General Relativity and Gravitational Waveforms · Spacetime And Geometry: An Introduction To...

General Relativity and GravitationalWaveforms

Deirdre Shoemaker

Center for Relativistic AstrophysicsSchool of Physics

Georgia Institute of Technology

Kavli Summer Program in Astrophysics 2017Astrophysics with Gravitational Wave Detections

Copenhagen Niels Bohr Institute

Deirdre Shoemaker General Relativity and Gravitational Waveforms

References

Spacetime And Geometry: An Introduction To General Relativity, Sean Carroll, Pearson (2016), ISBN-10:9332571651, ISBN-13: 978-9332571655

Gravity: An Introduction to Einstein’s General Relativity, James B. Hartle, Pearson (2003), ISBN-10:0805386629, ISBN-13: 978-0805386622

Numerical Relativity: Solving Einstein’s Equations on a Computer. Thomas Baumgarte and Stuart Shapiro,Cambridge University Press, ISBN: 9780521514071

Introduction to 3+1 Numerical Relativity. Miguel Alcubierre, Oxford University Press, ISBN13:9780199205677

Relativistic Hydrodynamics. Luciano Rezzolla, Oxford University Press, ISBN: 978-0-19-852890-6

Astro-GR Online Course on GWs http://astro-gr.org/online-course-gravitational-waves/

2nd Fudan Winter School on Astrophysics Black Holes Pablo Laguna’s and DS’s Courseshttp://bambi2017.fudan.edu.cn/bh2017/Program.html


Goals

By the end of these three lectures, I intend for you to

understand the connection between the gravitational waveform seen inthe figure to Einstein’s General Theory of Relativity,

recognize the techniques employed to predict theoretical gravitationalwavesforms, and what the best use practices are for each,

and develop some intuition on how the waveform depends on thephysical parameters of the black holes.


Lecture 1: General Relativity


Gravity as Geometry

Where do gravitational wave come from?Hint: Einstein is the stork.

According to Einstein:

The metric tensor describing the curvature of spacetime is thedynamical field responsible for gravitation.

Gravity is not a field propagating through spacetime but rather aconsequence of curved geometry.

Gravitational interactions are universal (Principle of equivalence)


Index Notation


The Metric: gµ⌫

The metric gµ⌫ :

(0, 2) tensor,

gµ⌫ = g⌫µ (symmetric)

g = |gµ⌫ | 6= 0 (non-degenerate)

g

µ⌫ (inverse metric)

g

µ⌫ is symmetric and g

µ⌫g⌫� = �µ� .

gµ⌫ and g

µ⌫ are used to raise and lower indices on tensors.


gµ⌫ properties

The metric:

provides a notion of “past” and “future”

allows the computation of path length and proper time:ds

2 = gµ⌫ dx

µdx

⌫

determines the “shortest distance” between two points

replaces the Newtonian gravitational field

provides a notion of locally inertial frames and therefore a senseof “no rotation”

determines causality, by defining the speed of light faster thanwhich no signal can travel

replaces the traditional Euclidean three-dimensional dot productof Newtonian mechanics ds · ds = ds

2 = gµ⌫ dx

µdx

⌫


Example: Space-time interval of flat spacetime

(ds)2 = �c

2(dt)2 + (dx)2 + (dy)2 + (dz)2 .

Notice:

ds

2 can be positive, negative, or zero.

c is some fixed conversion factor between space and time (NB:relativists drive people nuts by setting c = 1 and G = 1)

c is the conversion factor that makes ds

2 invariant.

The minus sign is necessary to preserve invariance.

Using the summation convention,

ds

2 = ⌘µ⌫dx

µdx

⌫ ,

where dx

µ = �x

µ = (�t ,�x ,�y ,�z) or (dt , dx , dy , dz) and ⌘µ⌫ is a4 ⇥ 4 matrix called the metric:

⌘µ⌫ =

0

BB@

�c

2 0 0 00 1 0 00 0 1 00 0 0 1

1

CCA .


Dot Product

A vector is located at a given point in space-time

A basis is any set of vectors which both spans the vector space and islinearly independent .

