General Relativity and Gravitational Waveforms

Deirdre Shoemaker

Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology

Kavli Summer Program in Astrophysics 2017 Astrophysics 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, ISBN 13:9780199205677

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 Courses http://bambi2017.fudan.edu.cn/bh2017/Program.html

Deirdre Shoemaker General Relativity and Gravitational Waveforms Goals

By the end of these three lectures, I intend for you to understand the connection between the gravitational waveform seen in the figure to Einstein’s General Theory of Relativity, recognize the techniques employed to predict theoretical gravitational wavesforms, and what the best use practices are for each, and develop some intuition on how the waveform depends on the physical parameters of the black holes.

Deirdre Shoemaker General Relativity and Gravitational Waveforms Lecture 1: General Relativity

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 the dynamical field responsible for gravitation. Gravity is not a field propagating through spacetime but rather a consequence of curved geometry. Gravitational interactions are universal (Principle of equivalence)

Lecture 1: General Relativity Index Notation

Lecture 1: General Relativity The Metric: gµ⌫

The metric gµ⌫ : (0, 2) tensor,

gµ⌫ = g⌫µ (symmetric) g = g = 0 (non-degenerate) | µ⌫ |6 gµ⌫ (inverse metric) µ⌫ µ⌫ µ g is symmetric and g g⌫ = . µ⌫ gµ⌫ and g are used to raise and lower indices on tensors.

Lecture 1: General Relativity gµ⌫ properties

The metric:

provides a notion of “past” and “future” allows the computation of path length and proper time: 2 µ ⌫ ds = gµ⌫ dx dx determines the “shortest distance” between two points replaces the Newtonian gravitational field provides a notion of locally inertial frames and therefore a sense of “no rotation” determines causality, by defining the speed of light faster than which no signal can travel replaces the traditional Euclidean three-dimensional dot product of Newtonian mechanics ds ds = ds2 = g dx µ dx ⌫ · µ⌫

Lecture 1: General Relativity Example: Space-time interval of flat spacetime

(ds)2 = c2(dt)2 +(dx)2 +(dy)2 +(dz)2 . Notice: ds2 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 ds2 invariant. The minus sign is necessary to preserve invariance. Using the summation convention,

2 µ ⌫ ds = ⌘µ⌫ dx dx , µ µ where dx =x =(t, x, y, z) or (dt, dx, dy, dz) and ⌘µ⌫ is a 4 4 matrix called the metric: ⇥ c2 000 0100 ⌘µ⌫ = 0 1 . 0010 B 0001C B C @ A

Lecture 1: General Relativity 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 is linearly independent . µ Consider at each point a basis eˆ(µ) adapted to the coordinates x ; that is, eˆ(1) pointing along the x-axis, etc. Then, any abstract vector A can be written as µ A = A eˆ(µ) . The coefficients Aµ are the components of the vector A. The real vector is the abstract geometrical entity A, while the components Aµ are just the coefficients of the basis vectors in some convenient basis.

The parentheses around the indices on the basis vectors eˆ(µ) label collection of vectors, not components of a single vector.

Lecture 1: General Relativity Definition of metric: g = eˆ eˆ µ⌫ (µ) · (⌫) Inner or dot product: Given the metric g,

g(V , W )=g V µW ⌫ = V W µ⌫ ·

If g(V , W )=0, the vectors are orthogonal. Since g(V , W )=V W is a scalar, it is left invariant under · Lorentz transformations. norm of a vector is given by V V . · < 0 , V µ is timelike µ ⌫ µ if gµ⌫ V V is = 0 , V is lightlike or null 8 µ <> 0 , V is spacelike . :

Lecture 1: General Relativity Example in flat spacetime: when gµ⌫ = ⌘µ⌫ in Cartesian coordinates

c2 000 0100 2 ⌘µ⌫ = 0 1 = diag( c , 1, 1, 1) 0010 B 0001C B C @ A µ ⌫ t t t x t y t z V W = ⌘µ⌫ V W = ⌘tt V W + ⌘tx V W + ⌘ty V W + ⌘tz V W · x t x x x y x z + ⌘xt V W + ⌘xx V W + ⌘xy V W + ⌘xz V W y t y x y y y z + ⌘yt V W + ⌘yx V W + ⌘yy V W + ⌘yz V W z t z x z y z z + ⌘zt V W + ⌘zx V W + ⌘zy V W + ⌘zz V W = c2V t W t + V x W x + V y W y + V z W z Example in flat spacetime: in spherical polar coordinates

⌘ = diag( c2, 1, r 2, r 2sin2✓) µ⌫

V W = ⌘ V µW ⌫ = c2V t W t + V r W r + r 2V ✓W ✓ + r 2sin2✓V W · µ⌫

Lecture 1: General Relativity 2 µ ⌫ (ds) = ⌘µ⌫ dx dx

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 as seen by an observer moving on a straight path between events. That is c2(d⌧)2 = (ds)2 i 2 2 2 Notice: if dx = 0 then c (d⌧) = ⌘µ⌫ (dt) , thus d⌧ = dt.

