Surprising Structures Hiding at Penrose’s Future Null Infinity 7 1 Ezra .T. Newman∗ 0 2 1.25.17 n a J 9 Abstract 2 Sincethelate1950s,almostalldiscussionsofAsymptoticallyFlat(Einstein- ] c Maxwell) Space-Times have taken place in the context of Penrose’s Null q Infinity, I+. In addition, almost all calculations have used the Bondi - coordinateandtetradsystems. Weshow-first,thatthereareothernat- r g ural coordinate systems, nearI+, (analogous to light-conesin flat-space) [ that are based on (asymptotically) shear-free null geodesic congruences (analogoustotheflat-spacecase). Usingthesenewcoordinatesandtheir 1 associated tetrad,wedefine the complex dipole moment, i.e., as the mass v dipole plus i times angular momentum, from the l = 1, harmonic coeffi- 6 cient of a component of the asymptotic Weyl tensor. Second, from this 0 4 definition,fromtheBianchiIdentitiesandfromtheBondimassandlinear 8 momentum, we show that there exists a large number of results - iden- 0 tifications and dynamics - identical to those of classical mechanics and . electrodynamics. Theyinclude,amongmanyothers,P=Mv+...,L=rxP, 1 spin, Newtons 2nd Law with the Rocket force term (M˙v) and radiation 0 7 reaction,angularmomentumconservationandothers. Alltheserelations 1 take place in therather mysterious H-Space. : Thisleadstotheenigma: ”whydothesewellknownrelationsofclassi- v calmechanictakeplaceinH−space?”and”Whatisthephysicalmeaning i X of H−space?” r a 1 Introduction The modern era of the study of Gravitational Radiation began in the 1950s with the pioneering work of Hermann Bondi[1]. This was quickly expanded by the major contributions of Rainer Sachs[2] and Roger Penrose[3][4][5][9] among many others. After years of further developments, theoretical, numerical and observational,we had its culmination with the observation and analytic under- standingofthecollisionandmergerofthepairofblackholesthatproducedthe gravitationalwavesignal,GW105,[6], thatwasseenby LIGOin 2016. Gravita- tional wave theory now could play a major role in astrophysics and physics. ∗UniversityofPittsburgh 1 Bondi’s work began with integrating the Einstein Equations in the asymp- totic region- inthe vicinityoffuture nullinfinity. This involvedthe important step of using special null surfaces as part of the coordinate system referred to as Bondi coordinates, (r,u,ζ,ζ). (The Bondi coordinates are defined uniquely up to a group of transformations known as the BMS group.[3][7]) The idea of working near or even at infinity, (though at the beginning was slightly nebu- lous), was formalized (by Penrose[3]) by bringing infinity into a finite region of the space-time by its conformal compactification, (rescaling and contrac- tion). FuturenullInfinitywasthenrepresentedbyanullthree-surfaceinspace- time (referred to as +, vocalized by SCRI+). +, a null 3-surface with the I I topology of 2 , (visualized as a light-cone at future null infinity, apex at S ×R time-like infinity) is coordinatized by the complex stereographic coordinates, (ζ = eiφcotθ,ζ = e−iφcotθ), on the 2 part, (labeling the null generators) 2 2 S and with u on the part (labeling the cross-sectionsof +). The BMS group R I can be described as coordinate transformationsamong the coordinates (u,ζ,ζ). In addition to the introduction of null surfaces, Bondi’s other insight was the realization that several components of the asymptotic Weyl tensor could be identified with the mass/energy and linear momentum of the source and their loss - in analogy, in Maxwell theory, to charge and charge conservation as in- tegrals of the fields at infinity. These Weyl tensor components come from the harmonic components of the leading coefficients in the r−1 expansion in the spin-coefficientversionoftheWeyltensorandarethusfunctionsjuston +,i.e., I are functions of (u,ζ,ζ). An important idea to recognize is that the