant spin extended for effectsmemory ultra-high the in the in form the of gyratons. gyraton The background nature of was gravitational clearly illustrated through the evolution of the i.e. gravitational vaciua. The shockwaveinequivalent’ scattering flat phase reflects regions. thisgravitational map The memory between ‘gauge on full one webon hand of the and other connections scattering is, amplitudes between however, and left symmetries, soft for vacua future graviton and work. theorems spacetime may be viewedthe as past and Minkowski future spacetimespacetime halves cut glued regions along back are with the described aby coordinate the in metric displacement Rosen coefficient ∆ coordinatesu by different metrics,domain distinguished wall separating diffeomorphic but physically inequivalent copies of flat spacetime, and soft gravitongravity theorems, (for and a more review,asymptotic generally symmetries of see with radially propagating [ the gravitationalSachs waves, infra-red described spacetime. by physics the Bondi- of Here,ideas quantum we to gravitational have memory established in shockwave the spacetimes. geometric In foundations particular, the to Aichelburg-Sexl apply similar JHEP12(2018)133 = are n 0 then (A.4) (A.5) (A.6) (A.7) (A.1) (A.2) (A.3) b E,E with ) in terms n s, t ( = A 2 p , where 0 . . . ) ) is/M EE  qb qb . ( ν ( 0 ). . = 4 0
+ J 2 J ), as well as the extremely p s 2 p 
# A.2 2 1  p , − s, t 1 1 1 ( 2 2 0 − b r is/M  − ) A − . ) Γ is/M − . s,b 2 p / 4 2 log 1 t E,b s 2 2 0 Θ(  Γ ( i b r v ν 4 e p is/M Γ GE ∆ 1  0 − + M 2 p log 2 as the momentum cut-off, we find p b  iE . 2 2 2 2 0 ) 2 q 0 p i b r − ) = 2 s π 1 + e is/M is the (vector) impact parameter: e b /r M  (    b b " f − 2 2 Γ t – 37 – d = (8 1 2 p − − 2 4Λ db b − 1

Z ) = db b ) −  ∞ a ∞ is 0 p t s s, b 0 ) = s, t 2 Z ( Z − Θ( πi A

) = 2 E, b πis ( πis 4 az v ) = 4 ( ν − ) = 8 ∆ − and setting Λ = 1 J s, t ), p ( 2 p s, t 13 ( ) = is the exchanged transverse momentum. A ) = ). Evaluating the integral over the angular dependence of dz z A /M q s, t ∞ s, t ( 0 ( as a function of the impact parameter, we can therefore determine the E, b Z A ( ), which depends on the CM energy through = 1 A v v G ∆ s, b 0 , where E 2 2 q − − is given by implied by the gamma functions in the amplitude = v t ) = It is remarkable that the complex pole structure, with poles at For the Aichelburg-Sexl shockwave, the discontinuous shift in the Brinkmann coordi- In the shockwave picture, the phase is identified (with our metric conventions) as To see what is involved, recall the formula for the scattering amplitude The required Hankel tansform is ,... s, b 2 13 , cut and paste construction.inequivalent It copies is of in flat this spacetime. sense that the scattering phase reflects the map between these two It follows directly that 1 The integral is standard, We therefore have nate ∆ implying (since Given the shift ∆ scattering amplitude by performing the Hankel transform in ( Here, Θ( gives, of particle spin would influence Planck energy scattering amplitudes. of the phase Θ( the energies of the scattering particles and geodesics in the gyraton metric is to develop some insight into how the gravitational effects JHEP12(2018)133 ).   2 (A.8) (A.9) u, L across (A.10) , while !! v v , ), we can # π . ) = Θ( ) in analytic . After some GEL ) A.8 t u are varied, as 2 8 p  ( ) s, t ) for ∆ 2 ) F p J ( r 6 χ b 1 A A.3 , this will also give /b is/M ) for the precessing

b 3 u and is/M 1 ( − L r ( − r E + O r − erf ) on the impact parameter

Γ( + Γ(1 − E, b 4   ( 1 r b v across the extended shockwave is  2 , we easily find the shift ∆ 0 2 0 . v exp b L E E 2 p 3.1 ! 4 p 2 s t L s M describing the duration of the extended M  π L GEL 1 r . Naturally, we can still obtain numerical 15 8 a , section r 10 − and metric parameters 8 =1 ∞ b 1 – 38 – r X b 2 1

b 2 p 1 0 s − M L E i ) after substituting the solution 2 erf p u s ( ) and performing the Hankel transform, we find an expan- M v , is reproduced so elegantly by the classical calculation of + 2 2 ) in footnote L 1 3 GE

s, t u ) 2 ) p 8 ( ( 2 p ) v A − ) for ˙ 2 where the test geodesic interacts with the shockwave. This is r 2 s, t ) for the Aichelburg-Sexl shockwave. It is therefore not obvious p ( is/M b b 4.7 A  ) for the Aichelburg-Sexl shockwave with profile is/M

