THE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH. Scholarpedia 11(12):33528 (2016) with Thomas Mädler

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1 THE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH Scholarpedia 11(12):33528 (2016) with Thomas Mädler

2 NULL HYPERSURFACES u = const Normal u is null u =0 Normal vector k = u is also tangent vector u = u@ u =0 Null rays x = k Null rays are null geodesics k r k = k r r u = k r r u = k r k = 1 2 r (k k )=0 The local null cone If T is timelike or null then T k 6=0unless T = k If V is tangent to u = const so that u =0then V is spacelike unless V = k

3 MINKOWSKI SPACE Cartesian inertial coordinates x =(t, x i )=(t, x, y, z) Minkowski metric = diag( 1, 1, 1, 1) Future nullcone: u = t r, r 2 = ij x i x j Cartesian components of normal: k u =( 1,x i /r) Tangent vector k = k =(1,x i /r) k k =0 Null coordinates x =(u, r, x A ), x A =(, ) ds 2 = dt 2 + dr 2 + r 2 q AB dx A dx B = du 2 2dudr + r 2 q AB dx A dx B Unit sphere metric q AB dx A dx B = d 2 +sin 2 d 2 In null coordinates u = x A =0, r

4 Curved space Bondi Sachs coordinates Null coordinates: x =(u, r, x A ) u = const: Family of outgoing null hypersurfaces r: Areal radial coordinate along null rays x A =(, ): Angular coordinates constant along null rays Future pointing tangent vector to null rays k = u Coordinate Conditions on Metric Components 0= g (@ u)@ u ) g uu =0 x A = g (@ u)@ x A = g (@ u) A ) g ua =0 Areal coordinate r ) det[g AB ]=r 4 det[q AB ]=r 4 sin 2 ) g AB = r 2 h AB det[h AB ]=det[q AB ]=q(x A ) = g g u r =0=g ur g rr ) g rr =0 u A =0=gur g ra ) g ra =0 Covariant Bondi-Sachs metric g dx dx = V r e2 du 2 2e 2 dudr+r 2 h AB (dx A U A du)(dx B U B du) det[h AB ]=q so conformal 2-metric h AB has only two degrees of freedom Non-zero contravariant components g ur = e 2, g rr = V r e 2, g ra = U A e 2, g AB = 1 r 2hAB

5 The Electromagnetic Analogue Minkowski metric in outgoing null coordinates (u, r, x A ) dx dx = du 2 2drdu + r 2 q AB dx A dx B Assume sources of electromagnetic field are inside a worldtube, with spherical cross-sections of radius r = R. The outgoing null cones N u from r =0intersect in spheres S u coordinatized by x A The Maxwell field F is represented by vector potential A, F = r A r A with gauge freedom A! A + r Choice (u, r, x A )= R r 0 A rdr 0 leads to null gauge A r =0, which is analogue of Bondi-Sachs coordinate conditions g rr = g ra =0 Remaining freedom (u, x A ) used to set lim r!1 A u (u, r, x A )=0, i.e. A u I + =0where I + is future null infinity Remnant freedom A B! A B + r B (x C ) is analogue of Bondi-Metzner-Sachs supertranslation freedom E-mode gauge shift on radiation field A B I +

6 Bondi-Sachs Version of Maxwell Equations Source-free Maxwell equations M := r F =0 F = F ) IDENTITY 0 r M = r r F =0 r M = 1 p ( p gm )=@ u M u + 1 r 2@ r(r 2 M r )+ 1 p A ( p qm A ) Strategy Designate M u =0and M A =0as Main Equations Designate M r =0as Supplementary Condition If Main Equations satisfied then IDENTITY implies 0=@ r (r 2 M r ) so Supplementary Condition is satisfied if it is satisfied at any specified value of r, e.g.on or at I + Main Equations separate into Hypersurface Equation M u r (r r A u )=@ r (g B A B ) Evolution Equation M A u A B = 1 ra B 1 2 r2 g C (g B A C g C A B )+ 1 rg B A u g A is covariant derivative with respect to q AB, g A = q AB g B