Lecture 1: General Relativity Gravity is Universal

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

~ F = mi ~a F~ = m g gr

with mi and mg the inertial and gravitational masses, respectively.

According to the WEP: mi = mg for any object. Thus, the dynamics of a free-falling, test-particle is universal, independent of its mass; that is, ~a = r Weak Principle of Equivalence (WEP) The motion of freely-falling particles are the same in a gravitational field and a uniformly accelerated frame, in small regions of spacetime

Lecture 1: General Relativity Einstein Equivalence Principle In small regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field by means of local experiments.

Due to the presence of the gravitational field, it is not possible to build, as in SR, a global inertial frame that stretches through spacetime. Instead, only local inertial frames are possible; that is, inertial frames that follow the motion of individual free-falling particles in a small enough region of spacetime. Spacetime is a mathematical structure that locally looks like Minkowski or flat spacetime, but may posses nontrivial curvature over extended regions.

Lecture 1: General Relativity Physics in Curved Spacetime

We are now ready to address: how the curvature of spacetime acts on matter to manifest itself as gravity’ 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 = mi a .

with mi the inertial mass.

On the other hand, fg = mg . r with the gravitational potential and mg the gravitational mass.

In principle, there is no reason to believe that mg = mi .

However, Galileo showed that the response of matter to gravitation was universal. That is, in Newtonian mechanics mi = mg . Therefore, a = . r Lecture 1: General Relativity 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 µ r

Lecture 1: General Relativity Example: Newton’s 2nd Law and Special Relativity

d~p ~f = m ~a = dt in Special Relativity

d 2 d f µ = m x µ(⌧)= pµ(⌧) d⌧ 2 d⌧

Lecture 1: General Relativity Covariant Derivative of Vectors

Consider v(x ↵) and v(x ↵ + dx ↵) such that dx ↵ = t ↵ ✏ with t ↵ defining the direction of the covariant derivative. Parallel transport the vector v(x ↵ + t ↵ ✏) back to the point x ↵ and call it v (x ↵) k Covariant Derivative: ↵ ↵ ↵ v (x ) v(x ) tv(x )=lim k ✏ 0 ✏ r ! In a local inertial frame: ↵ ↵ ( v) = t @ v rt Thus, ↵ ↵ v = @ v r Notice: The above expression is not valid in curvilinear coordinates. In general, ↵ ↵ ↵ v (x )=v (x + ✏t )+v (x )(✏t ) k component changes basis vector changes Therefore ↵ ↵ ↵ v = @ v + v r Lecture 1: General Relativity Consequently:

Covariant differentiation of Vectors

↵ ↵ ↵ v = @ v + v r

Covariant differentiation of 1-forms

µ!⌫ = @µ!⌫ ! r µ⌫

Covariant differentiation of general Tensors

µ1µ2 µk µ1µ2 µk T ··· ⌫ ⌫ ⌫ = @T ··· ⌫ ⌫ ⌫ 1 2··· l 1 2··· l r µ1 µ2 µk µ2 µ1 µk + T ··· ⌫ ⌫ ⌫ + T ··· ⌫ ⌫ ⌫ + 1 2··· l 1 2··· l µ1µ2 µk µ1µ2 µk ··· ⌫ T ··· ⌫ ⌫ ⌫ T ··· ⌫ ⌫ 1 2··· l 2 1 ··· l ···

Lecture 1: General Relativity Christoffel symbols

From metric compatibility:

⇢gµ⌫ = @⇢gµ⌫ ⇢µg⌫ ⇢⌫ gµ = 0 r µg⌫⇢ = @µg⌫⇢ µ⌫ g⇢ µ⇢g⌫ = 0 r ⌫ g⇢µ = @⌫ g⇢µ gµ g⇢ = 0 r ⌫⇢ ⌫µ Subtract the second and third from the first,