leading far-field components of the Weyl tensor (in the spin-coefficient formulation[9]), depend very much on the choice of the tetrad and coordinatessystems to be used on +. In the past I almost all Weyl tensor components were chosen in a Bondi system. However, in the present work, the major new ingredient that leads to our results is the introductionoftotallynewcoordinatesandtetradsystemson + anditsneigh- I borhood, (very different from Bondi’s) that very closely mimic certain natural coordinate systems on the Minkowski space +. This allows us to describe I other functions on + (arising from other Weyl tensor coefficients) that yield - I with the Bianchi identities - a variety of additional physical quantities, such as angular momentum, center of mass, its position and velocity, their evolution, as well as force laws and electric and magnetic dipole moments and center of charge. In other words these results and relationships from classical mechanics simply appear as components of the Weyl tensor at infinity. The prime idea involved in the choice of these systems is use of the special null surfaces that are associated with asymptotically shear-free null geodesic congruences - as directly opposed to Bondi’s null surfaces which do have non- vanishing asymptotic shear - the time-integral of the Bondi news function. These special null surfaces, which are standard and easily understood surfaces in Minkowski space, are of two types. The first are the null cones (withgeneratorsautomaticallyshear-freeandtwist-free)withapexonarbitrary time-likecurves-aspecialcasebeingtime-likegeodesics. Thesecondtypearise 2 (formally)fromcomplexlight-coneswithapexoncomplexworld-lines[7][10]. In this case, the associated real congruence, though shear-free, is now twisting, They will be first reviewed and described in Sec. II. Sec.III will be devoted to their generalization (in asymptotically flat spaces) to asymptotically shear- free congruences, i.e., their definition and construction. Though at first it appears that there is a serious impediment to their construction, it turns out that by a slight zigzag or maneuver, the impediment can be overcome and the construction can be completed in exact analogy to the flat-space case. We willhavecoordinatesystems on(asymptotically flat) +, verycloselymatching I those in Minkowski space. The asymptotic generators (the null geodesics) of the complex surfaces (by construction) will be real and asymptotically shear- free but, in general, they will be twisting. In Secs.IV and V we will, by using earlier work, show how the asymptotic Weyl tensor components, expressed in thenewcoordinateswiththeirassociatedtetrads,naturallyyieldalargenumber offunctionsdeterminingtheinteriorspacetimepropertiesasmentionedearlier. In the discussion, the strange appearance of H-space-coordinates is addressed. We emphasize that though complex ideas are used, we are dealing with real space-times. For close to 50 years the coordinatization of + by Bondi coordinates has I been almost sacrosanct - nevertheless we present (what we believe is a strong argument)that other choices of coordinates on + have considerable value and I their use should be seriously considered. 2 + of Minkowski space I UsingstandardMinkowskicoordinates,xa,withmetricη andsignature(+,-,- ab ,-), the family of null cones with apex at the origin is described parametrically, (u,r,ζ,ζ), by xa =uta+rla(ζ,ζ) (1) with la(ζ,ζ) a null vector that sweeps outbthe null cone as ζ = cotθeiφ varies 2 over the sphere of null directions, i.e., b √2 ζ+ζ ζ ζ 1 ζζ √2 1 la(ζ,ζ) = (1, , i − , − )=( , Y0) (2) 2 1+ζζ − 1+ζζ −1+ζζ 2 2 1i b ta = δa. (3) 0 Ther istheaffineparameteralongthenullgeneratorsoftheconeanduthe time at the spatial origin (or the retarded time on the cone itself). AnalternateinterpretationofEq.