7 Supplementary Equation at I + M r u (r r A u )=g B (@ r A u A B + g B A u ) Integration over sphere at I + ) Conservation u I r r A u sin d d I + =0 INTERPRETATION Hypersurface r (r r A u )=@ r (g B A B ) Radial integral ) r r A u = Q(u, x A )+g B A B Null gauge A r =0so E-field component E r = F ru r A u Charge Aspect Q(u, x A ) is function of integration Integral over sphere at I + Q(u) := 1 4 H r r A u sin d d I + = 1 4 H Q(u, x A )sin d d Supplementary Condition at I + ) Charge Conservation dq(u) du =0

8 Worldtube-Null Cone Evolution Algorithm Initial data A B Nu0 Initial wordtube (boundary) r A u Su0 Worltube u A B A u (Dynamical degrees of freedom) (Gauge) Sequential radial integration scheme Hypersurface r (r r A u )=@ r (g B A B ) ) A u Nu0 Evolution u A B = 1 ra B 1 2 r2 g C (g B A C g C A B )+ 1 rg B A u u A B Nu0 Finite di erence approximation ) A B Nu0 + u Supplementary Condition r r A u = g B (@ r A u A B + g B A u ) r A u Su0 Finite di erence approximation r A u Su0 + u Given A B Nu0 + u ra u Su0 + u iteration gives finite di erence approximation for A B Nu0 +n u ra u Su0 +n u Convergent finite di erence scheme gives analytic solution

9 Bondi-Sachs Gravitational Evolution Algorithm Null coordinates x =(u, r, x A ) g uu = g ua =0, g AB = r 2 h AB, h AB h BC = A C det[h AB ]=q(x C ) ) h r h AB =0, h u h AB =0 Einstein equations: E := G 8 T =0 Bianchi identities r G =0and matter conservation r T =0 ) 0=r E = 1 p ( p ge )+ 1 2 (@ g )E Analogous to electromagnetic case, designate Main Equations: E r =0, E AB 1 2 g ABg CD E CD =0 Bianchi-matter identities imply r E r = 1 2 (@ rg )E = 1 2 (@ rg BC )E BC = 1 4 (@ rg BC )g BC (g AD E AD ) (@ r g BC )g BC = 4 r ) Trivial Equation: g AD E AD =0 Remaining components reduce r ( p ge r r ( p ge r A )=0 ) Supplementary Conditions: E r u = E r A =0 Satisfied if satisfied on worldtube or at I + At I + Supplementary Conditions give conservation laws for energy-momenum and angular momentum r 2 E r u I + =0gives rise to the famous Bondi mass loss equation

10 Main Equations Separate into Hypersurface Eqs: E r =0 Evolution Eqs: E AB 1 2 g ABg CD E CD =0 Bondi-Sachs metric g dx dx = V r e2 du 2 2e 2 dudr + r 2 h AB (dx A U A du)(dx B U B du) Hypersurface Eqs are sequence of radial ODEs for metric variables (,U r = r 16 hac h BD (@ r h AB )(@ r h CD )+2 rt r r 4 e 2 h AB (@ r U B ) =2r r ( 1 r 2 D A ) r 2 h EF D E (@ r h AF )+16 r 2 T ra 2e 2 (@ r V )=R 2[D A D A +(D A )(D A )] + e 2 r D A[@ 2 r (r 4 U A 1 )] 2 r4 e 4 h AB (@ r U A )(@ r U B ) +8 [h AB T AB r 2 T ] where D A is covariant derivative and R the Ricci scalar with respect to the conformal 2-metric h AB Vacuum Hypersurface Equations Given null data h AB Nu and worldtube data (,U r U A,V) the Hypersurface Eqs ) (,U r U A,V) Nu