@⇢gµ⌫ @µg⌫⇢ @⌫ g⇢µ + 2 g⇢ = 0 . µ⌫ Multiply by g⇢ to get

Christoffel Symbols

1 ⇢ = g (@µg⌫⇢ + @⌫ g⇢µ @⇢gµ⌫ ) µ⌫ 2

Lecture 1: General Relativity The Christoffel symbols vanish in flat space in Cartesian coordinates The Christoffel symbols do not vanish in flat space in curvilinear coordinates. For example, if ds2 = dr 2 + r 2d✓2, it is not difficult to show that r = r and ✓ = 1/r ✓✓ ✓r At any one point p in a spacetime (M, gµ⌫ ), it is possible to find a coordinate system for which µ⌫ = 0 (recall local flatness)

Lecture 1: General Relativity Example: Motion of freely-falling particles. In Flat spacetime

d 2x µ = 0 d2 Rewrite d 2x µ dx ⌫ dx µ = @ = 0 d2 d ⌫ d Substitute dx ⌫ dx µ dx ⌫ dx µ @ d ⌫ d ! d r⌫ d Thus d 2x µ dx ⇢ dx +µ = 0 . d2 ⇢ d d

Lecture 1: General Relativity The Newtonian Limit

Given a General Relativistic expression, one recover the Newtonian counterparts 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 dt << , d⌧ d⌧ so d 2x µ dt 2 +µ = 0 . d⌧ 2 00 d⌧ ✓ ◆ Static gravitational field implies

µ 1 µ = g (@ g + @ g @g ) 00 2 0 0 0 0 00 1 µ = g @g . 2 00 Weakness of the gravitational field implies

gµ⌫ = ⌘µ⌫ + hµ⌫ , hµ⌫ << 1 . | | Lecture 1: General Relativity Commutator of two covariant derivatives: it measures the difference between parallel transporting the tensor first one way and then the other, versus the

opposite ordering.

µ

µ That is: ⇢ ⇢ ⇢ [ µ, ⌫ ]V = µ ⌫ V ⌫ µV r r r r ⇢ rr ⇢ ⇢ = @µ( ⌫ V ) µ⌫ V +µ ⌫ V (µ ⌫) r ⇢ ⇢ r ⇢ r $ ⇢ ⇢ = @µ@⌫ V +(@µ⌫)V +⌫@µV µ⌫ @V µ⌫ V ⇢ ⇢ +µ@⌫ V +µ⌫V (µ ⌫) ⇢ ⇢ ⇢ ⇢ $ ⇢ =(@µ⌫ @⌫ µ +µ⌫ ⌫µ)V 2[µ⌫] V ⇢ ⇢ r = R µ⌫ V Tµ⌫ V r where Riemann Tensor

⇢ ⇢ ⇢ ⇢ ⇢ R µ⌫ = @µ @⌫ + ⌫ µ µ ⌫ ⌫ µ

Lecture 1: General Relativity The Ricci Tensor

Ricci Tensor

Rµ⌫ = R µ⌫ .

Because of R⇢µ⌫ = Rµ⌫⇢ Rµ⌫ = R⌫µ , Also Ricci scalar

µ µ⌫ R = R µ = g Rµ⌫ .

Lecture 1: General Relativity The Einstein Tensor

Contract twice the Bianchi identity

R⇢µ⌫ + ⇢Rµ⌫ + R⇢µ⌫ = 0 r r r to get

⌫ µ 0 = g g ( R⇢µ⌫ + ⇢Rµ⌫ + R⇢µ⌫ ) µ r ⌫r r = R⇢µ ⇢R + R⇢⌫ , r r r or µ 1 R⇢µ ⇢R = 0 r 2 r Define Einstein Tensor 1 Gµ⌫ = Rµ⌫ Rgµ⌫ , 2 Einstein Equation Gµ⌫ = 8⇡Tµ⌫

Einstein Equation in Vacuum

Gµ⌫ = 0

Lecture 1: General Relativity Lectures

L1: General Relativity

L2: Numerical Relativity

L3: Gravitational Waveforms

Deirdre Shoemaker General Relativity and Gravitational Waveforms Lecture 2: Numerical Relativity

Lecture 2: Numerical Relativity Numerical Relativity

From:

Gµ⌫ = 8 ⇡ Tµ⌫

To: t + t

Boundary'Condi$ons' t

Ini$al'Data'

Initial Value and Boundary Problem

Lecture 2: Numerical Relativity This Lecture

3+1 Decomposition: Foliations Tensor Projections ADM formulation BSSN formulation Choosing Coordinates Moving Puncture Coordinates Initial Data Boundary Conditions GW extraction

Lecture 2: Numerical Relativity Space-time Foliation

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

⌦ = t a ra such that its magnitude is given by

2 ab 2 ⌦ = g t t ↵ , | | ra rb ⌘ with ↵ the lapse function.

