The magnification theorem
Olaf Wucknitz
[email protected]
Three decades of gravitational lenses, JENAM, 21 April 2009
2.5 decades of the magnification theorem
General theorems in lensing
• odd image theorem
even image corollary
• cusp relation
flux ratio anomalies
• lensing is achromatic
chromatic microlensing
• magnification theorem ?
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
1
Light deflection
dz 1
α̃ = ∆ =
dl
c
Naive Newtonian calculation:
GM
Φ=−
R
titlepage
Z
dl
∇⊥Φ
c
∼
α
Soldner 1801
(Newton)
Einstein 1915
(general relativity)
GM
α̃ = 2 2
c r
GM
α̃ = 4 2
c r
introduction
summary
−→
contents
bonus
back
forward
previous next
fullscreen
2
Distortion/magnification
lensed
source
M
θs
θ
θ s = θ − α(θ)
lens equation
∂α
dθ = M−1 dθ
dθ s = dθ −
∂θ
first derivative
magnification / mapping matrix
M(θ) =
µ = ± det M
(area) magnification = amplification
titlepage
introduction
summary
contents
bonus
back
forward
∂α
1−
∂θ
−1
previous
next
fullscreen
3
Magnification — amplification
source
Ω
lens
observer
• observer’s view
• solid angles measure apparent size
Ωs
Ω
• magnification =
Ωs
_
Ω
• flux distributed over solid angle
source
lens
_
Ωs
observer
• source’s view
Ω̄
• amplification =
Ω̄s
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
4
Example: point-mass
µ+
µ−
4
2
0
-2
-4
-2
-1.5
-1
-0.5
0
θs
0.5
1
1.5
2
[ Wambsganss (1998), Liv. Rev. Rel. 1, 12 ]
titlepage
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
5
[ Einstein notebooks 1910–1912 ]
I’ve seen this before . . .
[ Einstein
(1936) ]
[ see also
Refsdal (1964) ]
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
6
Potential, light travel time
• potential ψ
α(θ) = ∇ψ(θ)
• Poisson equation
∇2ψ(θ) =: 2κ(θ) = 2σ = 2 ΣΣc
DdDs
∆t =
φ(θ)
c Dds
• light-travel time for virtual ray
(θ s fixed)
• Fermat-potential
(θ − θ s)2
φ(θ) =
− ψ(θ)
2
• Fermat’s principle: real images are
? minima,
(e.g. unperturbed image)
? maxima, or
? saddle-points of φ
titlepage
introduction
summary
contents
bonus
back forward
previous
next
fullscreen
7
Magnification theorem
[ Schneider (1984) ]
• Hessian of Fermat-potential is inverse magnification matrix
2 ∂ φ 1 − κ − γx
∂α
−γ
y
−1
=
=
µ = 1 −
−γy
1 − κ + γx
∂θ
∂θ 2
√
• diagonalise: rotate shear, γ = γx + γy
1 − κ − γ
0
−1
µ =
0
1 − κ + γ
• minimum: both eigenvalues positive
• Poisson: convergence κ = σ ≥ 0
• sum: 2(1 − κ) > 0
κ<1
0≤κ<1
• µ−1 = (1 − κ)2 − γ 2
titlepage
introduction
summary
contents
bonus
0 < µ−1 ≤ 1
back
forward
previous
next
fullscreen
8
An apparent paradox
• source in centre
• amplif. ≥ 1 in all directions
• integrate over sphere
• total flux amplification !
• conservation of energy ?
• solution to energy crisis ?
• lensing cannot create photons
titlepage
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
9
The standard explanation
• lens distorts geometry
• area of surface shrinks !
• have to compare with same
mean geometry
• compare with same mean
density in Universe
[ Weinberg (1976) ]
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
10
Bad excuse, because . . .
• equivalent: refraction
or Newtonian deflection
• does not change geometry
• same formalism
• same paradox !
• so far: tangential plane
? no problem in the plane
• now: do it on the sphere !
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
11
Deflection angle for the sphere
calculation for Ds → ∞
2GM
α = − 2 x0
c
Z∞
h
dz x20 +(z−z0)2
i−3/2
0
θ
m
α = cot
2
2
→
3
m
θ
m/θ
(m/2) cot (θ/2)
α
2
1
α
θs
M
θ
z
0
x
π/2
0
π
θ
Dd
[ compare Avni & Shulami (1988) and Jaroszynski & Paczynski (1996) ]
titlepage
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
12
Magnification for the sphere
i
2h
m
m
∂α
2
= 1+ −
1
+
O
θ
µ−1 = 1 −
∂θ 2
θ4
• for point-mass
m ∼ (100)2 = 2.35 · 10−11
2
m
µ−1 = 1 − 4
θ
µtot
µ+
−µ−
10
µtot−1
µ+−1
−µ−
3
2
1
0
1
[10−11]
• planar approximation
-1
-2
0.1
titlepage
1
θs [arcsec]
introduction
summary
contents
10
bonus
back
0
forward
500
1000
previous
1500
θs [arcsec]
next
2000
fullscreen
2500
-3
13
Lensing on the sphere
• far from optical axis: µ < 1
• in this situation: magnification theorem not valid
• integration over sphere: mean µ is 1
no paradox
• modified Poisson equation
h
∇2ψ(θ) =: 2κ(θ) = 2 σ(θ) − σ̄
• not always κ ≥ 0
i
failure of theorem
• field lines decay!
titlepage
introduction summary
contents
bonus
back
forward
previous
next
fullscreen
14
Field lines on the sphere
FIELD LINES
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
15
Back to gravitation and summary
• short summary
? flat spacetime with refractive medium (or Newtonian)
? magnification theorem not valid
? modified Poisson equation
• equivalent: gravity with appropriate reference situation
? constant area of sphere
magnification theorem invalid, no paradox
? advantage: magnification is function of deflection
• not equivalent: ‘inappropriate’ reference situation
? constant affine distance, area not constant
focusing theorem valid, apparent paradox
? planar proof does still not hold for sphere (e.g. mass shell)
[ Wucknitz (2008), MNRAS 386, 230 ]
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
16
Bonus: From deflection to magnification
• magnification on the plane
? derivative of deflection angle
? constant deflection not relevant
• magnification on the sphere
? derivatives still relevant
? ‘constant’ deflection too!
? parallel transport 6= rigid rotation
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
17
Exact magnification on the sphere (1)
• lens equation
?
=
Θsµ
Θµ − aµ(Θ)
? is not a vector equation (for finite a) !
• alternative description
?
?
?
?
start at Θµ
move in the direction of −aµ
along a geodesic
for a total distance of |a|
• geodesic equation
α µ ν
ẍα + Γµν
ẋ ẋ = 0
• affine parameter λ from 0 (Θ) to 1 (Θs)
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
18
Exact magnification on the sphere (2)
xµ(0) = Θµ
• start
ẋµ(0) = −aµ
xµ(1) = Θsµ
• end
• slightly displaced start: geodesic deviation
D2ξ α
β µ ν α
=
ẋ
ẋ ξ Rµβν
2
Dλ
• curvature tensor
α
α
α
α ρ
α ρ
Rµβν
:= Γµν,β
− Γµβ,ν
+ Γρβ
Γµν − Γρν
Γµβ
1
α
α
= 2 δβ gµν − δν gµβ
K
titlepage
introduction
summary
contents
bonus
back
forward
previous
next
fullscreen
19
Exact magnification on the sphere (3)
• coordinates k and ⊥ to aµ
• result for geodesic deviation
~
~
ξ(1)
= M−1ξ(0)
• magnification matrix

