Gravitational Wave Memories and Asymptotic Charges in General Relativity

Transcription

1 Gravitational Wave Memories and Asymptotic Charges in General Relativity Éanna Flanagan, Cornell General Relativity and Gravitation: A Centennial Perspective Penn State 8 June 2015 EF, D. Nichols, arxiv: ; EF, D. Nichols and A. Harte, in preparation

2 Motivations The Bondi-Metzner-Sachs asymptotic symmetry group (1960s) is a qualitatively new feature of general relativity. Implications for gravitational wave observations (memory) and numerical relativity. Relevant to quantum gravity, AdS/CFT etc Subject is not completely understood. This talk: highlights of some recent developments: New types of gravitational wave memory How to measure BMS charges

3 Gravitational Wave Memory: Review The permanent relative displacement of a pair of freely falling test masses caused the passage of a burst of gravitational waves (Zel dovich and Polnarev 1974) Geodesic deviation: Z Z h dt dt 0 (Riemann) î! ( îĵ + hîĵ ) ĵ with (non- Intuition: r radiative, non-stationary) h Q(t r,, ')! Q(, ') r

4 Two Types of Memory Derived by Christoudoulu (1991), clarified by Bieri and Garfinkle (2014), who performed a gauge invariant analysis in linearized gravity using the Bianchi identity. Ordinary Memory Computable in terms of change in lim between r!1 r3 C trtr i 0 and i + BMS supermomentum" Null Memory Computable in terms of de dtd 2 (u,, ') at null infinity Formerly called nonlinear memory

5 Gravitational Wave Spin Memory Pasterski, Strominger and Zhiboedov (2015) discovered a new type of memory called spin memory, related to angular momentum flux to null infinity rather than energy flux Normal memory associated with an infinite set of conserved charges (BMS group); spin memory similarly related to new infinite set of conserved charges associated with conjectured extension of BMS group. (Barnich and Troessaert 2009) Measurement procedure: Comoving observer in Bondi coordinates Sagnac interferometer Observable: Z dt t sagnac

6 Gravitational Wave Spin Memory Pasterski, Strominger and Zhiboedov (2015) discovered a new type of memory called spin memory, related to angular momentum flux to null infinity rather than energy flux Normal memory associated with an infinite set of conserved charges (BMS group); spin memory similarly related to new infinite set of conserved charges associated with conjectured extension of BMS group. (Barnich and Troessaert 2009) Measurement procedure: Observable: Z Comoving observer in Bondi coordinates Sagnac interferometer dt t sagnac Goal: find a gauge invariant description of spin memory Stragegy: use Bieri- Garfinkle method and search for new observables

7 Four Gauge Invariant Memories Setup: P A 1. Fix an event on worldline of freely falling observer, a spatial vector ~ at P, and a proper time interval A 2. Parallel transport ~ back by A to Q 3. Exponentiate ~ to get event R 4. Parallel transport ~u A to get ~u B at R 5. Solve for geodesic of observer B 6. Find unique spatial vector ~ 0 at P whose exponential intersects B s world line at an event S

8 Four Gauge Invariant Memories Displacement memory: ~ ~ 0 at P Proper time memory: A B Velocity memory: (Strominger & Zhiboedov 2014) u a A(P) g a a 0 u a0 B (S) (Grishchuk & Polnarev 1989, Tolish & Wald 2014, Harte 2015) Frame Dragging memory: The net relative rotation of gyroscopes carried by A and B, compared by parallel transport: i = B ij j at P

9 Four Gauge Invariant Memories Displacement memory: ~ ~ 0 at P Proper time memory: A B Velocity memory: (Strominger & Zhiboedov 2014) u a A(P) g a a 0 u a0 B (S) (Grishchuk & Polnarev 1989, Tolish & Wald 2014, Harte 2015) Frame Dragging memory: The net relative rotation of gyroscopes carried by A and B, compared by parallel transport: i = B ij j at P These observables are captured by a generalized holonomy around the loop PQRS, defined by solving t a r a apple b = t b, t a = dx a /d to give Poincare map apple a! a bapple b + apple a

10 Bieri-Garfinkle Computational Method Linear perturbation theory, extend Bieri & Garfinkle to subleading order in 1/r. Bianchi identities in terms of electric E ij and magnetic B ij pieces of Weyl tensor:

11 Bieri-Garfinkle Computational Method Linear perturbation theory, extend Bieri & Garfinkle to subleading order in 1/r. Coordinates (u, r, A ) and metric ds 2 = du 2 2dudr + r 2 AB d A d B Bianchi identities in terms of Expand all quantities as electric E ij and magnetic B ij f(u, r, A )= X 1 pieces of Weyl tensor: k=0 Displacement memory is Z Z du Two pieces du 0 EÂ ˆB O(1/r)+O(1/r 2 ) Electric parity r s+k f (k) (u, A ) E (0) AB =(D 1 AD B 2 D2 AB ) + C (A D B)D C Magnetic parity