Consider at each point a basis e(µ) adapted to the coordinates x

µ; thatis, e(1) pointing along the x-axis, etc. Then, any abstract vector A canbe written as

A = A

µe(µ) .

The coefficients A

µ are the components of the vector A.

The real vector is the abstract geometrical entity A, while thecomponents A

µ are just the coefficients of the basis vectors in someconvenient basis.

The parentheses around the indices on the basis vectors e(µ) labelcollection of vectors, not components of a single vector.


Definition of metric:gµ⌫ = e(µ) · e(⌫)

Inner or dot product: Given the metric g,

g(V ,W ) = gµ⌫V

µW

⌫ = V · W

If g(V ,W ) = 0, the vectors are orthogonal.

Since g(V ,W ) = V · W is a scalar, it is left invariant underLorentz transformations.

norm of a vector is given by V · V .

if gµ⌫V

µV

⌫is

8<

:

< 0 , V

µis timelike

= 0 , V

µis lightlike or null

> 0 , V

µis spacelike .


Example in flat spacetime: when gµ⌫ = ⌘µ⌫ in Cartesian coordinates

⌘µ⌫ =

0

BB@

�c

2 0 0 00 1 0 00 0 1 00 0 0 1

1

CCA = diag(�c

2, 1, 1, 1)

V · W = ⌘µ⌫V

µW

⌫ = ⌘tt

V

t

W

t + ⌘tx

V

t

W

x + ⌘ty

V

t

W

y + ⌘tz

V

t

W

z

+ ⌘xt

V

x

W

t + ⌘xx

V

x

W

x + ⌘xy

V

x

W

y + ⌘xz

V

x

W

z

+ ⌘yt

V

y

W

t + ⌘yx

V

y

W

x + ⌘yy

V

y

W

y + ⌘yz

V

y

W

z

+ ⌘zt

V

z

W

t + ⌘zx

V

z

W

x + ⌘zy

V

z

W

y + ⌘zz

V

z

W

z

= �c

2V

t

W

t + V

x

W

x + V

y

W

y + V

z

W

z

Example in flat spacetime: in spherical polar coordinates

⌘µ⌫ = diag(�c

2, 1, r2, r2sin

2✓)

V · W = ⌘µ⌫V

µW

⌫ = �c

2V

t

W

t + V

r

W

r + r

2V

✓W

✓ + r

2sin

2✓V

�W

�


(ds)2 = ⌘µ⌫dx

µdx

⌫

Light Cone: Events are

Time-like separated if (ds)2 < 0

Space-like separated if (ds)2 > 0

Null or Light-like separated if (ds)2 = 0

Proper Time ⌧ : Measures the time elapsed asseen by an observer moving on a straight pathbetween events. That is c

2(d⌧)2 = �(ds)2

Notice: if dx

i = 0 then c

2(d⌧)2 = �⌘µ⌫ (dt)2, thusd⌧ = dt .


Gravity is Universal

Weak Principle of Equivalence (WEP)The inertial mass and the gravitational mass of any object are equal

~F = m

i

~a

~F

g

= �m

g

r�

with m

i

and m

g

the inertial and gravitational masses, respectively.

According to the WEP: m

i

= m

g

for any object. Thus, the dynamics ofa free-falling, test-particle is universal, independent of its mass; thatis, ~a = �r�

Weak Principle of Equivalence (WEP)The motion of freely-falling particles are the same in a gravitationalfield and a uniformly accelerated frame, in small regions of spacetime


Einstein Equivalence PrincipleIn small regions of spacetime, the laws of physics reduce to those ofspecial relativity; it is impossible to detect the existence of agravitational field by means of local experiments.

Due to the presence of the gravitational field, it is not possible tobuild, as in SR, a global inertial frame that stretches throughspacetime.

Instead, only local inertial frames are possible; that is, inertialframes that follow the motion of individual free-falling particles ina small enough region of spacetime.

Spacetime is a mathematical structure that locally looks likeMinkowski or flat spacetime, but may posses nontrivial curvatureover extended regions.


Physics in Curved SpacetimeWe are now ready to address:

how the curvature of spacetime acts on matter to manifest itself asgravity’how energy and momentum influence spacetime to create curvature.

Weak Principle of Equivalence (WEP)

The inertial mass and gravitational mass of any object are equal.