A.6 2 ) for t log is/M − s, t 1) + −  4Λ ( A.2 Γ( − 2 p A  . We find, ), though it is not clear what insight this would bring, in contrast to < u < L/ Γ(1 s b t s 5 M 2 " s, t we have also found an analytic solution for the extended shockwave profile ( πi = × L/ (log A v − 3.1 are numerical coefficients. ∆ GE r 0 ) in the Aichelburg-Sexl spacetime. ) = 8 a ). In the case of the gyraton, however, the shift ∆ E s, t = 4 While we do not have an analytic form for the Hankel transform of ( As a first look at the effect of an extended profile on the scattering amplitude, however, Now of course our ability to perform the Hankel transform to find E, b ( ) can arise as the impact parameter ( v u, L A u v ( ∆ . For the impulsive shockwave profile, we have the simple solution ( where Substituting into ( sion of the form, make progress by expandingshockwave in interaction. the As parameter thisan is approximation equivalent to to the an scatteringreparametrisation, amplitude expansion for we in small find large momentum exchange giving the exact dependence on the impact parameter From the geodesic solution the range shown in figure the analytic solution ( at present how toscattering make using progress the gyraton in metric this to direction, future work. andwe we can leave calculate further investigation of Θ( determined by solving ( geodesic orbit. Evidently,v this is notillustrated so in straightforward the and numericalresults a plots for range in of figure behaviours for ∆ form depends on knowing theb functional dependence of ∆ in section simple final result for JHEP12(2018)133 . ) 0 r # ( E f (B.1) (B.2) (B.3) (B.4) ... , with (A.11) (A.12) n + + r -dependent ) 2 ) p u − 2 p , = . v v 2 p x is/M , , is/M x∂ y∂ y∂ #  − − − − is/M x y Γ( ...... y Γ(1 + u∂ u∂ . x∂ + +  = 0, where the pre-factors 2 and test particle energy 0 = = , s = t 2 1 2 E dy t L V J  P P  ∂ 0  + 0 K K K , the expansion parameter is the E 2 t L 2 4 = p p E t L 2 , must also be accompanied by a  s u Z dx  M ), there are two further i 6 K u i s/M 6 + 1 ( 36 2 ⇒ δ ⇒ ⇒ − − du ) 1 ) shows these are, ) u  2 1 + p ⇒ ( ) ) δ " B.1 2 p ) 2 p ) – 39 – r 4 2 ( 2 αx t s αu αu p is/M f 4 p + + + ). This makes clear how the corrections due to the is/M 0 term in the series has poles at − 1 is/M + y y x M α Γ( is/M th − 2 r → ) . → Γ(1 + → Γ( v π t L/E v dudv " Γ(1 characteristic of pp waves. Cylindrical symmetry of 2 2 p p → V = 2 ∂ = (8 v 2 ]. Inspection of ( is/M is/M ) also shows that, for fixed 2 = ds 65

  αy , y αy , y ) αx , x 2 2 Z t t − − − s, t A.10 K − − 4Λ 4Λ ( x v v ). As each new term in the series is included, an extra pole is added on A  

