Download Gravitational lensing by gravitational waves

Gravitational lensing by gravitational waves
Abraham Harte
Max-Planck-Institut für Gravitationsphysik
Albert-Einstein-Institut
Potsdam, Germany
March 13, 2013
Based on AIH CQG 30, 075011 (2013)
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1
Introduction
2
Exact gravitational wave solutions in GR
3
Some strong lensing effects
4
Wavepackets and flashes
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Gravitational lensing
Apparent bending of light rays (null geodesics) due to nontrivial geometry
Fairly intuitive for compact masses.
What about non-stationary or highly extended systems?
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Consider light propagating through gravitational waves
1
Gravitational waves are intrinsically dynamic and non-Newtonian.
2
Linearized theory of gravitational radiation misses a lot (caustics!).
3
Penrose limits: The metric near any null geodesic is the metric of an
appropriate plane gravitational wave.
Maybe lensing in a generic spacetime looks like lensing in a plane
wave if sources and observers are ultrarelativistic and nearly comoving.
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Gravitational waves in linearized GR
Vacuum metric perturbations gµν = ηµν + hµν in TT gauge:
hµν (x) = R[eµν exp(ikλ x λ )]
kµ is null and eµν has two DOFs in vacuum GR: h+ , h×
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Generalizing TT gauge
Rosen metric:
ds 2 = −2dUdV + [δij + hij (U)]dX i dX j
1
Simple generalization of linearized result
2
Geodesics can have constant X i
3
Vacuum Einstein is complicated (nonlinear ODE)
4
Many different hij represent the same wave
5
Coordinate singularities!
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Gravitational lensing by gravitational waves
(i = 1, 2)
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Exact plane waves
Coordinate transform of the Rosen metric allows global description via
Brinkmann metric:
ds 2 = −2dudv + δij dx i dx j + Hij (u)x i x j du 2
(u, v , x i ∈ R).
v = (affine parameter along rays)
u = (phase)
x i = (transverse coordinates)
Hij = (waveform)
Hij ≡ 0 is flat spacetime with u =
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√1 (t
2
+ z), v =
Gravitational lensing by gravitational waves
√1 (t
2
− z).
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Why call these plane waves?
1
2
`a = ∂/∂v is null and Killing.
∇a `b = 0, so there is no expansion, shear, or twist.
3
Surfaces with constant u, v are (transverse) 2-planes.
4
All rays are orthogonal to these transverse planes.
5
4-curvature is constant in transverse directions: L∂i Rabc d = 0.
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Interpreting the waveform
Specific waves are specified by 3 wavefunctions h+ (u), h× (u), hk (u):
H=
−h+ h×
h× h+
− hk
1 0
0 1
.
Rab = 2hk ∇a u∇b u, so
Vacuum Einstein ⇒ hk = 0: Two free polarizations.
Vacuum waves in the same direction obey linear superposition.
With matter (e.g. Einstein-Maxwell), energy conditions imply hk ≥ 0.
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Waveform interpretation II
Collections of test particle experience quadrupolar (h+ , h× ) and scaling
deformations (hk ):
Ruiuj = −Hij (u)
Transverse coordinates of a geodesic z(u) are solutions to
z̈ = Hz
“Coupled oscillators with variable (and possibly complex) stiffness”
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Focusing and caustics
Transverse coordinates of geodesics satisfy
z(u) = A(u, u 0 )z(u 0 ) + B(u, u 0 )ż(u 0 )
Families of geodesics with the same z(uo ) but different ż(uo ) can all focus
to the same z(τ ) if ∃τ, uo such that
det B(τ, uo ) = 0
(τ 6= uo ).
Observers at phase uo are conjugate to points at phase τ .
Caustics appear generically!
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Types of focusing
Null cone passing through a u = (const) hyperplane. . .
. . . collapses to a line if rank B = 1
(multiplicity 1 conjugate points)
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. . . collapses to a point if rank B = 0
(multiplicity 2 conjugate points)
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“Wavelike” solutions
Familiar plane waves from linearized theory go to
hω 2
H(u) =
2
cos ωu sin ωu
sin ωu − cos ωu
.