(1)is thatitisthe coordinatetransforma- tion between the xa and the null coordinates (u,r,ζ,ζ). We will refer to the coordinatetransformation(andcoordinates)givenbyEq.(1),aswellasitslater generalizationin Sec.III, as ”static null coordinates”. 3 Aside: The fullnulltetrad,(l,n,m,m),associatedwithEq.(2),is givenby √2 bb b b na = (1+ζζ, (ζ+ζ), i(ζ ζ),1 ζζ), (4) 2P − − − mba = ðla = √2(0,1 ζ2, i(1+ζ2), 2ζ), 2P − − mba = ðla = √2(0,1 ζ2, i(1+ζ2),2ζ), 2P − bP = 1+ζζ. (5) The metric tensor in these new coordinates, given by ds2 =du2+2dudr 4r2P−2dζdζ (6) − can be conformally transformed (rescaled) by Ω2 =r−2, leading to ds2 =g dxadxb =Ω2du2+Ω22dudr 4P−2dζdζ. (7) ab − The surfaceedefineed by Ω = 0, a null surface, is identified as the + of I Minkowski space. It can be thought of as the intersection of the endpoints of thefuturenullconesthathaveapexontheworld-linexa =uta,withfuturenull infinity. Itiscoordinatizedby(u,ζ,ζ)andisaspecialcaseofBondicoordinates. ∗ ∗ ∗ These null coordinates can be generalized to a new set, (t,r ,ζ ,ζ ), by basing them on null cones with apex on an arbitrary time-like world-line, xa = ξa(t), by the parametric form, (t,ζ∗,ζ∗), t a real parameter, xa =ξa(t) +r∗l∗a(ζ∗,ζ∗). (8) b l∗a is again a null vector sweeping out the null directions on the cone. By equating the right sides of Eqs.(1) and (8), multiplying by the four null tetrad b vectors associated with la(ζ,ζ), Eq.(4), and passing to the limit r => , (or ∞ Ω=0) we find the relationship, (the light-cone cuts)[11][12], b √2 1 u=G (t,ζ,ζ) ξa(t) l (ζ,ζ)= ξ0(t)+ ξi(t)Y0(ζ,ζ), (9) F ≡ a 2 2 1i that describes, in Bondi coordinates,the intersectionof the null cones, apex on ξa(t), with +. Eq.(9), thus defines a one real parameter, t, family of’slicings’ I of I+. At a point on +, (u,ζ,ζ), where the two null vectors l∗a and la meet, I the null angle (on their past light-cone) between them (stated in stereographic b b coordinates, L and L ), is given by L=ðG , L=ðG (10) F F 4 Asymptotically, the two null vectors (and associated tetrads), l∗a and la, are related by b b l∗a = la+bma+bma+bbna, (11) mb∗a = bma+bbna, b b n∗a = na, b b b L b b = b +0(r−2). −r ItisusefultodistinguishbetweenthenullcoordinatesbasedEq.(1),(u,r,ζ,ζ), ∗ ∗ ∗ referredtoasstaticnullcoordinatesandthosebasedonEq.(8),(t,r ,ζ ,ζ ),referred to as dynamic or comoving null coordinates. Animportantpointforusisto notethatthe cutsu=G (t,ζ,ζ)satisfythe F (so-called) flat-space good cut equation, ð2G =0, (12) F namely the condition for the null normals to the ’cuts’ to define null vectors thatareshear-free[7][3][9]. Inthe followingsectiondealingwithasymptotically flat spaces, this equation will be generalized to the good cut equation, ð2G=σ0(G,ζ,ζ) (13) where σ0(u,ζ,ζ) is the asymptotic shear (the time integral of the Bondi news function). Remark 1 For relevance and analogy with the following section, we point out that in this flat space discussion we could have taken the ξa(τ) to be a complex world-line[7][10]. The L = ðG and its complex conjugate, via Eq.(11), would F still lead to l∗a being shear-free but now it would be twisting. The cuts however would be intrinsically complex and their real parts would have to be chosen - but b only after the differentiation. 3 Sec.III, + of Asymptotically Flat Space I Turning from Minkowski space to asymptotically flat spaces, we begin with + constructed from fixed but arbitrary Bondi coordinates, (u,r,ζ,ζ), and I Bonditetrad,(la,ma,ma,na),withlatangenttotheBondinullsurfaces,ma,ma, tangenttotheBondicutsat + andna tangenttothe + nullgenerators,with I I n,m,m parallel propagateddown the null geodesics on u. The radiation free- data is σ0(u,ζ,ζ)=ξij(u)Y2 +.... 2ij Wenowmimictheconstructionoftheshear-freecongruencesoftheprevious section. 5 It is known[7][8] that the general regular solution of Eq.