→ → → t t s s -axis. That is, the translations, which must be linear in v s x v A.10 πi πi of the profile depend on the momentum transfer , with the exception that there is never a pole at x, y L = 8 ,... 2 ) = 8 , We focus on the Aichelburg-Sexl shockwave with metric, To complete the calculation keeping only the leading correction, we now find explicitly, This has an interesting effect on the pole structure, arising from the new gamma s, t ( = 1 A where the compensating transformation of However, for the impulsive profiletranslation proportional symmetries to [ Evidently, this has the symmetry with Killing vector immediately implies the rotational symmetry, pared to generic ppsimilar waves. issues for In the corresponding this plane appendix, waves we arising describe as their these Penrose symmetries limits. and discuss showing clearly the parametrisation of the correction due toB the extended profile. Symmetries of gravitationalA shockwaves gravitational shockwave with an impulsive profile exhibits an enhanced symmetry com- and so, extension functions in ( the imaginary n impose a zero. Eq.Lorentz ( invariant combination ( JHEP12(2018)133 – 2) , ]. 19 (B.8) (B.9) (B.5) (B.6) (B.7) 67 (B.10) (B.11) , i i = 1 ∂ ∂ 24 ) ) r r i ( i ( r, s f g ( . , + + Z, Z v v rs ∂ ∂ δ i i x ] = 0 ] = 0 x − ) ) and 2 r r i ( i ( ˙ r ˙ f , ,P g ] = Z,J P 1 [ v s − − P ∂ , [ r ,P = , = r = . Recall that the Euclidean Q 2 r . r . ) R Q , 2 Q P Z i [ i r δ × K K K dx , n R ) ], plane including the Ozsv´ath-Sch¨ucking ) = 0 ) = u + ( , 0 ( 0 2 δ u 53 u ⇒ ⇒ ⇒ , ( ( , for a plane wave with arbitrary profile ) ) ! du = 0 = 0 22 r r 1 i i j ( ( 1 ˙ ˙ ) ) 14 P f x f r , r . j i j ( , ( − − i i f g x α ) ) ) x x 1 0 0 ) ) u u u r r + ( ( ( i i ] = ] = 0 ) may possess a further symmetry. A notable case is the ( ( ˙ j j ] = 0 ] = 0 ˙ 2 u g v f ij i

i – 40 – s r ( ) ) ,J h h h r r h , ij ( 2 ( , → ,P h i r + − − α P α r Z,P δ [ [ Z,P v ) ) [ P ) = r r [ − − i i ( ( u ¨ ¨ f g v v ( ) = ) = 0 for ease of notation. dudv 0 0 ij u u → → h ( ( /b ) ) ) = 2 r r b i i ( ( 2 ( f g 0 f ds 2 1 , -translations with a compensating rotation of the transverse coordinates which , v , v , , u , 2 ) ) − r r P i i ( ( ]. The same symmetry also occurs in oscillatory polarised plane waves [ g f are independent solutions of the key oscillator equation, ) ) ]. The generic isometry group 22 ] = 0 ] = 0 ) = r r ] = ] = 0 ( ) ( s r b 1 r 22 ( i α α ( 00 ,J g ,Q 1 f + + r Z,Q 2 1 P Z,P i i [ ], [ [ Q x [ x and = 0, 53 ) , → → h r i ( i i < ] analysed in [ f 22 x ] for some particularly relevant recent discussions) and we follow here the approach x 0 66 ) is the 5-parameter Heisenberg group with generators u The corresponding symmetry transformations and Killing vectors are known The symmetries of general plane waves have been widely studied (see especially [ Now consider the Penrose limit. This is the plane wave with metric, The corresponding generators satisfy the commutation relations, u Plane wave metrics with specific forms for 53 ( , = 14 ij extra symmetry comprising arises in one of thewave two [ classes of homogeneous plane waves [ which are convenientlyu chosen to satisfy the canonical boundary conditions at some where to be [ and notation of [ h satisfying the commutation relations, defining 22 where for the particle shockwave, This determines the 4-parameter isometrygroup group ISO(2) as is ISO(2) the semi-direct product ISO(2) = SO(2) JHEP12(2018)133 - u v (B.18) (B.15) (B.16) (B.17) (B.12) (B.13) (B.14) ∂ ) u ), that is ( . h θ B.4
! ) characterising )) . ). The solutions )) . u u u 1 2 ( ( ( x x ij ! ∂ ∂ 3.37 h )

u h u θ . )) )) ( h u θ ), ( u u ir ( ( − 0 1 δ , 0 x i (1 + h u θ 2 3.36 (1 ∂ x 0 x 15 ) 0 . h u θ h u θ ∂ − u r factors in the Killing vectors shown )) -dependent translations of the u ( 1 u u h − )) rs α − ( − δ u  u ( (1 + (1 −

