General Relativity and Gravitational Waveforms

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 According to Einstein: Where do gravitational wave come from? Hint: Einstein is the stork. 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: g µ The metric g µ : (0, 2) tensor, g µ = g µ (symmetric) g = g µ 6= 0 (non-degenerate) g µ (inverse metric) g µ is symmetric and g µ g = µ. g µ and g µ are used to raise and lower indices on tensors. Lecture 1: General Relativity

g µ properties The metric: provides a notion of past and future allows the computation of path length and proper time: ds 2 = g µ dx µ 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 = ds 2 = g µ dx µ dx Lecture 1: General Relativity

Example: Space-time interval of flat spacetime Notice: (ds) 2 = c 2 (dt) 2 +(dx) 2 +(dy) 2 +(dz) 2. 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 µ dx, where dx µ = x µ =( t, x, y, z) or (dt, dx, dy, dz) and µ is a 4 4 matrix called the metric: 0 1 c 2 0 0 0 µ = B 0 1 0 0 C @ 0 0 1 0A. 0 0 0 1 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. Consider at each point a basis ê (µ) adapted to the coordinates x µ ; that is, ê (1) pointing along the x-axis, etc. Then, any abstract vector A can be written as A = A µ ê (µ). The coefficients A µ are the components of the vector A. The real vector is the abstract geometrical entity A, while the components A µ are just the coefficients of the basis vectors in some convenient basis. The parentheses around the indices on the basis vectors ê (µ) label collection of vectors, not components of a single vector. Lecture 1: General Relativity

Definition of metric: g µ = ê (µ) ê ( ) Inner or dot product: Given the metric g, g(v, W )=g µ V µ W = V W 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. 8 < < 0, V µ is timelike if g µ V µ V is = 0, V µ is lightlike or null : > 0, V µ is spacelike. Lecture 1: General Relativity

Example in flat spacetime: when g µ = µ in Cartesian coordinates 0 1 c 2 0 0 0 µ = B 0 1 0 0 C @ 0 0 1 0A = diag( c2, 1, 1, 1) 0 0 0 1 V W = µ V µ W = 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 2 V 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, r 2, r 2 sin 2 ) V W = µ V µ W = c 2 V t W t + V r W r + r 2 V W + r 2 sin 2 V W Lecture 1: General Relativity

(ds) 2 = µ dx µ 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 c 2 (d ) 2 = (ds) 2 Notice: if dx i = 0 then c 2 (d ) 2 = d = dt. µ (dt) 2, thus 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 = 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 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. with m i the inertial mass. f = m i a. On the other hand, with f g = m g r. 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 was universal. That is, in Newtonian mechanics m i = m g. 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 g µ the partial derivative @ µ by the covariant derivative r µ Lecture 1: General Relativity

Example: Newton s 2nd Law and Special Relativity in Special Relativity f µ = m d 2 ~ f = m ~a = d ~p dt d 2 x µ ( ) = d d pµ ( ) 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 k (x ) Covariant Derivative: In a local inertial frame: Thus, r t v(x )= lim!0 v k (x ) v(x ) (r t v) = t @ v r v = @ v Notice: The above expression is not valid in curvilinear coordinates. In general, Therefore v k (x ) = v (x + t ) + v (x )( t ) component changes r v = @ v + basis vector changes v Lecture 1: General Relativity

Consequently: Covariant differentiation of Vectors r v = @ v + v Covariant differentiation of 1-forms r µ! = @ µ! µ! Covariant differentiation of general Tensors r T µ 1µ 2 µ k 1 2 l = @ T µ 1µ 2 µ k 1 2 l + µ 1 T µ 2 µ k 1 2 l + µ 2 T µ 1 µ k 1 2 l + 1 T µ 1µ 2 µ k 2 l 2 T µ 1µ 2 µ k 1 l Lecture 1: General Relativity