Recall Newton’s Second Law.f = m

i

a .

with m

i

the inertial mass.

On the other hand,fg

= �m

g

r� .

with � the gravitational potential and m

g

the gravitational mass.

In principle, there is no reason to believe that m

g

= m

i

.

However, Galileo showed that the response of matter to gravitation wasuniversal. That is, in Newtonian mechanics m

i

= m

g

. Therefore,

a = �r� .


Minimal Coupling Principle

Take a law of physics, valid in inertial coordinates in flat spacetime

Write it in a coordinate-invariant (tensorial) form

Assert that the resulting law remains true in curved spacetime

Operationally, this principle boils down to replacing

the flat metric ⌘µ⌫ by a general metric gµ⌫

the partial derivative @µ by the covariant derivative rµ


Example: Newton’s 2nd Law and Special Relativity

~f = m

~a =

d

~p

dt

in Special Relativity

f

µ = m

d

2

d⌧2 x

µ(⌧) =d

d⌧p

µ(⌧)


Covariant Derivative of Vectors

Consider v(x↵) and v(x↵ + dx

↵) such that dx

↵ = t

↵ ✏ with t

↵ definingthe direction of the covariant derivative.Parallel transport the vector v(x↵ + t

↵ ✏) back to the point x

↵ and call itvk(x

↵)

Covariant Derivative:

rtv(x↵) = lim✏!0

vk(x↵)� v(x↵)

✏

In a local inertial frame:

(rtv)↵ = t

�@�v

↵

Thus,r�v

↵ = @�v

↵

Notice: The above expression is not valid in curvilinear coordinates. Ingeneral,

v

↵k (x

�) = v

↵(x� + ✏t�) +�↵��v

�(x�)(✏t�)

component changes basis vector changes

Thereforer�v

↵ = @�v

↵ + �↵��v

�


Consequently:

Covariant differentiation of Vectors

r�v

↵ = @�v

↵ + �↵��v

�

Covariant differentiation of 1-forms

rµ!⌫ = @µ!⌫ � ��µ⌫!�

Covariant differentiation of general Tensors

r�T

µ1µ2···µk

⌫1⌫2···⌫l

= @�T

µ1µ2···µk

⌫1⌫2···⌫l

+�µ1�� T

�µ2···µk

⌫1⌫2···⌫l

+ �µ2�� T

µ1�···µk

⌫1⌫2···⌫l

+ · · ·��

�⌫1 T

µ1µ2···µk

�⌫2···⌫l

� ��⌫2 T

µ1µ2···µk

⌫1�···⌫l

� · · ·


Christoffel symbols

From metric compatibility:

r⇢gµ⌫ = @⇢gµ⌫ � ��⇢µg�⌫ � ��

⇢⌫gµ� = 0rµg⌫⇢ = @µg⌫⇢ � ��

µ⌫g�⇢ � ��µ⇢g⌫� = 0

r⌫g⇢µ = @⌫g⇢µ � ��⌫⇢g�µ � ��

⌫µg⇢� = 0

Subtract the second and third from the first,

@⇢gµ⌫ � @µg⌫⇢ � @⌫g⇢µ + 2��µ⌫g�⇢ = 0 .

Multiply by g

�⇢ to get

Christoffel Symbols

��µ⌫ =

12

g

�⇢(@µg⌫⇢ + @⌫g⇢µ � @⇢gµ⌫)


The Christoffel symbols vanish in flat space in Cartesian coordinates

The Christoffel symbols do not vanish in flat space in curvilinearcoordinates.

For example, if ds

2 = dr

2 + r

2d✓2, it is not difficult to show that

�r

✓✓ = �r and �✓✓r

= 1/r

At any one point p in a spacetime (M, gµ⌫), it is possible to find acoordinate system for which ��

µ⌫ = 0 (recall local flatness)


Example: Motion of freely-falling particles. In Flat spacetime

d

2x

µ

d�2 = 0

Rewrited

2x

µ

d�2 =dx

⌫

d�@⌫

dx

µ

d�= 0

Substitutedx

⌫

d�@⌫

dx

µ

d�! dx

⌫

d�r⌫

dx

µ

d�

Thusd

2x

µ

d�2 + �µ⇢�dx

⇢

d�

dx

�

d�= 0 .