u = 0 0 ) from ( v h u θ = u u u  +

( ) h u θ s → a − − ), given by v i i ( i (2) = u ∂ g − u u E ( ) ) r i (2) i r )) ( i ˙ ( + (1 + u f + + f ). ( + (1 v θ v v − ∂ – 41 – ∂ ∂ 0 v 1 3.37 ) 1 2 ) is impulsive. Nevertheless, we can ask whether ∂ ). , v , f x s 2 x x u i ( ) i r h u ˙ ( x g u ! 14 )) )) ) ) ij , g ( − ) u δ r u u u i h ( ) ( ( ( u r ! f ( h θ θ θ ( correspond to those for a parallel congruence and we can ) systematically, we use the Wronskian condition,  0 0 α − h θ )) ) u )) (1 r ( u 0 i i u ( ) + ( = = h u h u h u θ ( r X f i i ( 1 2 ) with the zweibein x g − 0. Q Q u 1 + ( h u θ h u θ K K ) respectively, as in ( (1 (1 + → < r . i 0 i ( ] (see footnote 0 − −

/b f x Z u ) v b (1 + = = (1 + = 53 ( ∂ K 0 , 0 2 1 f − − − u P P i 1 2 (1) 22 f − = = K K ] = u 1 P ) and b

( 00 ,K = 1 f 1 2 Q i (1) K are satisfied by ‘spray’ boundary conditions, corresponding to geodesics emanating g [ The obvious approach is to look for analogues of the These expressions for the generators and Killing vectors have already made use of The explicit form for the Killing vectors is then, We therefore already have the solutions For a general source for the shockwave, we simply replace the ) 15 r i ( transverse coordinates shown for the original Aichelburg-Sexl shockwave in ( here by the fact that thethere metric are coefficient still morebra symmetries for for a this generica special plane homogeneous profile wave. compared plane totransformations For wave [ the example, is Heisenberg the known alge- particular to form give of rise to a further symmetry related to The commutation relations are readily checked, e.g. and A short calculation now shows that the required solutions are To determine the solutions therefore identify the g from a fixed point at The boundary conditions for JHEP12(2018)133 (C.2) (C.3) (C.4) (C.5) (C.1) . ]. 2 ). First, 51 dφ , in the metric. This C.1 2 ) r u ( 0 + α dudφ 2 ) = 0, the metric coefficient 16 dr r ) (C.6) = 0, which implies + ,uφ ,r J . u, r, φ u, φ J ( 2 ( ) 1 r 1 J r ). However, we see immediately 2 ,uφ du dφ − ˜ − ) and with no loss of generality take u, φ J ) − ( 2 2 , B.7 . ˜ 1 2 ) J ,rr r ) , ) ) C.3 J ,r u u u + ( J − ( ), (  ( 0 ( u, r, φ 0 2 α ) ( ,r r α α 2 ) J 2 J 2 B.6 ), leaving only + ). r 1 r u 2 r r 0 limit of the general transformations defining 1 2 ( φ ), we start with amore general gyraton metric, − ω − – 42 – C.2 u, r, φ B.8 = 0 and − + → ( ) + ,rφ 2 ) ) term in ( → J 0 4.1 F F ,rr u ,rφ u ) = du 2 ( J ∆ ∆ J φ ) r 1 ω 1 2 1 2 2  u, r, φ u, r, φ ( ( 1 2 − − − -dependence in the coeffcient of r F J u, r, φ ( = = = = u, r, φ ( J → → ur uu F uu uφ R R R R ) ) + u, r, φ u, r, φ dudv ( ( ), i.e. with no J F = 2 . No other extended symmetries are apparent. We therefore conclude that r 2 u, φ P ( ds ˜ J ) that these are simply the → = 0. In the vacuum region outside a source localised at ) ) is therefore constrained by B.16 uu Now consider the effect of coordinate redefinitions on the metric ( To motivate the choice of metric ( R the pp wave with metric A very clear presentation of these results for gyratons may be found in the paper [ u, r, φ u, r, φ 16 ( ( non-vanishing. It follows that we canJ eliminate the considerably simplifies the curvatures in ( changes the metric coefficients by and J The corresponding Ricci tensorderivatives), components are (with subscript commas denoting partial C Gyraton metrics In this appendix, we reviewwith briefly the more choice general of gyraton profile metrics functions and and discuss issues coordinate arising redefinitions. viz. from ( the generators even with an impulsive profile, theisometry plane group wave with metric exhibits Heisenberg only algebra the ( generic 5-parameter This is indeed a symmetry of the metric ( JHEP12(2018)133 ), at u ( ) to = 0, 0 uu u (C.7) (C.8) (C.9) δ . Since ( 0. (C.10) R ) can be ), then φ δ for some i dH d > β . Locally, ∼ ∧ i u, x u, L ∂ r uu ( ) i d u = /∂φ Θ( H ( ,r ) i J J . πGT ∼ H χ = ) ) u, φ ), so u i ( u ( = 8 ( dH J . . Without imposing J ∂β u, x χ uu j ( χ 4 ) ,i R dx β φ i exact, i.e. = 0, it follows that ( -dependence of the metric − and so dx J , u ) ,r . i 17 to be proportional to φ ij ) J ) δ d . u J ) = u, x + locally ). However, the Poincar´elemma does = 0) is topologically non-trivial uu ( ) ( u, φ i = 0. i 0 J ) to be embedded in a modified ) with = ( r R . χ r H C.7 ,u H so the integrability condition corresponds u, φ ) β C.1 C.1 ( dudx i , → u, r, φ = 0 φ ) ( x ( ) i ) ,φ i d ∂φ ,ji β i β ∂J ) + 2 u, x H u, φ u, x − ( 2 − i ( ( 2 ) i = r ). β H ,ij H u 2 − H β ( + u, r, φ u, φ , is entirely avoided in the case of cylindrical − 0 J ( ( = v – 43 – 2 χ ˜ uu F J ,J ) = j,i du u ) considered here, R ) ∼ ) ( → H i u ) → → F encircling the axis ( − C.1 u v χ term in the metric can be removed by a coordinate F ( u, x C ) ) ) then sends ( ) i,j χ F i ) and the Ricci tensor F ) H χ u ) is expressible as a partial derivative u, φ r, φ + , which necessarily gives a negative contribution to ( u, x ( ( (
F r, φ dudφ ˜ u, r, φ β J f 2 ( L ( χ u, φ f ( + ∆ F dudv ˜ − v J u = 2 → ) = 2 v δ ds ] if we consider the more general pp wave metric − ) applies (which excludes the source at
u, r, φ 51 where it would violate the null energy condition ( 2 L u F C.1 + if and only if u is a closed form. The Poincar´elemma now implies it is ) now imply δ H ∼ C.6 = 0. Now, with the simpler metric ( ) ) and can be removed locally by the coordinate redefinition ( This difficulty, which would require the metric ( In fact, this is problematic for a physical interpretation. If we take Now consider the special cases where we can factorise the Next, consider the redefinition u imply global exactness in the presence of non-contractible loops as we have here in the topologically This is clearest [ ( dH u, φ 17 F ( In the language ofto differential forms, we maywe define have established abovei.e. that that with noβ loss of generalitynot we can take non-trivial vacuum region around the gyraton. A coordinate redefinition eliminated if and only if it satisfies the integrability condition some values of spacetime allowing a positive definite The profiles are then related by be impulsive, this requires which is too singularχ for a physical source. On the other hand, if However, these profile functionstions are ( not independent, since the vacuum curvature equa- and admits non-contractible loops coefficients in terms ofcylindrical symmetry, the the profile metric functions is then introduced of in the section form ( This means thatredefinition the whole this is always true butregion not where not ( necessarily globally. This is the case here since the vacuum under which JHEP12(2018)133 ) ]. , 37 4.1 ]. (C.11) SPIRE edition, IN SPIRE st [ (1985) 744] ]. IN ]. , 1 ]. ][ 89 , M. Cahen and SPIRE (2010) 084036 SPIRE IN , SPIRE Class. Quant. Grav. Rev. Mod. Phys. IN ) 27 ][ , , IN u (1957) 1072 ( ][ Birthday edition, Cambridge J th 179 nd J χ 60 , 2 (1974) 30]. arXiv:0905.0771 ) = Differential Geometry and [ 51 (1989) 1153] [ Nature Zh. Eksp. Teor. Fiz. , , in 96 u, r, φ ( arXiv:1702.01759 [ Class. Quant. Grav. ) uncorrelated. This