11 Evolution Equations Evolution Eqs can be simplified by introducing a complex null dyad m a tangent to N u, m r u =0, whose components span the angular directions so that m = (0, 0,m A ) with h AB m A m B =0 Normalization m A m A =2, ) h AB = 1 2 (ma m B + m B m A ), m A = h AB m B ) Determines m A up to phase freedom m A! e i m A, which can be fixed by convention Evolution Eqs reduce to complex equation E AB 1 2 g ABg CD E CD =0) m A m B E AB =0 In terms of metric variables m A m B 1 { r@ r (r@ u h AB ) r h AB ) 2e D A D B e + h CA D r (r 2 U C 1 ) 2 r4 e 2 h AC h BD (@ r U C )@ r U D + r2 2 (@ rh AB )D C U C + r 2 U C D r h AB r 2 (@ r h AC )h BE (D C U E D E U C ) 8 e 2 T AB } =0 Radial ODE which determines the retarded time derivative of the two degrees of freedom in the conformal u h AB

12 Worldtube-Null-Cone Problem Given null data h AB Nu and worldtube u h AB and (,U r U A,V) which satisfy the supplementary conditions the Main Eqs determine a finite di erence approximation for h AB Nu+ u Cauchy-Characteristic Extraction Worldtube data on obtained from numerical solution of Einstein s equations carried out by Cauchy evolution of interior so that data satisfy the supplementary conditions Given the initial-boundary data the hypersurface equations can be integrated numerically in sequential order and the evolution equation can be solved using a finite di erence time-integrator Implemented as stable, convergent evolution code which propagates the exterior solution to I + The resulting waveform can be unambiguously computed at I + without near field contamination

13 Asymptotic Behavior Bondi-Sachs asymptotic solution based upon a 1/r expansion, with compact matter distribution Bondi-Sachs metric approaches Minkowski metric at I + ) lim =limu A V =0, lim r!1 r!1 r!1 r =1, lim r!1 h AB = q AB INTEGRATION OF MAIN EQUATIONS REQUIRES INITIAL DATA ON N 0 Conformal 2-metric h AB = q AB + 1 r c AB(u 0,x E )+ 1 r 2 d AB (u 0,x E )+... det(h AB )=q(x C ) ) q AB c AB =0, q AB d AB = 1 2 cab c AB Mass aspect M(u 0,x A )= 1 2 [V (u 0,r,x C ) r] I + Angular momentum aspect L A (u 0,x C )= 1 6 r U A (u 0,r,x C ) I + Schwarzschild data h AB = q AB, = L A =0, M = m ) Eddington-Finklestein ds 2 = (1 2m r )du2 2dudr + r 2 q AB dx A dx B RADIATION DATA ON I + News tensor N AB = 1 uc AB (u, x C ) Strain tensor c AB (u, x C )=r(h AB q AB ) I + Complex polarization dyad on unit sphere q AB = 1 2 (qa q B + q B q A ) q A =(1, i/ sin ) =m A I + Bondi news function N = q A q B N AB = N + in Radiation strain = 1 2 qa q B c AB = + i Asymptotic shear of N u = q A q B r A r B u I +

14 Integration of Main Equations = 1 32r 2 c AB c AB + O(r 3 ) Hypersurface r r 4 e 2 h AB (@ r U B ) = g E c AE + 1 r S A + O(r 2 ) where S A = g B (2d AB q FG c BG c AF ) Smoothness at I + ) Require S A =0 to avoid r 4 ln r term r U A Smoothness on sphere: S A =0 ) q A q B d AB =0 so that d AB = 1 2 q ABq CD d CD (See Scholarpedia article) Integration of smooth hypersurface data U A = 1 2r 2 g B c AB + 2 r 3 L A + O(r 4 ) V = r 2M(u 0,x A )+O(r 1 ) u (rh AB ) Evolution Equation Leading order ) q A q u d AB =0 Consistent with smoothness condition

15 Supplementary conditions at I + E r u supplementary equation 2@ u M = 1 2 g Ag B N AB 1 4 N ABN AB Radiation data - News tensor N AB (u, x A ) ) M(u, x A ) H BondiMass m(u) = 1 4 M(u,, )sin d d d Bondi mass loss formula du m(u) = H 1 4 N 2 sin d d E r A supplementary equation 3@ u L A = g A M 1 4 ge (g E g F c AF g A g F c EF ) c EFg A N EF 3 8 N EFg A c EF N AB g E c BE c AB g E N BE ) L(u, x A ) One motivation for calling L A (u, x A ) the angular momentum aspect results from coupling to matter where r 2 TA r is angular momentum flux of matter to I + Asymptotic integration approach shows how initial data and radiation data determine solution as 1/r expansion about I + However, this is not a well-posed evolution problem because the radiation data N AB (u, x C ), lies in the future of initial hypersurface at u 0 Assigning radiation data apriori is non-physical as opposed to determining N by evolving an interior system, as in nullcone-worldtube problems Nevertheless, approach led Bondi to the first clear understanding of mass loss due to gravitational radiation

