Gravitational waves Part I ICTP, 18-22 June 2018 Literature Some - - PowerPoint PPT Presentation

Stanislav (Stas) Babak.

AstroParticule et Cosmologie, CNRS (Paris)

Gravitational waves Part I

ICTP, 18-22 June 2018

Literature

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Some figures in these lectures are borrowed from theses books and these articles

LSC+Virgo, Phys. Rev. Lett. 116, 221101 (2016), LSC+Virgo, Phys. Rev. X 6, 041015 (2016), LSC+Virgo, Phys.Rev.Lett. 116 241102 (2016), LSC+Virgo, Phys.Rev. X6 041014 (2016), LSC+Virgo, Phys. Rev. Lett. 118, 221101 (2017), Berti et al., Class.Quantum Grav. 32, 243001 (2015), LSC+VIRGO, ArXiV: 1805.11579, S. Khan+, Phys.Rev. D93 (2016) 044007, LSC+Virgo arXiv: 1805.11579 , LIGO_Virgo, Astrophys.J. 848 (2017) L12, LIGO+Virgo Phys.Rev.Lett. 119 (2017) 161101, Babak+ Phys.Rev. D95 (2017) 103012, LISA consortium arXiv:1702.00786, Klein+ Phys.Rev. D93 (2016) 024003, LISA consortium, arXiv:1305.5720, Amaro-Seoane+ Class.Quant.Grav. 29 (2012) 124016 .

Lecture 1

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Notations and brief reminder of General Relativity Weak gravitational waves (GW) in vacuum Generation of GWs (quadrupole expression).

Gravitational Waves across frequency

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[Credits A. Sesana}

LISA

Main features of GWs

Gravitational waves (GWs) exist in every covariant theory of gravity In GR: GWs propagate with the speed of light GWs are acting as time varying tidal forces GWs have two polarization state (in case of linearly polarized wave: GWs weakly interact with the matter ( ) - are not scattered or absorbed, but hard to detect

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h+, h×

G = 6.674 × 10−11

Special relativity (reminder)

Existence of inertial coordinate systems (not accelerated) Equivalence of inertial coordinate frames Universality of the speed of light

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We use G=c=1 (mass, length, time in seconds)

interval: ∆s2 = −∆t2 + ∆x2 + ∆y2 + ∆z2 ≡ ηµν∆xµ∆xν

ηµν =     −1 1 1 1    

Minkowski metric,

ηµν∆xµ∆xν ≡

3

X

µ=0 3

X

ν=0

ηµν∆xµ∆xν

t x n u l l spacelike timelike worldline

~ u = dxµ d

Differential geometry (reminder)

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Non-flat geometry Metric defines the distances and angles:

ds2 = gµνdxµdxν

ds2 = ηµνdxµdxν

arbitrary metric flat (Minkowski) Local flatness theorem: the metric can be brought by a coordinate transformation at a point to Minkowski form, and all its first derivatives can be made = zero at that

• point. (but not the second derivatives!)

We can introduce Local Inertial Frame: approximately flat in a vicinity of a point (for the scale mauch less than radius of curvature of spacetime) dx ⌧ R

Differential geometry (reminder)

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Vector field: vector is defined continuously at each point: requires/involves comparison between vectors at two points.

d~ V dxµ = d(V ν~ eν) dxµ = dV ν dxµ ~ eν + V ν d~ eν dxµ

basis vector could be different at different points

d~ eν dxµ = Γα

µν~

eα

Christoffel symbols (not a tensor) Covariant derivative: V α

;β = V α ,β + V µΓα µβ

Γα

µν = 1

2gαβ(gβµ,ν + gβν,µ − gµν,β)

Covariant derivative of metric=0 =>

Differential geometry (reminder)

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Parallel transport of the vector along the curve: we transport the vector along the curve and we preserve its length and keep it parallel in nearby points. Local flatness: at a given point we can make Christoffel symbols=0: covariant derivative -> partial

dV α dλ = 0 = V α

,βuβ

parameter along the curve tangent to the curve Local inertial frame (LIF)