Lecture 2: Numerical Relativity Space-time Foliation

Define the unit normal vector na:

na ↵gab⌦ = ↵gab t ⌘ b rb and spatial metric: ab = gab + nanb

Lecture 2: Numerical Relativity Projection Operator and Covariant Differentiation

Space-like projection operator:

a ac a a a a b = g cb = g b + n nb = b + n nb a b It is easy to show that bn = 0.

Three-dimensional covariant derivative compatible with ab:

D T b d b f T e. a c ⌘ a e c rd f It is easy to show that Dabc = 0.

Lecture 2: Numerical Relativity Extrinsic Curvature

Kab is space-like and symmetric by construction and can be written in terms of the acceleration of normal observers a = nb n = D ln ↵ as a rb a a K = n n a . (1) ab ra b a b Kab can also be written in terms of the Lie derivative of the spatial metric along the normal vector na

1 K = (2) ab 2Ln ab

Kab is the “velocity” of the spatial metric.

Lecture 2: Numerical Relativity Lie Derivatives

f = X bD f = X b@ f (3) LX b b v a = X bD v a v bD X a =[X, v]a (4) LX b b ! = X bD ! + ! D X b (5) LX a b a b a T a = X c@ T a T c @ X a + T a @ X c (6) LX b c b b c c b

Lecture 2: Numerical Relativity Einstein Constraints

The Hamiltonian and momentum constraints:

R + K 2 K K ab = 16⇡⇢ (7) ab D K b D K = 8⇡j , (8) b a a a Only involve spatial quantities and their spatial derivatives.

They have to hold on each individual spatial slice ⌃t They are the necessary and sufficient integrability conditions for the embedding of the spatial slices (⌃,ab, Kab) in the space-time (M, gab).

Lecture 2: Numerical Relativity 3+1 time derivatives

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

a ab 1 n ⌦ = ↵g t t = ↵ . (9) a rb ra However, the vector ta = ↵na + a (10) a is dual to ⌦a for any spatial shift vector . That is

ta⌦ = ta t = 1 . (11) a ra

Lecture 2: Numerical Relativity 3+1 Foliations

t + t a

ta ↵ na t

ab

↵ and a determine how the coordinates evolve from one a slice ⌃t to the next along the time direction t . ↵ determines how much proper time elapses between time-slices along the normal vector na. a determines by how much spatial coordinates are shifted with respect to the normal vector na.

Lecture 2: Numerical Relativity ADM formulation in 3+1 Coordinates

Hamiltonian constraint

R + K 2 K K ij = 16⇡⇢, (12) ij Momentum constraint

D K j D K = 8⇡j , (13) j i i i

ab evolution equation:

@ = 2↵K (14) t ij L ab ij

Kab evolution equation:

@ K K = D D ↵ + ↵(R 2K K k + KK ) t ij L ab i j ij ik j ij 1 (15) ↵8⇡(S (S ⇢)) ij 2 ij

Lecture 2: Numerical Relativity Analogy with Electrodynamics

Constraint equation: i Di E = 4⇡⇢e (16) Evolution equations:

@ A = E D (17) t i i i @ E = Dj D A + D Dj A 4⇡J (18) t i j i i j i

The gauge quantity is the analogue of the lapse ↵ and shift i . The vector potential Ai is the analogue of the spatial metric ij The electric field E i is the analogue of the extrinsic curvature Kij

Lecture 2: Numerical Relativity BSSN formulation

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

4 ˜ij = e ij , (19)

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

1 K = A + K (20) ij ij 3 ij

with Aij = 0 and choose the following conformal rescaling

˜ 4 Aij = e Aij . (21)

Lecture 2: Numerical Relativity BSSN formulation

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

@ ln 1/2 = ↵K + D i , (22) t i Substitution of =(ln )/12 yields the evolution equation:

