Gravitational-wave memory observables and charges of the extended BMS algebra David A. Nichols 1 1 Dept. of Astrophysics, IMAPP, Radboud University Nijmegen In collaboration with ´ Eanna ´ E. Flanagan and Abraham I. Harte Yukawa Institute for Theoretical Physics—NPCSM Meeting 10 November 2016 David A. Nichols GW memory and extended BMS charges
Outline of introduction and summary of results Gravitational-wave (GW) observations as probe of nonlinear and dynamical regime of general relativity (GR) GW memory as an example of nonlinear, dynamical GR Qualitative review of GW memory Description of asymptotic symmetries and charges New memories from new symmetries of gravitational scattering Summary of work on computations of charges (“conserved” quantities) and memory observables More details about calculations after introduction David A. Nichols GW memory and extended BMS charges
Gravitational-wave (GW) detections of binary black holes (BBHs) LIGO Scientific Collaboration, arXiv:1606.04856 GW150914: > 5 σ , m 1 ≈ 36 M ⊙ , m 2 ≈ 29 M ⊙ , D L ≈ 420 Mpc GW151226: > 5 σ , m 1 ≈ 14 M ⊙ , m 2 ≈ 7 . 5 M ⊙ , D L ≈ 440 Mpc LVT151012: ∼ 2 σ , m 1 ≈ 23 M ⊙ , m 2 ≈ 13 M ⊙ , D L ≈ 1 Gpc David A. Nichols GW memory and extended BMS charges
Observational GR on new lengthscales. . . Yagi et al., arXiv:1603.08955 David A. Nichols GW memory and extended BMS charges
. . . and on new timescales Yagi et al., arXiv:1603.08955 David A. Nichols GW memory and extended BMS charges
Tests of general relativity (GR) with BBHs Ψ( f ) ∼ � h ( f ) ∼ Ae − i Ψ( f ) j ( p GR + δ p j ) f ( j − 5) / 3 j LIGO Scientific Collaboration, arXiv:1602.04856 David A. Nichols GW memory and extended BMS charges
Memory effect from GW150914 in LIGO Lasky et al., arXiv:1605.01415 David A. Nichols GW memory and extended BMS charges
Building evidence for memory by stacking detections Lasky et al., arXiv:1605.01415 David A. Nichols GW memory and extended BMS charges
Memory: Several perspectives on the phenomenon Spacetime quantity M. Favata, arXiv:0811.3451 David A. Nichols GW memory and extended BMS charges
Memory: Several perspectives on the phenomenon Spacetime quantity Measurable effect t M. Favata, arXiv:0811.3451 No Memory With Memory David A. Nichols GW memory and extended BMS charges
Memory: Several perspectives on the phenomenon Spacetime quantity Measurable effect t M. Favata, arXiv:0811.3451 Sources Null No Memory With Memory Ordinary � �� � � � � ���� + r 2 T GW “∆ h ” ∼ ∆ m + du T uu uu � �� � � �� � Linear Nonlinear David A. Nichols GW memory and extended BMS charges
Memory: Several perspectives on the phenomenon Spacetime quantity Measurable effect t M. Favata, arXiv:0811.3451 Sources Null No Memory With Memory Ordinary � �� � � � � ���� Symmetries + r 2 T GW “∆ h ” ∼ ∆ m + du T uu uu BMS supertranslation � �� � � �� � (next slide. . . ) Linear Nonlinear David A. Nichols GW memory and extended BMS charges
Overview of asymptotic symmetries Symmetry group of asymptotically flat spacetimes ( I + ) is the Bondi-Metzner-Sachs (BMS) group Bondi et al., 1962; Sachs, 1962 BMS has semidirect form: Supertranslations (ST) ⋊ Lorentz (Poincar´ e: Translations ⋊ Lorentz) ST: infinite-dimensional, abelian, 4D translation subgroup; roughly “angle-dependent translations” R. Penrose, Les Houches, 1963 Corresponding charges: Memory related to 4-momentum, supermomentum, supertranslation between and relativistic angular momentum early and late (spin and center of mass) non-radiative frames David A. Nichols GW memory and extended BMS charges
Memory effects and symmetries: Recent developments Extended symmetry groups Barnich & Troessaert ’09+: extend BMS algebra to include locally defined (but not globally defined) symmetries Extended algebra: ST ⋊ Virasoro Virasoro called “superrotations” (SR) in the context of 4D asymptotically flat case Intuition for SR: contains Lorentz subalgebra; “angle-dependent rotations and boosts” Showed certain charges are finite and well defined David A. Nichols GW memory and extended BMS charges