= uβV α

;β = 0

covariant extension Geodesic: the curve with the shortest distance connecting two points. Line in flat spacetime: how to extend? Line as straight as possible: tangent to a curve is parallel at nearby points - > parallel transport of a tangent vector.

uβuα

;β = 0

Differential geometry (reminder)

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Curvature tensor The vector parallel transported along the loop does not return to itself if space is curved A B C D

x1

x1 + δx1

x2 + δx2

x2

Vector propagated A->B->C->D->A ~ V

δV ∝ δx1δx2RV

curvature

Rα

βµν = Γα βν,µ − Γα βµ,ν + Γα ρµΓρ βν − Γα ρνΓρ βµ

Differential geometry (reminder)

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Rα

βµν = Γα βν,µ − Γα βµ,ν + Γα ρµΓρ βν − Γα ρνΓρ βµ

Properties of Riemann (curvature) tensor: Depends on the second derivatives of metric: cannot be eliminated by coord. transformation. Flat spacetime <-> Riemann=0 Second covariant derivatives do not commute (~ Riemann) Not all components independent (symmetries + differential Bianchi identity) In LIF:

Rαβµν = 1 2 [gαν,βµ − gαµ,βν + gβµ,αν − gβν,αµ]

Ricci tensor, Ricci scalar, Einstein tensor:

Rµν = gαβRαµβν

R = gαβRαβ

Gαβ = Rαβ − 1 2gαβR

Differential geometry (reminder)

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Geodesic deviation

ξα

ξα(λ)

x(λ0)

(x + δx)(λ0)

~ u|x+δx

~ u|x

Consider two geodesic, due to to local flatness ( ) they remain parallel for some time if started in the nearby points, but if spacetime is not flat they diverge/converge.

dx ⌧ R

ruruξα = Rα

µνβuµuνξβ

ruξα = uβξα

;β

• covar. deriv along the curve

General Relativity

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No particles neutral to grav. interaction Weak equivalence principle: freely falling particles move on timelike geodesics. Einstein equivalence principle: any local physical experiment not involving gravity has the same result in LIF and in flat spacetime Einstein equations: should be covariant, should have Newtonian limit

Rαβ − 1 2gαβR + Λgαβ = κT αβ

Gαβ

;β ≡ 0

T αβ

;β = 0

4 identities: choice of coordinate frame Bianchi identities: Covaiant conservation law Stress-energy tensor

Gauge transformation

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Consider weak gravitational field:

gµν = ηµν + hµν, |hµν| ⌧ 1

In the LIF

Rαβµν = 1 2 [hαν,βµ − hαµ,βν + hβµ,αν − hβν,αµ]

gauge transformation: we keep fixed Minkowski part but modify the field:

hnew

αβ

→ hold

αβ − ξα,β − ξβ,α

|hnew

µν | ⌧ 1

Rαβµν(hnew) = Rαβµν(hold)

x¯

α = xα + ξα(xν)

xi is arbitrary but preserving In terms of full metric it corresponds to the coordinate transformation:

GW in linearized gravity

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gαβ = gB

αβ + hαβ,

Rαβµν = RB

αβµν + RGW αβµν

gB

αβ =< gαβ >,

RB

αβµν =< Rαβµν >

We need to seprate the background geometry and GWs Averaging over several GW wavelength We will work with the metric, - GWs

hαβ

We use LIF for the background (!) geometry (LIF is also called sometimes local Lorentz frame)

l ⌧ R

λGW ⌧ R

Valid on the scales: still assume that

Weak GWs

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In LIF (w.r.t.) background:

gαβ = ηαβ + hαβ + O(h2)

Consider Einstein equations in vacuum (far from the source)