1 1 @ = ↵K + i @ + @ i (23) t 6 i 6 i

Lecture 2: Numerical Relativity BSSN formulation

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

@ K = D2↵ + ↵ K K ij + 4⇡(⇢ + S) + i D K , (24) t ij i h i 2 ij where D Di Dj . ⌘ ˜ 4 Substitution of Aij = e Aij in this equations yields the K evolution equation:

1 @ K = D2↵ + ↵ A˜ A˜ij + K 2 + 4⇡(⇢ + S) + i @ K . (25) t ij 3 i 

Lecture 2: Numerical Relativity BSSN formulation

Subtracting the @t and @t K evolution equations from the @t ij and @t Kij yields

2 @ ˜ = 2↵A˜ + k @ ˜ +˜ @ k +˜ @ k ˜ @ k . (26) t ij ij k ij ik j kj i 3 ij k and

@ A˜ = e 4 (D D ↵)TF + ↵(RTF 8⇡STF ) t ij i j ij ij +↵(KhA˜ 2A˜ A˜l ) i (27) ij il j +k @ A˜ + A˜ @ k + A˜ @ k 2 A˜ @ k . k ij ik j kj i 3 ij k where TF denotes BTF = B B/3. ij ij ij

Lecture 2: Numerical Relativity BSSN conformal connection

Define the conformal connection functions

˜i ˜jk ˜i = ˜ij , (28) ⌘ jk ,j

The Ricci tensor can be written as

R˜ = 1 ˜lm˜ +˜ @ ˜k + ˜k ˜ + ij 2 ij,lm k(i j) (ij)k lm ˜k ˜ ˜k ˜ (29) ˜ 2l(i j)km + imklj . ⇣ ⌘

Notice: The only second derivatives of ˜ij left over in this lm operator is the Laplace operator ˜ ˜ij,lm – all others have been absorbed in first derivatives of ˜i .

Lecture 2: Numerical Relativity Why bother introducing i ?

Recall the wave equation e

@tt = (30)

or

@t =⇧ (31)

@t ⇧= (32)

With i the BSSN equations have the structure of

e @ ˜ A˜ (33) t ij / ij @ A˜ ˜ (34) t ij / ij

Lecture 2: Numerical Relativity BSSN formulation: i evolution equation

From the time derivative of ˜i = ˜ij and the evolution ,j equation for ˜ij one gets

2 @ ˜i = @ 2↵A˜ij 2˜m(j i) + ˜ij l + l ˜ij . (35) t j ,m 3 ,l ,l h i

The divergence of the A˜ij can be eliminated with the help of the momentum constraint yielding the ˜i evolution equation:

@ ˜i = 2A˜ij @ ↵ + 2↵ ˜i A˜kj 2 ˜ij @ K 8⇡˜ij S + 6A˜ij @ t j jk 3 j j j +j @ ˜i ˜j @ ⇣i + 2 ˜i @ j + 1 ˜li j +˜lj i . ⌘ j j 3 j 3 ,jl ,lj (36)

Lecture 2: Numerical Relativity Analogy with Electrodynamics

Recall:

@ A = E D (37) t i i i @ E = Dj D A + D Dj A 4⇡J (38) t i j i i j i Auxiliary variable: i =D Ai . (39) New evolution equations:

@ A = E D (40) t i i i @ E = D Dj A + D 4⇡J (41) t i j i i i i i @t =@t D Ai = D @t Ai = Di E D Di i i = D Di 4⇡⇢ . (42) i e

Lecture 2: Numerical Relativity Moving Puncture Coordinates

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

Lecture 2: Numerical Relativity Moving Puncture Coordinates

Gauge Conditions

@ ↵ = 2↵K + i @ ↵ t i @ i ⇠B t ⌘ i @ B = @ ˜i ⌘Bi ⇣j @ ˜i t i t j with ⇠,,⌘, and ⇣ parameters. The conditions are modifications to the so-called 1+log slicing and Gamma-driver shift conditions [see, Gauge conditions for long-term numerical black hole evolutions without excision, Alcubierre et al, Phys.Rev. D67 (2003) 084023]

Lecture 2: Numerical Relativity Boundary Conditions

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

1 (t, x, y, z)= (r t) r n

Since @t + @r = 0, then 1 @ = @ [r n ] t r n r

Lecture 2: Numerical Relativity Initial Data

Recall the constraints:

R + K 2 K K ab = 16⇡⇢ ab D K b D K = 8⇡j , b a a a Notice: 4 equations and 12 variables , K { ij ij } York et.al suggested conformal and transverse-traceless decompositions

