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Black Holes, LIGO and Geometric Analysis
Zoe Wyatt
Edinburgh University, PG ColloquiumDecember 2016
Motivation
LIGO=
Laser Interferometer Gravitational-Wave Observatory
Motivation
LIGO=
Laser Interferometer Gravitational-Wave Observatory
LIGO detected gravitational waves caused by two black holes 1.3billion light years away merging on 14 September 2015
First direct experimental proof of gravitational waves!
Outline
1 What is a black hole
2 What are gravitational waves
3 What is LIGO and what did it detect
4 Where is PDE analysis used in studying general relativity
Laws of Physics
Newton’s lawd2x
dt2=
F
m
Predictions:
inertial motion of particles
Laws of Physics
Maxwell’s equations:
∇ · ~E = ρ/ε0 ∇ · ~B = 0
∂ ~B
∂t= −∇× ~E
∂E
∂t=
1
µ0ε0∇× ~B − 1
ε0~J
Predictions:
propagation of electromagnetic field
Laws of General Relativity
Einstein equations:
Rµν [g ] =8πG
c4Tµν
2nd order, non-linear PDEs, with unknown matrix gµν
gµν =
g00 g01 · · · g03g01 g11 · · · g13
......
. . ....
g03 g13 · · · g33
The matrix gives us a metric: g = gµνdx
µdxν
Eg (Riemannian): g = dx2 + dy2 + dz2, or g = dθ2 + sin2 θdϕ2
Laws of General Relativity
Einstein equations:
Rµν [g ] =8πG
c4Tµν
2nd order, non-linear PDEs, with unknown matrix gµν
gµν =
g00 g01 · · · g03g01 g11 · · · g13
......
. . ....
g03 g13 · · · g33
The matrix gives us a metric: g = gµνdx
µdxν
Eg (Riemannian): g = dx2 + dy2 + dz2, or g = dθ2 + sin2 θdϕ2
Laws of General Relativity
(Vacuum) Einstein equations
Rµν [g ] = 0
gρτ∂ρ∂τgµν
=1
4gρτgσλ (−2∂νgρσ∂µgτλ + ∂µgσλ∂νgρτ )
+ gρτgσλ∂ρgσµ∂τgλν − gρτgσλ (∂ρgσµ∂λgτν − ∂λgσµ∂ρgτν)
+ gρτgσλ (∂µgτλ∂ρgσν − ∂ρgτλ∂µgσν)
+ gρτgσλ (∂νgτλ∂ρgσµ − ∂ρgτλ∂νgσµ)
+1
2gρτgσλ (∂λgτρ∂µgσν − ∂µgρτ∂λgσν)
+1
2gρτgσλ (∂λgρτ∂νgσµ − ∂νgρτ∂λgσµ)
Laws of General Relativity
(Vacuum) Einstein equations
Rµν [g ] = 0
Einstein summation convention:
XµYµ = X 0Y0 + X 1Y1 + X 2Y2 + X 3Y3
For each component of the metric matrix:(g00∂2tt + 2g01∂2tx + g11∂2xx + 2g12∂2xy + · · ·
)g00
= −1
2(g00)2(∂tg00)2 − g01g00∂tg00∂tg01 − g11g00(∂tg10)2 + · · ·
Laws of General Relativity
(Vacuum) Einstein equations
Rµν [g ] = 0
Predictions:
black holes
gravitational waves
Solutions of the Einstein equations
Geometry around a spherical object of mass M and radius Rdescribed by
g = −(
1− 2M
r
)−1
dt2 +
(1− 2M
r
)dr2 + dΩ2
for r > R, c ≡ 1.
R > 2M
Eg: the sun and earthEg: indirect proof of GR via gravitational lensing and binary pulsar(1974)
Solutions of the Einstein equations
Geometry around a spherical object of mass M and radius Rdescribed by
g = −(
1− 2M
r
)−1
dt2 +
(1− 2M
r
)dr2 + dΩ2
for r > R, c ≡ 1.R > 2M
Eg: the sun and earthEg: indirect proof of GR via gravitational lensing and binary pulsar(1974)
Solutions of the Einstein equations
Geometry around a spherical object of mass M and radius Rdescribed by
g = −(
1− 2M
r
)−1
dt2 +
(1− 2M
r
)dr2 + dΩ2
R < 2M⇒ Black hole
Schwarzschild radius
Rs = 2MG
c2
Eg: the sun Rs ' 3kmEg: the earth Rs ' 8.7mm
Solutions of the Einstein equations
Geometry around a spherical object of mass M and radius Rdescribed by
g = −(
1− 2M
r
)−1
dt2 +
(1− 2M
r
)dr2 + dΩ2
R < 2M⇒ Black hole
Schwarzschild radius
Rs = 2MG
c2
Eg: the sun Rs ' 3kmEg: the earth Rs ' 8.7mm
Solutions of the Einstein equations
Why is this a black hole?