is the metric ( arXiv:1605.01415 , (2009) 089 u ]. [ ( The refractive index of curved spacetime II: Astron. Zh. J (1985) 427 [ 08 χ ]. Detecting gravitational wave memory without 62 SPIRE – 44 – ,J IN ) [ JHEP Radiation of gravitational waves by a cluster of ) and SPIRE u Kinematic resonance and memory effect in free mass , (2017) 181103 ( Gravitational wave pulses with ‘velocity coded memory u Theory of the detection of short bursts of gravitational IN (1974) 17 [ ( F (2016) 061102 Zh. Eksp. Teor. Fiz. Fermi coordinates and Penrose limits ][ F χ 118 18 χ ) Exact space-times in Einstein’s general relativity r ), which permits any use, distribution and reproduction in ( 117 ]. f (1971) 2191 ) = Sov. Phys. JETP SPIRE , D 4 (1989) 653 [ IN ]. Exact solutions of Einstein’s field equations ][ Sov. Astron. Detecting gravitational-wave memory with LIGO: implications of 69 CC-BY 4.0 u, r, φ , hep-th/0603109 ( [ Phys. Rev. Lett. This article is distributed under the terms of the Creative Commons F SPIRE , Phys. Rev. Lett. A Remarkable property of plane waves in general relativity Any space-time has a plane wave as a limit IN The gravitational-wave memory effect , [ Plane gravitational waves in general relativity Phys. Rev. , ]. ), we set C.1 (2006) 3993 constant and the profiles SPIRE J arXiv:1003.3486 IN [ GW150914 parent signals Cambridge University Press, Cambridge U.K. (2009). radiation 23 QED, Penrose limits and black holes University Press, Cambridge U.K. (2003). (1965) 215 Relativity: A Volume inM. Honour Flato of eds., Andr´eLichnerowicz on Springer, his Germany (1976). [ Sov. Phys. JETP super-dense stars gravitational antennas T.J. Hollowood, G.M. Shore and R.J. Stanley, H. Stephani et al., R. Penrose, M. Blau, D. Frank and S. Weiss, H. Bondi, L.P. Grishchuk and A.G. Polnarev, R. Penrose, Ya.B. Zel’dovich and A.G. Polnarev, V.B. Braginsky and L.P. Grishchuk, P.D. Lasky et al., L.O. McNeill, E. Thrane and P.D. Lasky, J.B. Griffiths and J. Podolsky, G.W. Gibbons and S.W. Hawking, M. Favata, [8] [9] [6] [7] [3] [4] [5] [1] [2] [13] [14] [10] [11] [12] References Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. metric ( with studied in detail in the main text. symmetry. This seems in any case to be the most natural physical situation. Then, in the JHEP12(2018)133 ]. B D , ]. ] ]. Phys. , SPIRE Phys. (1987) 61 IN (2015) , SPIRE (2017) 039 ][ SPIRE ]. IN ]. Gen. Rel. 32 IN Phys. Rev. ][ , ]. Nucl. Phys. ]. 09 , ][ , B 198 SPIRE Class. Quant. SPIRE , IN IN SPIRE SPIRE JHEP ][ IN ][ IN , arXiv:1705.01378 ]. ][ ][ [ ]. Phys. Lett. ]. , SPIRE arXiv:1803.09640 Sturm–Liouville and Carroll: at [ IN SPIRE arXiv:1702.08284 arXiv:0806.1019 Class. Quant. Grav. Carroll symmetry of plane Velocity memory effect for The memory effect for plane Soft gravitons and the memory ][ [ [ IN , SPIRE ]. ][ IN (2017) 064013 ][ ]. arXiv:1802.09061 arXiv:1704.05997 [ (2018) 107 arXiv:0707.2303 [ SPIRE arXiv:1807.00990 [ [ IN (2008) 091 D 96 50 SPIRE Gravitational-wave memory: waveforms and ][ IN (2017) 175003 12 ][ Superstring collisions at Planckian energies Memory effect for impulsive gravitational waves 34 – 45 – (2018) 030 arXiv:1709.02299 (2017) 743 (2008) 138 arXiv:1111.3174 [ ]. JHEP 05 [ , Phys. Rev. , arXiv:1512.04952 (2018) 064031 The refractive index of curved spacetime: the fate of The Causal Structure of QED in Curved Spacetime: The effect of gravitational tidal forces on renormalized Causality violation, gravitational shockwaves and UV [ On the gravitational field of a massless particle B 772 ]. SPIRE Gen. Rel. Grav. B 795 High-energy unitarity of gravitation and strings ]. JCAP , IN ]. , D 98 ][ arXiv:1807.00765 The gravitational shock wave of a massless particle (2012) 120 SPIRE [ arXiv:1210.1449 ]. (2018) 065011 SPIRE [ IN ]. 