The Newtonian LimitGiven a General Relativistic expression, one recover the Newtoniancounterparts by

particles move slowly with respect to the speed of light.the gravitational field is weak, namely a perturbation of spacetime.the gravitational field is static.

Consider the geodesic equation.Moving slowly implies

dx

i

d⌧<<

dt

d⌧,

sod

2x

µ

d⌧ 2 + �µ00

✓dt

d⌧

◆2

= 0 .

Static gravitational field implies

�µ00 =

12

g

µ�(@0g�0 + @0g0� � @�g00)

= �12

g

µ�@�g00 .

Weakness of the gravitational field implies

gµ⌫ = ⌘µ⌫ + hµ⌫ , |hµ⌫ | << 1 .Lecture 1: General Relativity

Commutator of two covariant derivatives: it measures the difference betweenparallel transporting the tensor first one way and then the other, versus theopposite ordering.

�

µ

��

�

µ

�

�

That is:

[rµ,r⌫ ]V⇢ = rµr⌫V

⇢ �r⌫rµV

⇢

= @µ(r⌫V

⇢)� ��µ⌫r�V

⇢ + �⇢µ�r⌫V

� � (µ $ ⌫)

= @µ@⌫V

⇢ + (@µ�⇢⌫�)V

� + �⇢⌫�@µV

� � ��µ⌫@�V

⇢ � ��µ⌫�

⇢��V

�

+�⇢µ�@⌫V

� + �⇢µ��

�⌫�V

� � (µ $ ⌫)

= (@µ�⇢⌫� � @⌫�

⇢µ� + �⇢

µ��⌫� � �⇢

⌫��µ�)V

� � 2��[µ⌫]r�V

⇢

= R

⇢�µ⌫V

� � Tµ⌫�r�V

⇢

where

Riemann Tensor

R

⇢�µ⌫ = @µ�

⇢⌫� � @⌫�

⇢µ� + �⇢

µ��⌫� � �⇢

⌫��µ�


The Ricci Tensor

Ricci Tensor

Rµ⌫ = R

�µ�⌫ .

Because of R⇢�µ⌫ = Rµ⌫⇢�

Rµ⌫ = R⌫µ ,

Also

Ricci scalar

R = R

µµ = g

µ⌫Rµ⌫ .


The Einstein TensorContract twice the Bianchi identity

r�R⇢�µ⌫ +r⇢R��µ⌫ +r�R�⇢µ⌫ = 0

to get

0 = g

⌫�g

µ�(r�R⇢�µ⌫ +r⇢R��µ⌫ +r�R�⇢µ⌫)= rµ

R⇢µ �r⇢R +r⌫R⇢⌫ ,

orrµ

R⇢µ � 12r⇢R = 0

Define Einstein Tensor

Gµ⌫ = Rµ⌫ � 12

R gµ⌫ ,

Einstein EquationGµ⌫ = 8⇡Tµ⌫

Einstein Equation in Vacuum

Gµ⌫ = 0


Lectures

L1: General Relativity

L2: Numerical Relativity

L3: Gravitational Waveforms


Lecture 2: Numerical Relativity


Numerical Relativity

From:

Gµ⌫ = 8⇡ Tµ⌫

To: t + �t

t

Ini$al'Data'

Boundary'Condi$ons'

Initial Value and Boundary Problem


This Lecture

3+1 Decomposition:FoliationsTensor ProjectionsADM formulationBSSN formulationChoosing CoordinatesMoving Puncture CoordinatesInitial DataBoundary ConditionsGW extraction


Space-time Foliation

Foliate the space-time (M, gab) into a family of non-intersecting,space-like, three-dimensional hyper-surfaces ⌃t leveled by ascalar function t with time-like normal:

⌦a = rat

such that its magnitude is given by

|⌦|2 = gabratrbt ⌘ �↵�2,

with ↵ the lapse function.


Space-time Foliation

Define the unit normal vector na:

na ⌘ �↵gab⌦b = �↵gabrbt

and spatial metric:�ab = gab + nanb


Projection Operator and Covariant Differentiation

Space-like projection operator:

�ab = gac�cb = ga

b + nanb = �ab + nanb

It is easy to show that �abnb = 0.

