Nonlinear gravitational waves Optics, scattering, and Huygens - PowerPoint PPT Presentation

Nonlinear gravitational waves Optics, scattering, and Huygens’ principle Abraham Harte Max-Planck-Institut f¨ ur Gravitationsphysik Albert-Einstein-Institut Potsdam, Germany September 24, 2014 Astrophysical relativity seminar Abraham Harte Nonlinear gravitational waves September 24, 2014 1 / 30

Perspectives on gravitational waves in GR 1 Perturbation theory 2 Scri 3 Exact solutions Each of these provides different insights. Focus on option 3. . . Abraham Harte Nonlinear gravitational waves September 24, 2014 2 / 30

This talk explores the physics of exact gravitational plane waves. What can plane waves model? Generic gravitational waves far from a source Penrose limit : Any spacetime looks like a plane wave along a null geodesic [Penrose (1976), Blau, Frank, & Weiss (2006)] Used to find universal effects of caustics on field propagation [AIH & Drivas (2012)] Abraham Harte Nonlinear gravitational waves September 24, 2014 3 / 30

Plane waves are interesting 1 Dictionary for linearized theory isn’t trivial O ( h 2 ) coefficients can be enormous uniqueness 2 “Unique counterexamples to almost everything,” yet also simple 3 Rich phenomenology: free functions vs. parameters Abraham Harte Nonlinear gravitational waves September 24, 2014 4 / 30

Organization Two parts: 1 Perturbative vs. non-perturbative plane waves 2 Wave-wave scattering in GR Abraham Harte Nonlinear gravitational waves September 24, 2014 5 / 30

Part I: What are plane waves? Exact vs. approximate Abraham Harte Nonlinear gravitational waves September 24, 2014 6 / 30

Plane waves in linearized GR Vacuum plane wave in + Z direction in transverse-traceless (TT) gauge: ds 2 = − dT 2 + ( δ ij + h ij ) dX i dX j + dZ 2 + O ( h 2 ) , i , j = 1 , 2 � h + ( u ) � 1 h × ( u ) h ij ( X λ ) = √ , u = ( T − Z ) h × ( u ) − h + ( u ) 2 1 2 free functions in one variable → 2 polarizations 2 Objects at fixed X , Y , Z are in free-fall → geodesic coordinates Abraham Harte Nonlinear gravitational waves September 24, 2014 7 / 30

Exact plane waves Generalizing the linear result directly is a bad idea Exact solution found in 1923 [Brinkmann] , but not interpreted for 30+ years 1 Find geometries with “the same” symmetries as EM plane waves in flat spacetime [Bondi, Pirani, Robinson (1959)] : ds 2 = − 2 dudv + H ij ( u ) x i x j du 2 + | d x | 2 2 Impose exact Einstein’s equation: tr H ( u ) = − 8 π T uu Abraham Harte Nonlinear gravitational waves September 24, 2014 8 / 30

Exact plane waves II Arbitrary vacuum plane waves described by STF matrices H ij ( u ): 1 Again, there are 2 polarizations 2 H ij is local ( ∝ R abcd ) and almost gauge-invariant Relation to linear-wave h ij is not obvious. . . Abraham Harte Nonlinear gravitational waves September 24, 2014 9 / 30

Exact vs. perturbative plane waves Find coordinates s.t. constant X , Y , Z worldlines are geodesic: ds 2 = − 2 dudv + H ij ( u ) x i x j du 2 + | d x | 2 (Brinkmann) = − dT 2 + H ij ( u ) dX i dX j + dZ 2 (Rosen) �� In perturbation theory, H ij = δ ij + 2 H ij + . . . H ( u ) = (transverse 2-metric) = E ⊺ ( u ) E ( u ) d 2 E du 2 = HE Abraham Harte Nonlinear gravitational waves September 24, 2014 10 / 30

Which waveform? Curvature H ij TT-like H ij Einstein’s equation linear, algebraic nonlinear ODE Coordinate singularities no yes Locality local nonlocal Uniqueness simple complicated(!) Generalizable only slightly yes Although TT-like gauges have problems, people have intuition for them Lots of observables are nicer in terms of H instead of H Abraham Harte Nonlinear gravitational waves September 24, 2014 11 / 30

TT-like waveform was H = E ⊺ E with ¨ E = HE Almost everything in plane wave spacetimes is governed by this equation. Essentially coupled parametric oscillators Similar things show up everywhere: 1 Schr¨ odinger equation 2 celestial mechanics 3 geodesic deviation in general spacetimes 4 kids on swings Abraham Harte Nonlinear gravitational waves September 24, 2014 12 / 30

Lots of powerful methods Runaway instabilities in parts of parameter space More modestly, linear approximations always fail on large scales: 1.2 1.0 0.8 0.6 0.4 0.2 5 10 15 20 One eigenvalue of H for linearly-polarized monochromatic wave with ω = 2 π , h = [0 . 01 , 0 . 04] Abraham Harte Nonlinear gravitational waves September 24, 2014 13 / 30