These usually contain closely-spaced pairs of caustics with multiplicity 1:
det BHu, 0L
15
10
5
2Π
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4Π
6Π
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8Π
u
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Focusing with sinusoidal plane waves
Linear polarization
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Circular polarization
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Dividing up generic plane waves
Given a preferred uo , the spacetime naturally divides into regions Nn (uo )
separated by phases τn (uo ) where det B(τn (uo ), uo ) = 0.
Caustics of po appear on the boundaries Sτn (uo ) and nowhere else.
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Gravitational lensing
Point source on a timelike worldline Γ and observation event po
Multiple imaging
For almost all sources, there is exactly one
image from each Nn (n ≤ 0).
Odd number theorem doesn’t apply. The number of images can be even.
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Number of images depends only on the waveform H(u) and the phase
parameter uo associated with the observation event po .
Solve
∂u2 B(u, uo ) = H(u)B(u, uo )
lim B(u, uo ) = 0,
u→uo
lim ∂u B(u, uo ) = I.
u→uo
for all u ≤ uo .
Then,
N(uo ) = (number of images) = [number of zeros of det B(·, uo )]
With the right waveform, N can be anything (including ∞)
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Other lensing effects
Different images of the same object appear with
1
different colors
2
different shapes
3
different brightnesses
4
different positions on the observer’s sky
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Frequency shifts
For geodesic sources and instantaneously comoving observers,
ωo
− 1 = F 2 (B−| δxo )| (AA| − I) (B−| δxo ).
ωe
Qualitative frequency shifts depends on the eigenvalues of AA| − I:
1
Both eigenvalues negative: All sources redshifted.
2
Both eigenvalues positive: All sources blueshifted.
3
Eigenvalues have opposite signs: Nature of frequency shift depends
on the direction of B−| δxo (source location).
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Wavepackets
Spacetime is curved only in between two null hyperplanes.
Γo
Γs
Everywhere else is flat.
Almost all interesting optical effects can be understood using a few
numbers (not functions!) constructed from integrals of H.
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Do sandwich waves form caustics?
For an observer not yet hit by the wave: No (just flat spacetime).
Observer inside the wave: Yes if the amplitude is large enough.
Late times: At most two caustics appear behind the wave.
N depends on the number of caustics, so N = N(uo ).
Even very weak wavepackets produce multiple images if you wait
long enough...
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Short and weak wavepackets
Everything important is contained in the 2 numbers (or just their ratio)
Z
H(u)du
rotation
→
−
H1 0
0 H2
.
Weak energy condition ⇒ H1 + H2 ≥ 0 .
Vacuum Einstein ⇒ H1 = −H2 .
(All vacuum wavepackets are the same up to scale)
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Multiple imaging from weak vacuum wavepackets
One caustic (and a second image) appears from u → −∞ when H1 uo = 1:
2
H1 ue
2
4
6
H1 uo
8
�2
�4
�6
Once this forms, it persists for all time
The second image always stays behind the wave.
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Frequency shifts for the second image
When the second image first appears, an infinite amount of the source’s
history is observed in finite time ⇒ Arbitrarily large blueshifts!
At late times, ue can only advance so much without running into the wave
⇒ Arbitrarily large redshifts
1000
100
ωo
ωe
10
1
0.1
0.01
0.1
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0.5
1.0
5.0 10.0
Gravitational lensing by gravitational waves
H1 uo
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Distance measures
How far away does an image look?
(observed flux) =
106
(luminosity)
,
2
4πdlum
H1 dlum
(angular size) =
1000
500
(physical size)
.
dang
H1 dang
4
10
100
50
100
1
10
5
0.01
1
0.1
0.5
1.0
5.0 10.0
H1 uo
0.1
0.5
1.0
5.0 10.0
H1 uo
dlum → 0 and dang → ∞!
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Transient flashes
All sources focused to the same spot on the observer’s sky at uo = H1−1 .
Initially,
1
Infinite blueshift
2
Infinite brightness
3
Zero angular size
4
Images from the infinitely-distant past
At later times,
1
Sources separate on the sky
2
Ever-increasing redshift
3
Decreasing brightness
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Conclusions
1
Optical effects of gravitational waves can be dramatic
2
Short wavepackets generically produce multiple images if you wait
long enough
3
Secondary images initially appear as bright high-frequency flashes
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Questions
1
To what extent are the more interesting features of plane wave
lensing preserved for “realistic” waves which decay at infinity?
2
Can Penrose limits be used to show that plane wave lensing is
relevant for generic spacetimes with large boosts?
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