Rαβ = 0

¯ hµν = hµν − 1 2ηµνh, h = hµ

µ

¯ hµν

,ν = 0

⇤¯ hµν = 0

Introduce trace reverse metric Choose gauge (coordinate frame): Einstein’s field equations (linear): Remaining gauge freedom:

¯ hµν

new,ν = ¯

hµν

,ν − ⇤ξµ = 0,

⇤ξµ = 0

if

¯ h0α = 0, h = 0

T(ransverse)T(raceless) gauge: For the GW travelling in z-direction hµν = hT T

µν =

    h+ h× h× −h+    

<— harmonic gauge

Weak GWs

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Consider the plane wave solution:

¯ hµν = Aµνeikαxα, kα = {ω, ki}

4-wave vector Einstein equations:

kαkα = −!2 + ~ k.~ k = 0

Aµνkν = 0

transverse & traceless

A0µ = 0

propagates with tthe speed of light.

Aµ

µ = 0

Moreover: Note that in general the linearized Einstein equations have also static (non-radiative) solution (Newtonian potential). To get the radiative part we can take the “TT” part of the spatial metric by applying the projection operator:

hT T

jk = Prjklmhlm,

Prjklm = PjlPmk − 1 2PjkPlm, Pjk = δjk − njnk,

ni = ki/|~ k|

Weak GW, polarization

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We can choose polarization basis vectors arbitrary in the plane

• rthogonal to the direction of GW propagation

~ k

~ p

~ q

Introduce basis:

✏ij

+ = pipj − qiqj

✏ij

× = piqj + pjqi

hij = ✏ij

+h+ + ✏ij ×h×

Two polarization

RGW

0j0k = −1

2 ¨ hT T

jk

finally, Riemann tensor for GW:

Geodesic deviation in GW

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Consider two observers (A, B) moving on geodesic. We use LIF associated with the observer A, which is at rest. Consider GW propagating in z-direction, then the distance between A & B:

d2ξi dt2 = Ri

00jξj = 1

2 ¨ hjk

T T ξk

δξj = ξj − ξj

Introduce:

δξj = 1 2hjk

T T ξk

δx = 1 2x0h+ δy = −1 2y0h+ δx = 1 2y0h× δy = 1 2x0h×

Generation of GWs

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Consider isolated source with weak internal gravity. Observer (field point) Assume that the observer is far away (far zone), we can use LIF of the observer As before we use trace-reverse form of the metric We use harmonic gauge (coordinates)

⇤¯ hµν = −16π(T µν + tµν)

Einstein equations: Contains all non-linear terms from the r.h.s We start with the linear order (neglecting the red term)

¯ hµν = 4 Z T µν(x0, t0 = t − |~ x0 − ~ x|) |~ x0 − ~ x| d3~ x0

Solution in form of retarded potentials

Generation of GWs

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Assume slow motion (v<<c) Take observer far away (far zone) We are interested in the radiative part of gravitational potentials => take “TT” part

|~ x ~ x0| ⇡ R GW

hT T

jk =

 4 R Z Tjk(x0, t0 = t − R)d3x0 T T

Use the conservation law

T µν

,ν = 0

hT T

jk =

 2 R d2 dt2 Mjk(t − R) T T

Mjk = Z T 00 ✓ xjxk − 1 3δjkr ◆ d3x

Quadrupole formula (Landau & Lifshitz) mass quadrupole moment

Generation of GWs

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Besides leading order (mass quadrupole) other moments also give contribution. There are two types of moments: mass-moments and current-moments

Il ∼ MLl Sl ∼ MvLl

mass moments current moments

h+,× ∼ 1 R d2I2 dt2 &d3I3 dt3 &....&d2S2 dt2 &d3S3 dt3 ...

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1 R dlIl dtl ∼ M R vl

Einstein equations are non-linear: grav field is its own source (the red term which we have neglected). Post-Newtonian expansion:

gµν = ηµν + εh(1)

µν + ε2h(2) µν + ...

ε = v/c ⌧ 1

Solving Einstein equations iteratively updating the equation of motion at each step