Weak, monochromatic, linearly-polarized waves Through O ( h 3 ), � 1 − 1 � � 1 � 0 8( h ω u ) 2 + . . . H = 0 1 �� 1 − 1 � � � 1 � 0 32( h ω u ) 2 + h cos ω u + . . . 0 − 1 Rapidly-growing trace at O ( h 2 ) [mimics alternative gravities] Rapidly-growing oscillation at O ( h 3 ) Complete breakdown of linear theory when (# oscillations) � h − 1 Abraham Harte Nonlinear gravitational waves September 24, 2014 14 / 30

Gravitational lensing Interesting optics when observing distant objects “through” a GW: Frequency shifts (cf. pulsar timing arrays) Wiggling on the sky Twinkling Exact formulae for all of these involve H − 1 ( u o ), H − 1 ( u s ), � H − 1 � [AIH (2013), AIH & S. Babak (in prep.)] Much better to use H instead of H − I Abraham Harte Nonlinear gravitational waves September 24, 2014 15 / 30

Pulsar timing � ω s det � H − 1 � λ o � = 1 + 1 2(1 + cos ψ ) − 1 (det H − 1 o H − 1 λ s ω o s ) 1 / 2 × [( H − 1 o − ( H − 1 s ) ij n i o n j o ) ij n i s n j s ] 1 � H − 1 � − 1 ( X o − X s → o ) n o ≡ d ang ψ = (angle between source and GW) d ang = (angular diameter distance) λ o /λ s ∼ (Doppler along wave direction) [ . . . ] ∼ (Doppler transverse to wave direction and curvature) Abraham Harte Nonlinear gravitational waves September 24, 2014 16 / 30

Part II: Wave(-wave) effects How do gravitational plane waves affect other waves? Abraham Harte Nonlinear gravitational waves September 24, 2014 17 / 30

Wave propagation in general We’re used to waves traveling essentially without distortion: Everyday life would be very different without this. But it’s not always true: There can be dispersion, scattering, echoes, etc. Abraham Harte Nonlinear gravitational waves September 24, 2014 18 / 30

Spacetime viewpoint Waves generically travel both along characteristics and behind them : The leftovers are usually called tails in relativistic contexts. Abraham Harte Nonlinear gravitational waves September 24, 2014 19 / 30

Huygens’ principle (Hadamard’s minor premise) There are no tails. This holds for the usual flat-spacetime wave equation: ∂ t 2 + ∂ 2 ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 � − 1 � φ ( t , x , y , z ) = 0 . c 2 ∂ z 2 It fails for almost everything else: Huygens’ principle is very special. Abraham Harte Nonlinear gravitational waves September 24, 2014 20 / 30

Waves propagating on curved spacetimes For scalar, electromagnetic, spin- 1 2 fields, Gravitational plane waves don’t scatter non-grav. fields [McLenaghan (1969), G¨ unther & W¨ unsch (1974)] . . . the only curved vacuum backgrounds for which Huygens’ principle holds are plane waves. But what about gravitational waves? ( sort of... ) Abraham Harte Nonlinear gravitational waves September 24, 2014 21 / 30

Gravitational wave-wave scattering Do weak gravitational waves (of arbitrary geometry) pass through strong plane waves without distortion? It’s sufficient to consider a single impulsive burst: Abraham Harte Nonlinear gravitational waves September 24, 2014 22 / 30

Impulsive solutions → Green functions In Lorenz gauge, linearized Einstein is h cd = − 16 π T ab . ∇ c ∇ c ¯ h ab + 2 R acbd ¯ So consider ∇ c ∇ c G aba ′ b ′ + 2 R acbd G cda ′ b ′ = − 4 π g ( a a ′ g b ) b ′ δ ( x , x ′ ) . h ab computable by integrating G aba ′ b ′ (even vacuum solns). Arbitrary ¯ Abraham Harte Nonlinear gravitational waves September 24, 2014 23 / 30

Green functions in general [Hadamard, DeWitt, . . . ] Retarded Green functions always look like: G aba ′ b ′ = [∆ 1 / 2 g aa ′ g bb ′ δ ( σ ) + V aba ′ b Θ( − σ )] ret with σ ( x , x ′ ) = 1 2(squared geodesic distance) g aa ′ ( x , x ′ ) = (parallel propagator) ∆( x , x ′ ) = (measure of focusing for null geodesics) V aba ′ b ′ ( x , x ′ ) = (tail) Abraham Harte Nonlinear gravitational waves September 24, 2014 24 / 30

V aba ′ b ′ satisfies a characteristic initial value problem. It does not vanish for plane wave backgrounds. But this only means that h ab has a tail in Lorenz gauge . Could the tail be pure gauge? [In Lorenz-gauge EM, F tail = 2 ∇ [ a A tail = 0 even though A tail � = 0] a ab b ] Abraham Harte Nonlinear gravitational waves September 24, 2014 25 / 30

Checking for gauge effects Scalars like ∇ a R bcdf ∇ a R bcdf , R abcd R abcd , R abcd R ∗ abcd , . . . vanish in the background, so their perturbations are gauge-invariant. Tail perturbations vanish for all curvature scalars. All geometries with vanishing curvature scalars have been classified [Pravda et al. (2002)] Abraham Harte Nonlinear gravitational waves September 24, 2014 26 / 30