12 Assumptions on Stress-Energy Falloff

13 Results for Memory Observables All new observables are subleading in E type displacement memory: with D = D 2 + D 4 /2 Z Z 1/r, except one = D 1 apple E (0) rr + Z T (0) uu + O 1 r Proper time, velocity and frame dragging memories: all all computable from T uu (0), E rr (0), E (1) ra O(1/r) and B type displacement memory: T (0) Z Z = D 1 h B (0) rr i 1 r D 1 applez ua = D A B (0) rr +2D 2 Z + AB D B B (1) rr + O 1 r 2

14 Results for Memory Observables All new observables are subleading in E type displacement memory: with D = D 2 + D 4 /2 Z Z 1/r, except one = D 1 apple E (0) rr + Z T (0) uu + O 1 r Proper time, velocity and frame dragging memories: all all computable from T uu (0), E rr (0), E (1) ra O(1/r) and B type displacement memory: T (0) Z Z = D 1 h B (0) rr vanishes (Winicour) Strominger observable is really Z i.e. dt h ij (t), measurable by LIGO i 1 r D 1 ua = D A Z Z Z + AB D B = D 1 Z B (0) rr Disagreement on contribution by angular momentum flux?? applez B (0) rr +2D 2 Z B (1) rr + O + O 1 r 2 1 r

15 Memory Effects: Summary Spin Memory: there is a leading order, non-vanishing magnetic parity effect; the observable is the time integral of the strain Other memory observables likely too small to be observationally important.

16 BMS Conserved Charges: Review Spacetimes which are asymptotically flat at null infinity have a universal background structure there: a vector field n a and a (0, +, +) metric h ab such that n a h ab =0 and is a conformal Killing vector. Pairs, are equivalent if h ab =! 2 h ab n a =! 1 n a n a (h ab,n a ) ( h ab, n a ) Diffeomorphisms of null infinity which preserve this structure give the BMS group. Explicit parameterization: (u,, ') =(u, A ), ds 2 = d 2 +sin 2 d' 2 ~n u ad bc =1 z 0 = az + b cz + d u 0 = 1 w [u + (z, z)] w 1 = z = cot( /2) exp[i'] 1+ z 2 az + b 2 + cz + d 2

17 BMS Conserved Charges: Review Spacetimes which are asymptotically flat at null infinity have a universal background structure there: a vector field n a and a (0, +, +) metric h ab such that n a h ab =0 and is a conformal Killing vector. Pairs, are equivalent if h ab =! 2 h ab n a =! 1 n a n a (h ab,n a ) ( h ab, n a ) Diffeomorphisms of null infinity which preserve this structure give the BMS group. Explicit parameterization: (u,, ') =(u, A ), ds 2 = d 2 +sin 2 d' 2 ~n u ad bc =1 z 0 = az + b cz + d u 0 = 1 w [u + (z, z)] w 1 = z = cot( /2) exp[i'] 1+ z 2 az + b 2 + cz + d 2 There is a generalization of Noether s theorem that allows the construction of conserved charges and fluxes for each generator (Wald 2000, Dray & Streubel 1984) charge = P a t a 1 2 J ab! ab + X l 2,m P lm lm

18 How to measure BMS Conserved Charges? In principle can be measured by a set of observers on a 2-surface who measure the geometry ( surface integral ). What happens for observes who try to use methods of linearized general relativity to measure charges? Analogous to Newtonian observers making measurements in Special Relativity. There, inconsistencies (Lorentz contraction etc) arise due to unexpected dependence on observer s Lorentz frame. So expect inconsistencies due to dependence on observer s asymptotic Bondi frame. How does this work in detail? Local measurement algorithm: At an event P, observer measures R abcd (P), r a R bcde (P), r a r b R cdef (P). From these she can compute P a (P), J ab (P) which work in linearized, stationary spacetimes

19 How can Observers Compare Charges? The inhomogeneous parallel transport law t a r a v b = t a can be used to transport along curves. Equivalent to Associated generalized holonomy (i.e. memory) around measures obstruction to consistency of measurements: (P, J) r ~ ~ t P =0, r ~ t J = P ~ ^ ~t C (P, J)! (P 0,J 0 ) 6= (P, J) Not good enough: the generalized holonomy is not trivial near null infinity in stationary regions, where preferred charges exist. Improved transport law P a = R a bcdt b J cd /2, J a = 2P [a t b] (J. Vines; A. Harte) Conjecture: For this law, transport around curves becomes trivial near null infinity in stationary regions. Framework: Observers measure charges locally and transport them to give consistent answers in stationary regions. Comparisons between different stationary regions yield inconsistencies as measured non-locally by holonomies.

20 BMS Charge Measurements: Summary We have a operational understanding of measurements for stationary to stationary transitions (assuming validity of a conjecture). We would like to generalize this to nowhere stationary spacetimes, i.e. to give an operational definition of asymptotic Bondi frame and an operational definition of supermomentum measurements.