16 Penrose compactification of I + Geometric picture of asyptotic behavior Metric of a conformally compactified spacetime ĝ = 2 g related to physical metric g by conformal factor satisfying =0at I + Distance to I + is finite measured by ĝ Asymptotic flatness requires I + has topology R S 2 and ˆr I + 6= 0, ˆr is covariant derivative associated with ĝ Einstein Equations Conformal and physical Ricci tensors related by 2 R = 2 ˆR +2 ˆr ˆr + ĝ [ ˆr ˆr 3( ˆr ) ˆr ] Trace of vacuum Einstein equations R =0at I + ) 0=(ˆr ) ˆr I + ) I + is null with tangent n = ˆr Next order in ) 0=[ˆr ˆr 1 4ĝ ˆr ˆr ] I + Preferred Conformal Factors Existence of a conformal transformation!! such that ˆr ˆr!! ˆr ˆr + ˆr ˆr! =0 (ODE for!) ) Preferred conformal factors such that ˆr ˆr I + =0 I + is shear and divergence free

17 Bondi-Sachs Compactification Coordinates ˆx =(u, `, x A ) of compactified space can be obtained from Bondi-Sachs coordinates of physical space by transformation ˆx =(u, `, x A )=(u, 1/r, x A ) Inverse areal coordinate ` =1/r is also preferred conformal factor =` which leads to the conformal Bondi-Sachs metric ĝ ab dˆx a dˆx b = `3Ve 2 du 2 +2e 2 dud` + h AB (dx A U A du)(dx B U B du) Smoothness of ĝ ab (at least C 3 ) implies 1/r = ` expansion at I + which rules out troublesome ln behavior and leads to peeling property Ĉ = O(`) Einstein equations determine general leading coe the conformal space metric h AB = H AB (u, x C )+`c AB (u, x c )+O(`2) = H(u, x C )+O(`2) U A = H A (u, x C )+2`e 2H H AB D B H + O(`2) `2V = D A H A + `( 1 2 R + DA D A e 2H )+O(`2) cients of where R is Ricci scalar and D A the covariant derivative associated with H AB Leading coe cients H, H A and H AB do not correspond to an asymptotic inertial frame - Pure gauge

18 Construction of Bondi-Sachs Inertial Coordinates General leading coe cients of the conformal space metric h AB = H AB (u, x C )+`c AB (u, x c )+O(`2) det[h AB ]=q(x C ) = H(u, x C )+O(`2) U A = H A (u, x C )+2`e 2H H AB D B H + O(`2) `2V = D A H A + `( 1 2 R + DA D A e 2H )+O(`2) ĝ I + = (H, H A,H AB ) Gauge terms Tangent to null geodesics on I + 0 e 2H 0 1 e 2H 0 H A e 2H A 0 H A e 2H H AB n I + = ĝ ab ˆrb` I + =(e 2H, 0, e 2H H A ) Inertial angular coordinates a x A I + =0) H A =0 Inertia retarded time coordinate u - a ne parameter u I + =1 ) H =0 ) I + u p ` is preferred conformal factor: det[ĝ ]=e p 2 q(x C ) ) ˆr ˆn I + = 1 e 2 p (e 2 p qĝ` ) I + = 1 e 2 p u(e 2 p qe 2 ) I + =0 )I + is divergence free and shear free u H AB =0 Allows u-independent conformal transformation `!!(x C )` such that H AB! q AB ) Cross-sections of I + have unit sphere geometry )R=2 Bondi inertial conformal frame H = H A =0, H AB = q AB