−1
M

=

k
k
−a;⊥
1 − a;k
−a⊥
;k

sin(a/K)
a/K


⊥ sin(a/K) 
cos(a/K) − a;⊥
a/K


k
k
1 − a;k −a;⊥


≈
⊥
−a⊥
1
−
a
;⊥
;k
titlepage
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
20
Affine distance, light travel time
ds2 = (1 + 2Ψ ) dt2 − (1 − 2Ψ ) dx2
• metric (c = 1)
• affine distance, light travel time: measured at observer’s position
R
• light travel time:
T = (1 + Ψ0) dx (1 − 2Ψ )
R
• affine distance:
L = (1 − Ψ0) dx
• general focusing theorem:
titlepage
introduction summary
contents
bonus
µ > 1 for constant affine distance
back
forward
previous
next
fullscreen
21
Lensing by a spherical shell
• shell of radius (metric) r0 with mass M
GM/c2
• σ≈
for σ 1
r0
• limit r0, M → 0 with σ = const
no change of global geometry
• unlensed situation
? (re-)move sphere or . . .
r
? affine distance Λ = √
constant
1 − 2σ
• focusing theorem: compare with constant Λ
titlepage
introduction
summary
contents
bonus
back
forward
previous
−→
next
focusing
fullscreen
22
Contents
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
titlepage
The magnification theorem
Light deflection
Distortion/magnification
Magnification — amplification
Example: point-mass
I’ve seen this before . . .
Potential, light travel time
Magnification theorem
An apparent paradox
The standard explanation
Bad excuse, because . . .
Deflection angle for the sphere
Magnification for the sphere
Lensing on the sphere
Field lines on the sphere
Back to gravitation and summary
Bonus: From deflection to magnification
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
23
18
19
20
21
22
23
titlepage
Exact magnification on the sphere (1)
Exact magnification on the sphere (2)
Exact magnification on the sphere (3)
Affine distance, light travel time
Lensing by a spherical shell
Contents
introduction
summary
contents
bonus
back
forward
previous next
fullscreen
24