Download pdf - Black Holes, LIGO and Geometric AnalysisBlack Holes, LIGO and Geometric Analysis Zoe Wyatt Edinburgh University, PG Colloquium December 2016. Motivation LIGO = Laser Interferometer

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


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


(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)



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 + · · ·



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)



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)



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



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


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.


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


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


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



h =∆`

`∼ Rs

rRs

r∼ 30 · Rs [Sun]

1.3× 109ly · c∼ 10−23


⇒ ∆` = h · ` ∼ 4× 10−20m







h =∆`

`∼ Rs

rRs

r∼ 30 · Rs [Sun]

1.3× 109ly · c∼ 10−23


⇒ ∆` = h · ` ∼ 4× 10−20m





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


4 stage pendulum

360 kg weight

glass fibers 0.4 mm(400 microns) thick

masses held in placeby EM forces →keeps arm lengthconstant


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)


Recall the vacuum Einstein equations

Rµν [g ] = 0

Or in compact notation:

ggµν = B(∂g , ∂g) + N(∂g , ∂g)


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



ggµν = B(∂g , ∂g) + N(∂g , ∂g)


ghµν = B(∂h, ∂h) + N(∂h, ∂h) + Q(h)(∂h, ∂h)


|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



ggµν = B(∂g , ∂g) + N(∂g , ∂g)


ghµν = B(∂h, ∂h) + N(∂h, ∂h) + Q(h)(∂h, ∂h)


|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!