Particle travelling along geodesic xµ(s) parametrised by s.
Claim: if r(s0) < 2M then r(s) < 2M for all s ≥ s0 and r(s) ismonotonically decreasing for s ≥ s0.
Proof comes from ODE:If V µ = dxµ
ds is the tangent vector, then
−2dv
ds
dr
ds= −V 2 +
(2M
r− 1
)(dv
ds
)2
+ r2(dΩ
ds
)2
≥ 0 for r ≤ 2M
Also dvds > 0 since − ∂
∂r and V µ are future-directed causal vectors.
Solutions of the Einstein equations
Why is this a black hole?
Particle travelling along geodesic xµ(s) parametrised by s.
Claim: if r(s0) < 2M then r(s) < 2M for all s ≥ s0 and r(s) ismonotonically decreasing for s ≥ s0.
Proof comes from ODE:If V µ = dxµ
ds is the tangent vector, then
−2dv
ds
dr
ds= −V 2 +
(2M
r− 1
)(dv
ds
)2
+ r2(dΩ
ds
)2
≥ 0 for r ≤ 2M
Also dvds > 0 since − ∂
∂r and V µ are future-directed causal vectors.
Black hole mergers
Rotating black hole: Kerr
g =−(
1− 2Mr
ρ2
)dt2 − 4Mra sin2 θ
ρ2dϕdt +
ρ2
∆dr2
+ ρ2dθ2 +
(r2 + a2 +
2Mra2 sin2 θ
ρ2
)sin2 θdϕ2
where
a =J
Mρ2 = r2 + a2 cos2 θ
∆ = r2 − 2Mr + a2
Black hole mergers
Rotating black hole: Kerr
Merging of two rotating black holes
What are gravitational waves
Sources of GWs: stars orbiting each other, stars dying in explosionsor collapsing to create black holes
Massive merging black holes: 36M + 29M = 62M
accelerating massive objects → fast changing large curvature inspacetime → 3M into grav radiation → gravitational waves
What are gravitational waves
Sources of GWs: stars orbiting each other, stars dying in explosionsor collapsing to create black holes
Massive merging black holes: 36M + 29M = 62Maccelerating massive objects → fast changing large curvature inspacetime → 3M into grav radiation → gravitational waves
What are gravitational waves
Linearised gravity gives transverse waves(−∂2t + ∆
)hij = 0
Two polarizations of GWs: h+, h−
GW speed = c
GWs do not get blocked
How does LIGO work
Observatories in Livingston, Louisiana, and Richland, Washington.
Sites separated by 3,000 km ' 10 milliseconds by GWs
How does LIGO work
How sensitive does LIGO have to be
Rough order-of-magnitude argument:
h =∆`
`∼ Rs
rRs
r∼ 30 · Rs [Sun]
1.3× 109ly · c∼ 10−23
ie, perturbations in spacetime are very small
⇒ ∆` = h · ` ∼ 4× 10−20m
Reality: ∆` ∼ 10−18m
How do we measure such tiny scales?
Interferometry with high laser power!
Eg: c = const by Michaelson and Morley
How sensitive does LIGO have to be
Rough order-of-magnitude argument:
h =∆`
`∼ Rs
rRs
r∼ 30 · Rs [Sun]
1.3× 109ly · c∼ 10−23
ie, perturbations in spacetime are very small
⇒ ∆` = h · ` ∼ 4× 10−20m
Reality: ∆` ∼ 10−18m
How do we measure such tiny scales?
Interferometry with high laser power!