(13) depends on fourcomplex parameters,za, (defining H-space)thatcanbe takenas functions ofthe complex parameter τ andwrittenasza =ξa(τ),. i.e.,asanarbitrary(to be determined) complex world-line in H-space. The solution (via coordinate conditions on the first four harmonics, l =0,1) can be written in the form u = G∗(τ,ζ,ζ) G(ξa(τ),ζ,ζ) zal (ζ,ζ)+ξij(za)Y2(ζ,ζ)+...(14) ≡ ≡ a ij za = ξa(τ) (15) with the quadrupole term ξij, arising from the data, σ0 = ξijY2(ζ,ζ) + ... ij From the freedom to rescale τ, i.e., τ∗ = F(τ), we set ξ0(τ) = τ. This is the velocity normalization and the slow motion approximation, with ξa ′ va, ξ0 ′ =√1+vi2 1 . Eq.(14) then becomes ≡ ≈ τ 1 u= ξi(τ)Y0(ζ,ζ)+ξij(ξa(τ))Y0(ζ,ζ)+.. (16) √2 − 2 1i ij (Theξij turnouttobethetime-derivativesofthegravitationalquadrupoles: ξij = √2G(Qij′′ +iQij′′ ). 24c4 Mass spin The idea is now to generalizethe flat-spacecuts, Eq.(9), to a one-parameter family of real cuts in the asymptotically flat situation, via Eq.(16). Unfortu- natelythisdoesnotworkimmediatelysince,ingeneral,forarbitraryσ0(u,ζ,ζ),the ∗ G (τ,ζ,ζ) willbecomplexandtherewillessentiallybenorealcuts. Beforewe see a way around this problem we mention that IF the σ0 was of pure electric type[13]then realcuts couldbe found andthe situationwouldresembleEq.(9). For general type of σ0 the Remark of the previous section becomes relevant, i.e., it is the analogue of the present case. The way around the problem of the complexity of the cuts, Eq. (14, is the following: treating τ as complex, we first construct the null angles, L = ðG(ξa,ζ,ζ), (17) L = ðG(ξa,ζ,ζ), to produce an asymptotically shear-free null vector field, l∗a, via Eq.(11), l∗a = la+bma+bma+0(r−2), (18) L b = +0(r−2). −r Note: We use L and do not use L=ðG(ξa,ζ,ζ). Next, using e τ = t+iλ, (19) ξa(τ) = ξa(t,λ)+iξa(t,λ), (20) R I 6 ∗ we decompose G (τ,ζ,ζ) into its real and imaginary parts, G(ξa(τ),ζ,ζ)=G (ξa(t,λ),ξa(t,λ),ζ,ζ)+iG (ξa(t,λ),ξa(t,λ),ζ,ζ). R R I I R I By setting G =G (ξa(t,λ),ξa(t,λ),ζ,ζ)=0 and solving it for λ, i.e., I I R I λ=Λ(t,ζ,ζ), (21) andsubstitutingλintoG ,wehavethe one real parameter family of real cuts, R u=G (ξa(t,Λ),ξa(t,Λ),ζ,ζ). (22) R R I This construction can be done under fairly general conditions assuming ∂G /∂λ=0. I 6 With the L and L, of Eqs.(17) and (18) evaluated on the real cuts, the associated real but twisting shear-free null vector field is, L L l∗a =la ma ma+0(r−2). (23) − r − r We now have the situation in asymptotically flat spaces that is totally anal- ogous to the situation we had in Minkowski space. We refer to the H-space coordinates, za, as complex pseudo-Minkowski coordinates, since in the flat- limit they are the complexified Minkowski coordinates. The + coordinates I associated with these pseudo-Minkowski coordinates i.e., the (τ,ζ,ζ), will be referred to as pseudo-Minkowskicuts. Fortherealityconsiderations,wehavetwoseparatecases: (1) thecomoving choice of an arbitrary ξa(τ), in Eq.(16), to be determined by choosing it to be the complex center of mass or (2) by the choice of the static pseudo-Lorentzian + coordinates, i.e., by taking (a ‘straight’ H-space world-line) ξa = τδa, for I 0 the cuts: τ b u= +ξij(ξa(τ))Y2(ζ,ζ)+.. (24) √2 ij b b If weassumethatboth λis smallandthe slowmotionapproximation,these constructionscanbe explicitly carriedout. They lead,viathe assumedformof the asymptotic shear, σ0 =ξij(u)Y2 (ζ,ζ)+..., (25) 2ij to the linearized expressions (that we need), case 1 (26) √2 λ = Λ(t,ζ,ζ) ξi(t)Y0(ζ,ζ) √2ξij(t)Y0 (ζ,ζ), (27) ≡ 2 I 1i − I 2ij t 1 u = G = ξi (t)Y0(ζ,ζ)+ξij(t)Y0 (ζ,ζ), (28) R R √2 − 2 R 1i R 2ij ξij(u) = ξij(u)+iξij(u). (29) R I 7 case 2 (30) τ = t+iλ (31) λb = bΛ(t,ζb,ζ)≡−√2ξIij(t)Y20ij(ζ,ζ), (32) ub = Gb b= t +ξij(t)Y0b(ζ,ζ). (33) R R √2 R 2ij b b 4 Review and Further Developments Asmentionedearlier,muchofthematerialdescribedherewillinvolvefunctions orstructuresthat‘live’on + andoriginatewiththeleadingWeylandMaxwell