19 Universal Structure of I + Conformal metric in inertial Bondi-Sachs coordinates ĝ dˆx dˆx = `2g dx dx ˆx =(u, `, x A ) ` =1/r ĝ dˆx dˆx = `3Ve 2 du 2 +2e 2 dud` + h AB (dx A U A du)(dx B U B du) h AB = q AB + `c AB (u, x C )+O(`2) = O(`2) U A = 2`2g 1 B c AB +2L A`3 + O(`4) `3V = `2 2M`3 + O(`4) ĝ I + = A 00q AB Shear-free property of I + ) ` 1 ˆr ˆr ` has finite limit News tensor N = ` 1 ˆr ˆr ` I + N is unchanged under =`!!`,!>0 News tensor is geometrically defined tensor field on I + independent of choice of conformal factor or u-foliation

20 Bondi-Metzner-Sachs (BMS) Group BMS group is asymptotic symmetry group Generators satisfy physical space Killing s equation at I + 2 L g I + = 2 2 r ( ) I + =0 L is Lie derivative Bondi inertial conformal metric, =` [ ˆr ( ) ` `]`=0 =0 ` =0 is tangent to I + and ` ` I + ` I + ) +ĝ ĝ `] I + =0 where ĝ I A 00q AB Only ĝ I + enters ) simple to analyze Example `A-component A +ĝ `] I + u A I + =0 ) A I + = f A (x C ) General solution for BMS `=0 =[ (x C )+ u 2 g Bf B (x C )]@ u + f A (x C )@ A f A (x C ) is a conformal killing vector on unit sphere g (A f B) 1 2 qab g C f C =0

21 BMS Group `=0 =[ (x C )+ u 2 g Bf B (x C )]@ u + f A (x C )@ A Supertranslations f A =0 u! u + (x C ) Conformal transformations of unit sphere =0 Isomorphic to the orthochronous Lorentz transformations Supertranslations: infinite dimensional invariant subgroup Supertranslations consisting of l =(0, 1) spherical harmonics, = a + a x sin cos + a y sin sin + a z cos, form an invariant 4-dimensional group of time and space translations ) Unambiguous definition of energy-momentum Poincare group consisting of Lorentz transformations and space-time translations is not invariant subgroup There is a supertranslation freedom in Lorentz transformations analogous to translation freedom (change in origin) in special relativity Only in special cases, such as non-radiative spacetimes, can a preferred Poincare group be singled out

22 The Supertranslation Ambiguity A given cross-section + of I + picks out a preferred rotation subgroup which maps + into itself A supertranslation can map any given cross-section into any other cross-section ) Supertranslation ambiguity in rotation subgroup Without introducing a preferred structure there is no invariant way to extract a preferred rotation group, Lorentz group or Poincare group from BMS group Supertranslation Ambiguity in the Radiation Strain Under transformation ũ = u + (x A )+O(`), with x A = x A, the radiation strain has supertranslation gauge freedom (u, x C )=q A q B r A r B ũ I + = (u, x C )+q A q B g A g B (x C ) At a non-radiative time a preferred Poincare group can be singled out by requiring electric (E-mode) component of the shear to satisfy E =0 Determines up to time and space translation freedom q A q B g A g B = g 2 =0) = a + a x sin cos + a y sin sin + a z cos

23 Supertranslation Ambiguity In Angular Momentum For any cross-section + of I + there is geometrically defined surface integral Q ( + ) for each BMS generator (Wald-Zoupas) Time and space translations lead to unambiguous construction of Bondi energy-momentum 4-vector AMBIGUITY IN ANGULAR MOMENTUM IN TRANSITION BETWEEN NON-RADIATIVE u (u, x C ) u=±1 =0 Initial state at u = 1 picks out inertial frame (u = 1) =0 and preferred Poincare group with rotation generators = R with associated angular momentum Q R Initial inertial frame and its preferred rotation generators extend uniquely to all I + but (u =+1) 6= 0in general Final state picks out inertial frame (ũ = +1) = 0 and Poincare group with rotation generators = R+ with associated angular momentum Q R+ If ũ and u di er by a supertranslation the rotation generators R + and R di er and the corresponding angular momentum di er by supermomenta similar to the Poincare relation L! L + R P under a translation of origin by R Does this give rise to a distinctively general relativistic mechanism for angular momentum loss via supermomenta?