Three-dimensional covariant derivative compatible with �ab:

DaT bc ⌘ � d

a �b

e �f

c rdT ef .

It is easy to show that Da�bc = 0.


Extrinsic Curvature

Kab is space-like and symmetric by construction and can bewritten in terms of the acceleration of normal observersaa = nbrbna = Da ln↵ as

Kab = �ranb � naab. (1)

Kab can also be written in terms of the Lie derivative of thespatial metric along the normal vector na

Kab = �12Ln�ab (2)

Kab is the “velocity” of the spatial metric.


Lie Derivatives

LXf = X bDbf = X b@bf (3)LXva = X bDbva � vbDbX a = [X , v ]a (4)LX!a = X bDb!a + !bDaX b (5)LXT a

b = X c@cT ab � T c

b@cX a + T ac@bX c (6)


Einstein Constraints

The Hamiltonian and momentum constraints:

R + K 2 � KabK ab = 16⇡⇢ (7)

DbK ba � DaK = 8⇡ja, (8)

Only involve spatial quantities and their spatial derivatives.They have to hold on each individual spatial slice ⌃t

They are the necessary and sufficient integrabilityconditions for the embedding of the spatial slices(⌃, �ab, Kab) in the space-time (M, gab).


3+1 time derivatives

To derive the evolution equations for �ab and Kab one needs atime derivative. The Lie derivative along Ln is not a natural timederivative orthogonal to ⌃t (e.g. na is not dual to ⌦a). That is,

na⌦a = �↵gabrbtrat = ↵�1. (9)

However, the vectorta = ↵na + �a (10)

is dual to ⌦a for any spatial shift vector �a. That is

ta⌦a = tarat = 1 . (11)


3+1 Foliations

t + �t

t

↵ na

�ab

ta

�a

↵ and �a determine how the coordinates evolve from oneslice ⌃t to the next along the time direction ta.↵ determines how much proper time elapses betweentime-slices along the normal vector na.�a determines by how much spatial coordinates are shiftedwith respect to the normal vector na.


ADM formulation in 3+1 Coordinates

Hamiltonian constraint

R + K 2 � KijK ij = 16⇡⇢, (12)

Momentum constraint

DjKji � DiK = 8⇡ji , (13)

�ab evolution equation:

@t�ij � L��ab = �2↵Kij (14)

Kab evolution equation:

@tKij � L�Kab = �DiDj↵+ ↵(Rij � 2KikK kj + KKij)

�↵8⇡(Sij �12�ij(S � ⇢))

(15)


Analogy with Electrodynamics

Constraint equation:DiEi = 4⇡⇢e (16)

Evolution equations:

@tAi = �Ei � Di� (17)

@tEi = �DjDjAi + DiDjAj � 4⇡Ji (18)

The gauge quantity � is the analogue of the lapse ↵ andshift � i .The vector potential Ai is the analogue of the spatial metric�ij

The electric field Ei is the analogue of the extrinsiccurvature Kij


BSSN formulation

Introduced first by Shibata and Nakamura and re-introducedlater by Baumgarte and Shapiro.Start with the conformal transformation

�ij = e�4��ij , (19)

and choose �ij = 1.Split the extrinsic curvature as

Kij = Aij +13�ijK (20)

with Aij = 0 and choose the following conformal rescaling

Aij = e�4�Aij . (21)


BSSN formulation

From @t ln � = � ij@t�ij and the trace of the �ij evolution equation:

@t ln �1/2 = �↵K + Di�i , (22)

Substitution of � = (ln �)/12 yields the � evolution equation:

@t� = �16↵K + � i@i�+

16@i�

i (23)


BSSN formulation

Similarly, combining the trace of the Kij evolution equations withthe Hamiltonian constraint gives

@tK = �D2↵+ ↵hKijK ij + 4⇡(⇢+ S)

i+ � iDiK , (24)

where D2 ⌘ � ijDiDj .Substitution of Aij = e�4�Aij in this equations yields the Kevolution equation:

@tK = �D2↵+ ↵

Aij Aij +

13

K 2 + 4⇡(⇢+ S)

�+ � i@iK . (25)


BSSN formulation

Subtracting the @t� and @tK evolution equations from the @t�ijand @tKij yields

@t �ij = �2↵Aij + �k@k �ij + �ik@j�k + �kj@i�

k � 23�ij@k�

k . (26)

and

@t Aij = e�4�h�(DiDj↵)

TF + ↵(RTFij � 8⇡STF

ij )i

+↵(K Aij � 2Ail Alj)

+�k@k Aij + Aik@j�k + Akj@i�

k � 23 Aij@k�

k .