02 (2016) 129 [ SPIRE IN 35 Class. Quant. Grav. Phys. Lett. IN ][ 03 , , Ion traps and the memory effect for periodic gravitational waves [ Nucl. Phys. SPIRE , SPIRE IN Phys. Rev. JHEP IN [ , , [ Graviton dominance in ultrahigh-energy scattering A new twist on the geometry of gravitational plane waves (1987) 81 JHEP Optics in a nonlinear gravitational plane wave Strong lensing, plane gravitational waves and transient flashes (2018) 044037 , (2013) 075011 ]. ]. (1971) 303 arXiv:1502.03658 [ 2 30 B 197 D 98 (1985) 173 (1988) 359 SPIRE SPIRE arXiv:1705.09533 IN IN causality in QED Analyticity and the Refractive Index quantum fields 37 Lett. Grav. 253 [ the heart of the memory effect Rev. [ Class. Quant. Grav. polarized gravitational waves gravitational waves effect for plane gravitational[ waves completion gravitational waves phenomenology Grav. 175017 T.J. Hollowood and G.M. Shore, T.J. Hollowood and G.M. Shore, I.J. Muzinich and M. Soldate, D. Amati, M. Ciafaloni and G. Veneziano, T.J. Hollowood and G.M. Shore, T. Dray and G. ’t Hooft, G. ’t Hooft, P.M. Zhang, M. Elbistan, G.W. Gibbons and P.A. Horvathy, P.M. Zhang et al., P.C. Aichelburg and R.U. Sexl, P.M. Zhang, C. Duval and P.A. Horvathy, P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, G.M. Shore, T.J. Hollowood and G.M. Shore, C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, A.I. Harte, A.I. Harte, C. Talbot, E. Thrane, P.D. Lasky and F. Lin, [33] [34] [30] [31] [32] [28] [29] [25] [26] [27] [23] [24] [21] [22] [18] [19] [20] [16] [17] [15] JHEP12(2018)133 , ]. , Phys. , Nucl. , , ]. SPIRE Phys. Rev. IN ]. , ]. [ ]. ]. SPIRE , , Cambridge (2003) 135 IN (1990) 1105. SPIRE ][ SPIRE IN SPIRE (2008) 089905] IN 22 SPIRE IN ][ ][ IN (1992) 377 ][ B 654 [ ]. D 77 ]. B 383 ]. SPIRE IN ]. ]. ]. (1989) 462 ][ Nucl. Phys. SPIRE gr-qc/9405053 SPIRE , Gen. Rel. Grav. ]. arXiv:1601.06989 IN gr-qc/0703127 [ , IN [ [ ][ SPIRE SPIRE hep-th/0504027 ][ Erratum ibid. SPIRE ]. [ IN IN B 232 [ Nucl. Phys. IN ]. ][ ][ SPIRE , [ IN SPIRE ][ (1995) 707 IN SPIRE Apparent horizon formation in the head-on gr-qc/0605131 Gyratonic pp-waves and their impulsive limit ][ – 46 – IN [ 12 particles Gravitational field of relativistic gyratons (2016) 215401 (2007) 124005 (1989) 55 Phys. Lett. ][ 2 , / (2005) 104034 gr-qc/9404027 (2005) 104028 gr-qc/9508044 1 [ Ultrarelativistic limits of boosted dilaton black holes [ Classical black hole production in high-energy collisions Causality, renormalizability and ultra-high energy A 49 The ultrarelativistic Kerr geometry and its energy Boosting the Kerr geometry into an arbitrary direction gr-qc/0412071 gr-qc/0201034 D 75 The Ultrarelativistic limit of the boosted Kerr-Newman Gravitational shock waves of ultrahigh energetic particles on The Ultrarelativistic limit of the Kerr-Newman geometry and [ [ B 220 Gravitational field of a spinning radiation beam-pulse in Improved analysis of black hole formation in high-energy D 71 D 71 Homogeneous plane waves ]. Boosting the Kerr metric Black hole formation in the head-on collision of ultrarelativistic Gravitational shock waves for Schwarzschild and Kerr black ]. ]. arXiv:1406.3227 (2006) 044003 gr-qc/9801097 [ J. Phys. [ (1996) 731 (1994) 3666 , SPIRE hep-th/0506001 Phys. Rev. SPIRE SPIRE 13 IN [ D 74 , [ Phys. Lett. Class. Quant. Grav. IN IN