Physical relevance of extended symmetries Digression on charges, memories, symmetries of gravitational scattering Strominger,+ ’13+: Identify BMS subgroup of past ( I − ) and future ( I + ) null infinity in a class of spacetimes α = Q + Supertranslation charges related: Q − α S matrix satisfies: � out | ( Q + α S − SQ − α ) | in � = 0 In particle basis: lim ω → 0 M n +1 = S (0) M n with M n n -particle amplitude and S (0) related to memory effect “Triangle” of relations: soft theorem ⇔ BMS symmetry ⇔ memory effect Similar types of relations between subleading soft theorem, extended BMS symmetry, and a new “spin” memory effect (Pasterski+, ’15) David A. Nichols GW memory and extended BMS charges
Overview of Results 1 Review asymptotic flatness, symmetries, and charges Show how supermomentum charges are related to “ordinary” memory 2 Compute charges conjugate to superrotation symmetries in more general contexts than before Find charges contain information about the total memory and the ordinary spin memory 3 Investigate the spin memory Show it can be measured inertially, but not locally in space 4 Look for other intertial memory effects Besides displacement effect, there are proper-time, rotation, and velocity memory effects Relative displacement is the only effect that is locally measurable at O (1 / r ) David A. Nichols GW memory and extended BMS charges
Outline of details 1 Asymptotically flat spacetimes, in brief 2 Charges (“conserved” quantities) of the BMS group 3 Memory effects and charges 4 Extended BMS algebra and its charges 5 Relation of extended charges and memory effects 6 Search for additional classical memory observables David A. Nichols GW memory and extended BMS charges
Bondi-Sachs framework Work in Bondi coordinates ( u , r , θ A ): ds 2 = − du 2 − 2 dudr + r 2 h AB d θ A d θ B + 2 m r du 2 + rC AB d θ A d θ B + D B C AB d θ A du + . . . David A. Nichols GW memory and extended BMS charges
Bondi-Sachs framework Work in Bondi coordinates ( u , r , θ A ): ds 2 = − du 2 − 2 dudr + r 2 h AB d θ A d θ B + 2 m r du 2 + rC AB d θ A d θ B + D B C AB d θ A du + . . . θ A : coordinates on S 2 with 2-metric h AB and covariant derivative operator D A m = m ( u , θ A ): Bondi mass aspect C AB = C AB ( u , θ C ): shear tensor (symmetric trace-free) David A. Nichols GW memory and extended BMS charges
Bondi-Sachs framework Work in Bondi coordinates ( u , r , θ A ): ds 2 = − du 2 − 2 dudr + r 2 h AB d θ A d θ B + 2 m r du 2 + rC AB d θ A d θ B + D B C AB d θ A du + . . . θ A : coordinates on S 2 with 2-metric h AB and covariant derivative operator D A m = m ( u , θ A ): Bondi mass aspect C AB = C AB ( u , θ C ): shear tensor (symmetric trace-free) N AB = ∂ u C AB : news tensor (vanishes when stationary) N A = N A ( u , θ B ): Bondi angular-momentum aspect David A. Nichols GW memory and extended BMS charges
Einstein equations and initial data m = ∂ u m and ˙ Einstein equations (evolution equations for ˙ N A ) T uu − 1 8 N AB N AB + 1 m = 4 π ˆ 4 D A D B N AB ˙ where T uu = ˆ T uu ( u , θ A ) / r 2 David A. Nichols GW memory and extended BMS charges
Einstein equations and initial data m = ∂ u m and ˙ Einstein equations (evolution equations for ˙ N A ) T uu − 1 8 N AB N AB + 1 m = 4 π ˆ 4 D A D B N AB ˙ where T uu = ˆ T uu ( u , θ A ) / r 2 T rr + D A m + 1 N A = − 8 π ˆ ˙ T uA + π D A ∂ u ˆ 4 D B D A D C C BC − 1 4 D B D B D C C CA + 1 4 D B ( N BC C CA ) + 1 2 D B N BC C CA . and T uA = ˆ T uA ( u , θ B ) / r 2 , T rr = ˆ T rr ( u , θ A ) / r 4 David A. Nichols GW memory and extended BMS charges
Einstein equations and initial data m = ∂ u m and ˙ Einstein equations (evolution equations for ˙ N A ) T uu − 1 8 N AB N AB + 1 m = 4 π ˆ 4 D A D B N AB ˙ where T uu = ˆ T uu ( u , θ A ) / r 2 T rr + D A m + 1 N A = − 8 π ˆ ˙ T uA + π D A ∂ u ˆ 4 D B D A D C C BC − 1 4 D B D B D C C CA + 1 4 D B ( N BC C CA ) + 1 2 D B N BC C CA . and T uA = ˆ T uA ( u , θ B ) / r 2 , T rr = ˆ T rr ( u , θ A ) / r 4 At u = u 0 , specify m ( u 0 , θ C ), N A ( u 0 , θ C ), C AB ( u 0 , θ C ) News N AB unconstrained; also certain components of T ab David A. Nichols GW memory and extended BMS charges
ξ τ Nearby freely falling observers Primary geodesic observer, P: u ( ) 4-velocity � u P ( τ ) + δτ ( ) P 2 u ξ 2 S τ 2 Secondary geodesic observer, S: 4-velocity � u S ( τ ) At τ 1 , P and S co-moving; S at location ξ ˆ i S P 1 u ( ) τ 1 u ( ) P 1 τ 1 S u ( τ ): 4-velocity � ξ = ξ ˆ � i � i ( τ ): “separation” e ˆ David A. Nichols GW memory and extended BMS charges
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