(27)

where TF denotes BTFij = Bij � �ijB/3.


BSSN conformal connection

Define the conformal connection functions

�i ⌘ � jk �ijk = �� ij

,j , (28)

The Ricci tensor can be written as

Rij = �12 �

lm�ij,lm + �k(i@j)�k + �k �(ij)k+

� lm⇣

2�kl(i �j)km + �k

im�klj

⌘.

(29)

Notice: The only second derivatives of �ij left over in thisoperator is the Laplace operator � lm�ij,lm – all others have beenabsorbed in first derivatives of �i .


Why bother introducing e�i?

Recall the wave equation

@tt� = �� (30)

or

@t� = ⇧ (31)@t⇧ = �� (32)

With e�i the BSSN equations have the structure of

@t �ij / Aij (33)

@t Aij / ��ij (34)


BSSN formulation: �i evolution equation

From the time derivative of �i = �� ij,j and the evolution

equation for �ij one gets

@t �i = �@j

h2↵Aij � 2�m(j� i)

,m +23� ij� l

,l + � l � ij,l

i. (35)

The divergence of the Aij can be eliminated with the help of themomentum constraint yielding the �i evolution equation:

@t �i = �2Aij@j↵+ 2↵⇣�i

jk Akj � 23 �

ij@jK � 8⇡� ijSj + 6Aij@j�⌘

+� j@j �i � �j@j�

i + 23 �

i@j�j + 1

3 �li� j

,jl + � lj� i,lj .

(36)


Analogy with Electrodynamics

Recall:

@tAi = �Ei � Di� (37)

@tEi = �DjDjAi + DiDjAj � 4⇡Ji (38)

Auxiliary variable:� = DiAi . (39)

New evolution equations:

@tAi = �Ei � Di� (40)@tEi = �DjDjAi + Di�� 4⇡Ji (41)

@t� = @tDiAi = Di@tAi

= �DiEi � DiDi�

= �DiDi�� 4⇡⇢e. (42)


Moving Puncture Coordinates

Requirements:Lapse collapses to zero at the puncture, hiding the blackhole singularity.Non-vanishing shift to advect the frozen puncture throughthe domain


Moving Puncture Coordinates

Gauge Conditions

@t↵ = �2↵K + �� i@i↵

@t�i ⌘ ⇠Bi

@tBi = �@t �i � ⌘Bi � ⇣� j@j �

i

with ⇠, �, ⌘, � and ⇣ parameters.The conditions are modifications to the so-called 1+log slicingand Gamma-driver shift conditions [see, Gauge conditions forlong-term numerical black hole evolutions without excision,Alcubierre et al, Phys.Rev. D67 (2003) 084023]


Boundary Conditions

Far away from the sources, in the wave-zone, all quantitieshave the following asymptotic behavior

�(t , x , y , z) =1r n�(r � t)

Since @t�+ @r� = 0, then

@t� = � 1r n @r [r n �]


Initial Data

Recall the constraints:

R + K 2 � KabK ab = 16⇡⇢

DbK ba � DaK = 8⇡ja,

Notice: 4 equations and 12 variables {�ij , Kij}York et.al suggested conformal and transverse-tracelessdecompositions

�ij = 4�ij

K ij = �10Aij +13� ijK


Initial Data

Aij = AijTT + Aij

L,

where the transverse part is divergenceless

Dj AijTT = 0

and where the longitudinal part satisfies

AijL = DiW j + DjW i � 2

3� ij DkW k ⌘ (LW )ij .