4 ij = ¯ij

ij 10 ij 1 ij K = A¯ + K 3

Lecture 2: Numerical Relativity Initial Data

¯ij ¯ij ¯ij A = ATT + AL, where the transverse part is divergenceless

¯ ¯ij Dj ATT = 0 and where the longitudinal part satisfies

2 A¯ij = D¯ i W j + D¯ j W i ¯ij D¯ W k (LW¯ )ij . L 3 k ⌘

Lecture 2: Numerical Relativity Initial Data

Hamiltonian Constraint

2 2 5 2 7 ij 5 8 D¯ R¯ K + A¯ A¯ = 16⇡ ⇢, 3 ij Momentum constraint 2 (¯ W )i 6¯ij D¯ K = 8⇡ 10ji . L 3 j

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

Lecture 2: Numerical Relativity Initial Data

8 Assumptions

¯ij = ⌘ij K = 0 ¯ij ATT = 0 Then Hamiltonian Constraint

2 7 ij 5 8 D¯ + LW¯ LW¯ = 16⇡ ⇢, ij Momentum constraint

¯ i 10 i (LW ) = 8⇡ j .

Lecture 2: Numerical Relativity Initial Data: Binary Black Holes

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

¯ 2 7¯ ¯ ij 8 D + LWij LW = 0,

Lecture 2: Numerical Relativity Gravitational Wave Extraction

The Weyl tensor scalar 4 is related to the grav. wave strain polarizations: ¨ ¨ 4 = h+ h ⇥ How does one construct 4 from the numerical relativity simulations? Start with an orthonormal tetrad eˆa and build the null-tetrad: { (N)} 1 la = eˆa + eˆa p2 (0) (1) 1 ⇣ ⌘ k a = eˆa eˆa p2 (0) (1) 1 ⇣ ⌘ ma = eˆa + ieˆa p2 (2) (1) 1 ⇣ ⌘ m¯ a = eˆa ieˆa p2 (2) (1) ⇣ ⌘

Lecture 2: Numerical Relativity Gravitational Wave Extraction

Then

a b c d 4 = Cabcd k mˆ k mˆ

Lecture 2: Numerical Relativity Spherical Harmonics

rM 4(◆,, t)= 2Y`,m(◆,)C`,m(t) X`,m

Lecture 2: Numerical Relativity Higher Order Modes

Lecture 2: Numerical Relativity Conversion to Strain

t t h(t)=h (t) ih (t)= dt0 dt00 . + x 4 Z1 Z1

fundamental uncertainties in producing strain from 4 due to integration of finite length, discretely sampled, noisy data results in large secular non-linear drifts most groups use a method developed by Pollney and Reisswig (arXiv:1006.1632) that integrates in the frequency domain

Lecture 2: Numerical Relativity Lecture 3: Waveforms

Lecture 3: Waveforms Gravity as Geometry

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

Lecture 3: Waveforms The post-Newtonian (PN) Approximation

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

v 2 @ h 2 ✏ h 0 ⇠ c2 ⇠| µ⌫ |⇠ @ h i

Lecture 3: Waveforms In The Know

Key definitions and lingo In progress = M = m1 + m2 q = m1 m2 ⌘ =

Lecture 3: Waveforms IMR Waveforms

IMR: Ispiral Merger Ringdown Waveforms Left: GW signal from q=1 nonspinning BH binary as predicted at 2.5PN order by Buonanno and Damour (2000) The merger is assumed almost instantaneous and one QNM is included Right: GW signal from q=1 BH binary with a small spin 1 = 2 = 0.06 obtained in full general relativity by Pretorius

0.3 0.3 numerical relativity

0.2 0.2

0.1 0.1

0 0 h(t) h(t)

-0.1 -0.1

-0.2 inspiral-plunge -0.2 merger-ring-down

-200 -100 0 100 -200 -100 0 100 t/M t/M

Lecture 3: Waveforms IMR Waveforms

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

20

numerical relativity 4 10 PN inspiral 15

10 3 10 SNR at 3Gpc

SNR at 100 Mpc numerical relativity 5 PN inspiral

2 0 10 5 6 7 30 60 90 120 150 180 10 10 10 M (M ) sun M (Msun)

Lecture 3: Waveforms Horizons & Merger

MORE COMING!

Lecture 3: Waveforms

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.

1. Introduction to the EinsteinToolkit: https://arxiv.org/abs/1305.5299 2. Numerical Relativity Review: https://arxiv.org/abs/gr-qc/0106072

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