I tensor components. Our major interest will center on the asymptotic behav- ior, the physical meaning, the evolution and transformation properties of these tensors. Using Bondi coordinates and tetrad, the five complex self-dual NP components of the Weyl tensor and three complex Maxwell components are[4]: Ψ = C lamblcmd = C , (34) 0 abcd 1313 − − Ψ = C lanblcmd = C , (35) 1 abcd 1213 − − Ψ = C lambmcnd = C , (36) 2 abcd 1342 − − Ψ = C lanbmcnd = C , (37) 3 abcd 1242 − − Ψ = C nambmcnd = C . (38) 4 abcd 2442 − − φ = F lamb, 0 ab 1 φ = F (lanb+mamb), 1 ab 2 φ = F namb. 2 ab From the radial asymptotic Bianchi identities and Maxwell equations, we have their asymptotic behavior (the ’peeling’ theorem)[4]: Ψ = Ψ0r−5+O(r−6), 0 0 Ψ = Ψ0r−4+O(r−5), 1 1 Ψ = Ψ0r−3+O(r−4), 2 2 Ψ = Ψ0r−2+O(r−3), 3 3 Ψ = Ψ0r−1+O(r−2). 4 4 φ = φ0r−3+O(r−4), 0 0 φ = φ0r−2+O(r−3), 1 1 φ = φ0r−1+O(r−2), 2 2 8 with Ψ0 = Ψ0(u,ζ,ζ), n n φ0 = φ0(u,ζ,ζ). n n Thenon-radialBianchiIdentitiesandMaxwellequationsyieldtheevolution equations for these leading terms (our basic variables): Ψ˙0 = ðΨ0 +σ0Ψ0 +kφ0φ0, (39) 2 − 3 4 2 2 Ψ˙0 = ðΨ0 +2σ0Ψ0 +2kφ0φ0, (40) 1 − 2 3 1 2 Ψ˙0 = ðΨ0 +3σ0Ψ0 +3kφ0φ0, (41) 0 − 1 2 0 2 k = 2Gc−4, (42) φ˙0 = ðφ0, (43) 1 − 2 φ˙0 = ðφ0+σ0φ0. (44) 0 − 1 2 The u-derivative is denoted by the overdot. After the final coordinate transformation to the static pseudo-Minkowski coordinates and static pseudo-Minkowski cuts, the Eqs. (39-44) are seen to contain our classical (mechanical) equations of motion. The quantity σ0(u,ζ,ζ) (referred earlier to as the asymptotic shear), is the leading term in the shear of the geodesic congruence, la; i.e.., σ =r−2σ0(u,ζ,ζ)+O(r−4), whileitsfirstu-derivativeistheBondinewsfunction. Weconsiderσ0(u,ζ,ζ)as afreefunctionbuttakeitonlyuptothequadrupoleterms,Eq.(25). It,assuch, playsasignificantroleinwhatlaterfollows. Fromthespin-coefficientequations one finds that Body Math Ψ0 = ð(σ0)·, (45) 3 Ψ0 = (σ0)·· . 4 − 4.1 Physical Identifications From the definition of the mass aspect, Ψ, (real from field equations) by Ψ=Ψ Ψ0 +ð2σ0+σ0(σ0)·, (46) ≡ 2 Bondi defined the asymptotic mass, M , and 3-momentum, Pi as the l = B B 0 & l =1 harmonic coefficients of Ψ. Specifically, 9 Definition 1 Ψ = Ψ0+ΨiY0 +ΨijY0 +. (47) 1i 2ij 2√2G Ψ0 = M (48) − c2 B 6G Ψi = Pi (49) − c3 B By rewriting Eq.(39), replacing the Ψ0 by Ψ via Eq.(46), we have 2 Ψ˙ = (σ0)·(σ0)·+ kφ0φ0. 2 2 Immediately we have the Bondi mass/energy loss theorem: M˙ = c2 ((σ0)·(σ0)·+kφ0φ0)d2S <0, (50) B −2√2GZ 2 2 the integraltakenoverthe unit 2-sphereatconstant u. This relationshipis at the basis of almost all the contemporary work on the detection of gravitational radiation. Definition 2 Though it has been a controversial subject and there is nogeneralagreement,weadopt the definition (whichcomesfromlineartheory) of the complex mass dipole moment, (Di = Di +ic−1Ji), as the (complex) (mass) l=1 harmonic component of Ψ0, 1 Ψ0 = 6√2Gc−2(Di +ic−1Ji)Y1 +.... (51) 1 − (mass) 1i Di the mass dipole and Ji, the total angular momentum, as ”seen” at null infinity. The main defense of this definition is that it works extremely well. Definition 3 Ouridentification-whichisstandard-forthecomplexE&M dipole, (electric and magnetic dipoles, (Di +iD )) as the l =1 harmonic Elec Mag component of φ0 is: 0 φ0 = 2(Di +iD )Y1, (52) 0 Elec Mag 1i Di = (Di +iD )=qξi. (53) E&M Elec Mag Later we will connect these three physical identifications with the complex center of mass. Actually, for the general situation there is the independent complex center of charge. Here, however for simplicity, we assume that they coincide. This is not necessary but is a simplifying restriction.[7] Comment. For later use we note that from the asymptotic Maxwell equa- tions, Eqs.(43) and (44), we have that[7] φ0 = q+√2qξi ′Y0 +Q +... (54) 1 1i 1 φ0 = 2qξi ′′Y−1+Q +..., 2 − 1i 2 10