Eg: c = const by Michaelson and Morley
How sensitive does LIGO have to be
Rough order-of-magnitude argument:
h =∆`
`∼ Rs
rRs
r∼ 30 · Rs [Sun]
1.3× 109ly · c∼ 10−23
ie, perturbations in spacetime are very small
⇒ ∆` = h · ` ∼ 4× 10−20m
Reality: ∆` ∼ 10−18m
How do we measure such tiny scales?
Interferometry with high laser power!
Eg: c = const by Michaelson and Morley
How is LIGO that sensitive
Fabry Perot cavities → bounce the light signals ∼ 280 times⇒ 4km long arms → 1120km long
Power recycling mirror boosts laser power ×3750 times ⇒P ∼ 750kWPhotodetector measures change in power ∆P ∼ ∆`
LIGO results – hearable frequency
How does LIGO remove experimental errors?
Eg: earthquakes, trucks driving nearby, sneezing
⇒ isolate LIGO
distance ↔ 10 milliseconds
vacuum, air pressure 10−12 of sea level
mirrors, absorb 1 in 3 million photons
active damping, vibration senses 2x10−13m
How does LIGO remove experimental errors?
4 stage pendulum
360 kg weight
glass fibers 0.4 mm(400 microns) thick
masses held in placeby EM forces →keeps arm lengthconstant
How does LIGO remove experimental errors?
Where’s all the analysis?
Recall the vacuum Einstein equations
Rµν [g ] = 0
We could write these as:
gρτ∂ρ∂τgµν
=1
4gρτgσλ (−2∂νgρσ∂µgτλ + ∂µgσλ∂νgρτ )
+ gρτgσλ∂ρgσµ∂τgλν − gρτgσλ (∂ρgσµ∂λgτν − ∂λgσµ∂ρgτν)
+ gρτgσλ (∂µgτλ∂ρgσν − ∂ρgτλ∂µgσν)
+ gρτgσλ (∂νgτλ∂ρgσµ − ∂ρgτλ∂νgσµ)
+1
2gρτgσλ (∂λgτρ∂µgσν − ∂µgρτ∂λgσν)
+1
2gρτgσλ (∂λgρτ∂νgσµ − ∂νgρτ∂λgσµ)
Where’s all the analysis?
Recall the vacuum Einstein equations
Rµν [g ] = 0
Or in compact notation:
ggµν = B(∂g , ∂g) + N(∂g , ∂g)
Where’s all the analysis?
Idea: are solutions to this equation stable to perturbations?
ggµν = B(∂g , ∂g) + N(∂g , ∂g)
If g = g0 + h where ‘h << 1’, does h→ 0 as t →∞?
ghµν = B(∂h, ∂h) + N(∂h, ∂h) + Q(h)(∂h, ∂h)
properties of the non-linearityproperties of wave equations along the light cone
|φ(x , t)| . (1 + t + |x |)−n−12 (1 + |t − |x ||)
peeling estimates
|L(∂φ)| . (1 + t + r)−n−12 (1 + |t − r |)−3/2
|L(∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
|ei (∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
Where’s all the analysis?
Idea: are solutions to this equation stable to perturbations?
ggµν = B(∂g , ∂g) + N(∂g , ∂g)
If g = g0 + h where ‘h << 1’, does h→ 0 as t →∞?
ghµν = B(∂h, ∂h) + N(∂h, ∂h) + Q(h)(∂h, ∂h)
properties of the non-linearityproperties of wave equations along the light cone
|φ(x , t)| . (1 + t + |x |)−n−12 (1 + |t − |x ||)
peeling estimates
|L(∂φ)| . (1 + t + r)−n−12 (1 + |t − r |)−3/2
|L(∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
|ei (∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
Where’s all the analysis?
Idea: are solutions to this equation stable to perturbations?
ggµν = B(∂g , ∂g) + N(∂g , ∂g)
If g = g0 + h where ‘h << 1’, does h→ 0 as t →∞?
ghµν = B(∂h, ∂h) + N(∂h, ∂h) + Q(h)(∂h, ∂h)
properties of the non-linearityproperties of wave equations along the light cone
|φ(x , t)| . (1 + t + |x |)−n−12 (1 + |t − |x ||)
peeling estimates
|L(∂φ)| . (1 + t + r)−n−12 (1 + |t − r |)−3/2
|L(∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
|ei (∂φ)| . (1 + t + r)−n+12 (1 + |t − r |)−1/2
Thanks for listening!