Phys. Rev. Phys. Rev. , , , D 50 (2005) 044032 (2002) 044011 ][ ][ , Lectures on the infrared structure of gravity and gauge theory (1998) 265 Lightlike limit of the boosted Kerr black holes in higher-dimensional spacetimes A relativist’s toolkit — The mathematics of black-hole mechanics (2014) 044050 D 71 D 66 Phys. Rev. , Phys. Rev. (2005) 084031 B 528 D 90 , hep-th/0212135 hep-th/0503171 University Press, Cambridge U.K. (2004). [ arXiv:1703.05448 higher dimensions Rev. charges collision of gyratons D 72 Phys. Rev. particle collisions [ Phys. Rev. Phys. particle scattering at the Planck scale geometry and the scattering of spin momentum tensor Class. Quant. Grav. holes gravitational scattering curved space-times M. Blau and M. O’Loughlin, A. Strominger, V.P. Frolov and D.V. Fursaev, J. Podolsky, R. Steinbauer and R. Svarc, E. Poisson, H. Yoshino, A. Zelnikov and V.P. Frolov, V.P. Frolov, W. Israel and A. Zelnikov, H. Yoshino and V.S. Rychkov, H. Yoshino and R.B. Mann, H. Yoshino, R.-G. Cai, J.-Y. Ji and K.-S. Soh, D.M. Eardley and S.B. Giddings, C.O. Lousto and N.G. Sanchez, C.O. Lousto and N.G. Sanchez, H. Balasin and H. Nachbagauer, K. Hayashi and T. Samura, C.O. Lousto and N.G. Sanchez, V. Ferrari and P. Pendenza, H. Balasin and H. Nachbagauer, T.J. Hollowood and G.M. Shore, [53] [54] [50] [51] [52] [48] [49] [46] [47] [43] [44] [45] [41] [42] [39] [40] [36] [37] [38] [35] JHEP12(2018)133 , B 116 ] Class. Class. , ]. , Phys. Rev. , Phys. Lett. , (2017) 161101 SPIRE IN arXiv:1501.00996 ][ 119 , Phys. Rev. Lett. , arXiv:1407.5597 [ (2003) 4169 Causality constraints on Observation of gravitational GW151226: observation of A 18 (2016) 061102 ]. GW170817: observation of (2016) 020 116 Phys. Rev. Lett. ]. Recent Developments in General 02 , SPIRE arXiv:1508.05303 IN [ waves with distributional profile ]. , in ][ SPIRE pp JHEP IN , ]. ][ ]. SPIRE Beamlike gravitational waves and their geodesics IN Int. J. Mod. Phys. (2015) 109 ]. , ][ – 47 – Phys. Rev. Lett. SPIRE SPIRE 11 , IN IN SPIRE ][ ][ IN Symmetries of arXiv:1202.0839 [ JHEP [ ]. ]. , An anti-Mach metric gr-qc/9509025 [ collaboration, B. Abbott et al., collaboration, B.P. Abbott et al., collaboration, B.P. Abbott et al., Graviton time delay and a speed limit for small black holes in ]. Low-frequency gravitational-wave science with eLISA/NGO SPIRE SPIRE Using atomic clocks to detect gravitational waves IN IN arXiv:1606.01859 [ SPIRE (1988) 1185 ][ ][ Gravitational wave detection with optical lattice atomic clocks IN (1996) 723 (2012) 124016 20 ][ arXiv:1804.07290 arXiv:1606.04855 [ 13 29 [ Constructing time machines Screw-symmetric gravitational waves: a double copy of the vortex , A. Masiello et al. eds., Pergamon Press, Oxford U.K. (1962). ]. ]. (2016) 124043 (2018) 22 SPIRE SPIRE arXiv:1710.05832 IN arXiv:1602.03837 gr-qc/0210048 IN Relativity 782 D 94 Quant. Grav. [ Quant. Grav. [ LIGO Scientific, Virgo gravitational waves from a(2016) 22-solar-mass 241103 binary black hole coalescence LIGO Scientific, Virgo gravitational waves from a binary neutron star inspiral Einstein-Gauss-Bonnet theory LIGO Scientific, Virgo waves from a binary[ black hole merger [ corrections to the graviton three-point[ coupling Gen. Rel. Grav. A. Ilderton, S. Kolkowitz et al., P.C. Aichelburg and H. Balasin, I. Ozsv´athand E. Sch¨ucking, P. Amaro-Seoane et al., A. Loeb and D. Maoz, G. Papallo and H.S. Reall, G.M. Shore, X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, V. Ferrari, P. Pendenza and G. Veneziano, [67] [64] [65] [66] [62] [63] [60] [61] [58] [59] [56] [57] [55]