Rodrigo Nemmen
July 25, 2018
95

# Gravitational waves

Lecture about gravitational waves targeted at undergraduate students and first year grad students in physics and astronomy. Prepared and taught by Prof. Rodrigo Nemmen at IAG USP.

Credit for the slides/figures belongs to Rodrigo Nemmen, unless otherwise stated.

https://blackholegroup.org

July 25, 2018

## Transcript

3. ### Electrodynamics accelerated charges produce electromagnetic radiation Gravitation accelerated masses produce

gravitational radiation 1 μ 0 E × B c c hμν electromagnetic wave gravitational wave
4. ### The Elegant Universe. Crédito: Nova Accelerated masses produce wave of

disturbance in spacetime curvature Gravitational waves

6. ### PSR 1913+16: A binary system of pulsars (Hulse & Taylor

pulsar) precisely determined orbital parameters r = 5 kpc M1 = M2 = 1.4 Msun
7. ### GWs emitted by binary system Accelerated masses emit GWs System

loses energy Distance between two bodies shrinks
8. ### GWs GWs emitted by binary system Accelerated masses emit GWs

System loses energy Distance between two bodies shrinks NS NS NS NS Distance Period
9. ### Hulse & Taylor pulsars r = 5Kpc, M1 ≈ M2

≈ 1.4M⊙ , T = 7h45min fGW = 7 × 10−5 Hz GR prediction h ∼ 10−23 · T = − 2.4 × 10−12 sec/sec Orbital shrinkage due to GW radiation τ = 3.5 × 108yr timescale for coalescence
10. ### First evidence of existence of GWs Weisberg & Taylor (2005)

GR prediction observations Monumental discovery in 1974 by Hulse & Taylor
11. ### Linearized gravitational waves Far away from the source, r ≫

M, gμν ≈ ημν Gravitational wave (GW) will be weak (space empty of matter) Einstein equation can be linearized and solved giving simpler solutions
12. ### Weak gravitational ﬁeld gμν = ημν + hμν Minkowski metric

Perturbation hμν = Aμν eikα xα Solution: plane wave traveling with c wave vector GW amplitude Einstein equation Rμν = 0 Ricci curvature (in vacuum) Reduces to wave equation ( ∂2 ∂t2 − ∇2 ) hμν = □ hμν = 0 Linearized gravitational waves
13. ### hμν = Aμν eikα xα Plane wave wave vector GW

amplitude Properties of linear GWs Transverse waves Aμν kμ = 0 kμ kμ = 0 Null wave vector: GWs move like light-rays homework
14. ### Polarization of GWs Aμν = h+ Aμν + + h×

Aμν × Wave moving in the z-direction = 0 0 0 0 0 h+ h× 0 0 h× −h+ 0 0 0 0 0 two polarizations Aμν + = 0 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 0 Aμν × = 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 Polarization components

16. ### Effect of GWs on test particles x y z L0

ds2 = gμν dxμdxν ⇒ L = ∫ gμν dxμdxν = ∫ g11 dx = L0 [1 + 1 2 h11 (t,0)] ∴ ΔL L0 = 1 2 h11 (t,0) strain produced by GW Particles oscillate after GW passes ⟂ k homework
17. ### Example 1: periodic wave ΔL L0 = 1 2 h(t,0)

h(t − z) = a sin[ω(t − z) + ϕ] x y z L0 GW propagating in z-direction ΔL L0 = a 2 sin[ωt + ϕ]

19. ### Example 2: wave packet ΔL L0 = 1 2 h(t,0)

h(t − z) = a exp [ − (t − z)2 σ2 ] x y z L0 GW propagating in z-direction ΔL L0 = a 2 exp [ − (t − z)2 σ2 ]
20. ### Example 3: Change in distance between two masses due to

GW x y z L0 ΔL L0 = h 2 ΔL = hL0 2 = 10−21 × 2 km = 2 × 10−18 m ~ 1 proton radii L0 = 4 km h = 10−21

24. ### black hole neutron star neutron star neutron star GW sources

High mass concentrations + Extreme accelerations star supermassive black hole GWs
25. ### Strain = h(t) ≡ Δx L Δx = 10−21 ×

4 km = 4 × 10−18 m a v v a = 3 proton radii Numerically solve Einstein’s ﬁeld equation Rμν − 1 2 gμν R = 8πG c4 Tμν spacetime curvature = constant× matter-energy

27. ### Importance of GW detection “I liken this to the ﬁrst

time we pointed a telescope at the sky, […] people realized there was something to see out there, but didn’t foresee the huge, incredible range of possibilities that exist in the universe.” Janna Levin
28. ### Importance of GW detection 100-yr wait for this: new astronomy

Direct, unabsorbed information about dynamics/ mass of relativistic sources  Black holes really do exist!

source

appears
35. ### A “kilonova” is born lighter elements (lanthanide-poor) heavier elements (lanthanide-rich)

(UV) (NIR) cf. also Cowperthwaite+2017 place where neutron stars merged
36. ### Importance of GW170817 Birth of multimessenger astronomy: photons + gravitons

Short gamma-ray bursts are due to neutron star collisions Neutron star collisions seem to produce most of the gold in the universe (lanthanides)