Initial Data

Hamiltonian Constraint

8 D2 � R � 23 5K 2 + �7Aij Aij = �16⇡ 5⇢,

Momentum constraint

(�LW )i � 23 6� ij DjK = 8⇡ 10j i .

4-equations for 4-unknowns { , W i}


Initial Data

8 Assumptions

�ij = ⌘ij

K = 0Aij

TT = 0

Then Hamiltonian Constraint

8 D2 + �7LWij LW ij = �16⇡ 5⇢,

Momentum constraint

(�LW )i = 8⇡ 10j i .


Initial Data: Binary Black Holes

For black holes there are well known solutions (Bowen-York) tothe momentum constraint (�LW )i = 0. Thus, constructinginitial data reduces to solving the Hamiltonian Constraint

8 D2 + �7LWij LW ij = 0,


Gravitational Wave Extraction

The Weyl tensor scalar 4 is related to the grav. wave strainpolarizations:

4 = h+ � h⇥

How does one construct 4 from the numerical relativitysimulations?Start with an orthonormal tetrad {ea

(N)} and build the null-tetrad:

la =1p2

⇣ea(0) + ea

(1)

⌘

ka =1p2

⇣ea(0) � ea

(1)

⌘

ma =1p2

⇣ea(2) + i ea

(1)

⌘

ma =1p2

⇣ea(2) � i ea

(1)

⌘


Gravitational Wave Extraction

Then

4 = Cabcdka mb kc md


Spherical Harmonics

rM 4(◆,�, t) =X

`,m�2Y`,m(◆,�)C`,m(t)


Higher Order Modes


Conversion to Strain

h(t) = h+(t)� ihx(t) =Z t

�1dt 0

Z t

�1dt 00 4 .

fundamental uncertainties in producing strain from 4

due to integration of finite length, discretely sampled, noisy dataresults in large secular non-linear driftsmost groups use a method developed by Pollney and Reisswig(arXiv:1006.1632) that integrates in the frequency domain


Lecture 3: Waveforms


Gravity as Geometry

What does the detector measure?How do we get NR waveforms in that formatwhy is NR not enough?what are waveform models


The post-Newtonian (PN) Approximation

The PN method involves an expansion around the Newtonian limitkeeping terms of higher order in the small parameter [?, ?]

✏ ⇠ v2

c2 ⇠ |hµ⌫ | ⇠��@0h@i h

��2


In The Know

Key definitions and lingoIn progress� =M = m1 + m2q = m1

m2⌘ =


IMR WaveformsIMR: Ispiral Merger Ringdown Waveforms

Left: GW signal from q=1 nonspinning BH binary as predicted at2.5PN order by Buonanno and Damour (2000)

The merger is assumed almost instantaneous and one QNM isincluded

Right: GW signal from q=1 BH binary with a small spin�1 = �2 = 0.06 obtained in full general relativity by Pretorius

-200 -100 0 100

t/M

-0.2

-0.1

0

0.1

0.2

0.3

h(t

)

inspiral-plungemerger-ring-down

-200 -100 0 100t/M

-0.2

-0.1

0

0.1

0.2

0.3

h(t

)

numerical relativity


IMR Waveforms

sky-averaged SNR for q=1, nonspinning binary with PN inspiralwaveform and full NR waveform for noise spectral density ofLIGO/LISA,

30 60 90 120 150 180M (M

sun)

0

5

10

15

20

SN

R a

t 100 M

pc

numerical relativityPN inspiral

105

106

107

M (Msun

)

102

103

104

SN

R a

t 3

Gp

c

numerical relativityPN inspiral


Horizons & Merger

MORE COMING!


For fun, I have included 4 problems in relativity. The solutions are at the end of these notes. I will not go into problems from Numerical Relativity, but I have included a couple of papers here if you would like to get started. The Einstein Toolkit is publicly available software but beyond the scope of these lectures. The Einstein Toolkit community runs schools of its own.

Introduction to the EinsteinToolkit: https://arxiv.org/abs/1305.52991.Numerical Relativity Review: https://arxiv.org/abs/gr-qc/01060722.

Problem 3: Which of the following are correct according to index notation?Problem 4:

Warning: Solutions Follow. Typos and mistakes should be expected.

Problem 1

Problem 4

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