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)]

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Plane waves are interesting

1 Dictionary for linearized theory isn’t trivial

O(h2) 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

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Plane waves in linearized GR

Vacuum plane wave in +Z direction in transverse-traceless (TT) gauge: ds2 = −dT 2 + (δij + hij)dX idX j + dZ 2 + O(h2), i, j = 1, 2 hij(X λ) = h+(u) h×(u) h×(u) −h+(u)

• ,

u = 1 √ 2 (T − Z)

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)]: ds2 = −2dudv + Hij(u)xixjdu2 + |dx|2

2 Impose exact Einstein’s equation:

tr H(u) = −8πTuu

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Exact plane waves II

Arbitrary vacuum plane waves described by STF matrices Hij(u):

1 Again, there are 2 polarizations 2 Hij is local (∝ Rabcd) and almost gauge-invariant

Relation to linear-wave hij is not obvious. . .

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Exact vs. perturbative plane waves

Find coordinates s.t. constant X, Y , Z worldlines are geodesic: ds2 = −2dudv + Hij(u)xixjdu2 + |dx|2 (Brinkmann) = −dT 2 + Hij(u)dX idX j + dZ 2 (Rosen) In perturbation theory, Hij = δij + 2

• Hij + . . .

H(u) = (transverse 2-metric) = E ⊺(u)E(u) d2E du2 = HE

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Which waveform?

Curvature Hij TT-like Hij Einstein’s equation linear, algebraic nonlinear ODE Coordinate singularities no yes Locality local nonlocal Uniqueness simple complicated(!) Generalizable

• nly 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

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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¨

• dinger 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:

5 10 15 20 0.2 0.4 0.6 0.8 1.0 1.2

One eigenvalue of H for linearly-polarized monochromatic wave with ω = 2π, h = [0.01, 0.04]

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Weak, monochromatic, linearly-polarized waves

Through O(h3), H =

• 1 − 1

8(hωu)2 + . . . 1 1

• + h
• 1 − 1

32(hωu)2

• cos ωu + . . .

1 −1

• Rapidly-growing trace at O(h2) [mimics alternative gravities]

Rapidly-growing oscillation at O(h3) Complete breakdown of linear theory when (# oscillations) h−1

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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(uo), H−1(us), H−1

[AIH (2013), AIH & S. Babak (in prep.)]

Much better to use H instead of H − I

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Pulsar timing

λo λs ωs ωo

• = 1 + 1

2(1 + cos ψ)−1 detH−1 (det H−1

• H−1

s )1/2

× [(H−1

s )ijni

• nj
• − (H−1
• )ijni

snj s]

no ≡ 1 dang H−1−1(Xo − Xs→o) ψ = (angle between source and GW) dang = (angular diameter distance) λo/λs ∼ (Doppler along wave direction) [. . .] ∼ (Doppler transverse to wave direction and curvature)

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Part II: Wave(-wave) effects

How do gravitational plane waves affect other waves?

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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.

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Spacetime viewpoint

Waves generically travel both along characteristics and behind them: The leftovers are usually called tails in relativistic contexts.

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Huygens’ principle (Hadamard’s minor premise)

There are no tails. This holds for the usual flat-spacetime wave equation:

• − 1

c2 ∂2 ∂t2 + ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2

• φ(t, x, y, z) = 0.

It fails for almost everything else: Huygens’ principle is very special.

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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...)

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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:

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Impulsive solutions → Green functions In Lorenz gauge, linearized Einstein is ∇c∇c¯ hab + 2Racbd ¯ hcd = −16πTab. So consider ∇c∇cGaba′b′ + 2RacbdG cda′b′ = −4πg(a

a′gb) b′δ(x, x′).

Arbitrary ¯ hab computable by integrating Gaba′b′ (even vacuum solns).

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Green functions in general [Hadamard, DeWitt, . . . ]

Retarded Green functions always look like: Gaba′b′ = [∆1/2gaa′gbb′δ(σ) + Vaba′bΘ(−σ)]ret with σ(x, x′) = 1 2(squared geodesic distance) gaa′(x, x′) = (parallel propagator) ∆(x, x′) = (measure of focusing for null geodesics) Vaba′b′(x, x′) = (tail)

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Vaba′b′ satisfies a characteristic initial value problem. It does not vanish for plane wave backgrounds. But this only means that hab has a tail in Lorenz gauge. Could the tail be pure gauge?

[In Lorenz-gauge EM, F tail

ab

= 2∇[aAtail

b]

= 0 even though Atail

a

= 0]

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Checking for gauge effects

Scalars like RabcdRabcd, RabcdR∗

abcd,

∇aRbcdf ∇aRbcdf , . . . 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)]

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Main result

Gravitational wave-wave scattering [AIH (2013)]

Locally, linear metric tails can only perturb a background plane wave into another plane wave. Initial data need not be planar at all, but all that is scattered is a “planar component” which changes the waveform and polarization.

[an “almost-Huygens’ principle”]

Only habℓaℓb|Σ matters, where ℓa = (background propagation direction).

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Observing a tail

1 2 3

O sees complicated (red, blue) wavefronts, but otherwise observes only a plane wave in regions 0, 1, 3. Noticing waveform changes requires nonlocal measurements.

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Conclusions I

1 Even if h ≪ 1, exact plane waves deviate strongly from approximate

• nes on lengthscales O(λ/h)

Astrophysics? Failure of pert. theory in general?

2 Although most fields propagate tail-free through gravitational plane

waves, linear metric perts do develop (subtle!) tails

Astrophysics (waveform bias)? Non-planar waves or nonlinear scattering?

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Conclusions II

An exact Gaba′b′ was computed. Several possible applications:

1 Particle motion in PWs: self-forces, self-torques 2 Penrose limits

Ultrarelativistic motion in general spacetimes Wave optics for GWs in general spacetimes

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