Christoffel symbols From metric compatibility: r g µ = @ g µ µ g g µ = 0 r µ g = @ µ g µ g µ g = 0 r g µ = @ g µ g µ µ g = 0 Subtract the second and third from the first, Multiply by g to get Christoffel Symbols @ g µ @ µ g @ g µ + 2 µ g = 0. µ = 1 2 g (@ µ g + @ g µ @ g µ ) 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 ds 2 = dr 2 + r 2 d 2, it is not difficult to show that r = r and r = 1/r At any one point p in a spacetime (M, g µ ), 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 2 x µ d 2 = 0 Rewrite Substitute Thus d 2 x µ d 2 dx d @ = dx d dx µ d d 2 x µ d 2 + µ @! dx d dx d dx µ d dx d r = 0 dx µ d = 0. 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 so dx i d Static gravitational field implies µ dt << d, d 2 x µ d + µ dt 2 00 d 2 = 0. 00 = 1 2 gµ (@ 0 g 0 + @ 0 g 0 @ g 00 ) 1 = 2 gµ @ g 00. Weakness of the gravitational field implies g µ = µ + h µ, h µ << 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 That is: [r µ, r ]V = r µ r V r r µ V = @ µ (r V ) µ r V + µ r V (µ $ ) = @ µ @ V +(@ µ )V + @ µ V µ @ V µ V + µ @ V + µ V (µ $ ) = (@ µ @ µ + µ µ )V 2 [µ ] r V = R µ V T µ r V where Riemann Tensor µ R µ = @ µ @ µ + µ µ Lecture 1: General Relativity

The Ricci Tensor Ricci Tensor R µ = R µ. Because of R µ = R µ R µ = R µ, Also Ricci scalar R = R µ µ = g µ R µ. Lecture 1: General Relativity

The Einstein Tensor Contract twice the Bianchi identity to get r R µ + r R µ + r R µ = 0 0 = g g µ (r R µ + r R µ + r R µ ) = r µ R µ r R + r R, or Define Einstein Tensor Einstein Equation r µ R µ 1 2 r R = 0 G µ = R µ 1 2 Rg µ, G µ = 8 T µ Einstein Equation in Vacuum G µ = 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: G µ = 8 T µ 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, g ab ) into a family of non-intersecting, space-like, three-dimensional hyper-surfaces t leveled by a scalar function t with time-like normal: a = r a t such that its magnitude is given by with the lapse function. 2 = g ab r a tr b t 2, Lecture 2: Numerical Relativity

Space-time Foliation Define the unit normal vector n a : n a g ab b = g ab r b t and spatial metric: ab = g ab + n a n b Lecture 2: Numerical Relativity

Projection Operator and Covariant Differentiation Space-like projection operator: a b = g ac cb = g a b + na n b = a b + na n b It is easy to show that a b n b = 0. Three-dimensional covariant derivative compatible with ab : D a T b c d a It is easy to show that D a bc = 0. b e f c r d T e f. Lecture 2: Numerical Relativity

Extrinsic Curvature K ab is space-like and symmetric by construction and can be written in terms of the acceleration of normal observers a a = n b r b n a = D a ln as K ab = r a n b n a a b. (1) K ab can also be written in terms of the Lie derivative of the spatial metric along the normal vector n a K ab is the velocity of the spatial metric. K ab = 1 2 L n ab (2) Lecture 2: Numerical Relativity

Lie Derivatives L X f = X b D b f = X b @ b f (3) L X v a = X b D b v a v b D b X a =[X, v] a (4) L X! a = X b D b! a +! b D a X b (5) L X T a b = X c @ c T a b T c b @ cx a + T a c@ b X c (6) Lecture 2: Numerical Relativity

Einstein Constraints The Hamiltonian and momentum constraints: R + K 2 K ab K ab = 16 (7) D b K b a D a K = 8 j a, (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 integrability conditions for the embedding of the spatial slices (, ab, K ab ) in the space-time (M, g ab ). Lecture 2: Numerical Relativity

3+1 time derivatives To derive the evolution equations for ab and K ab one needs a time derivative. The Lie derivative along L n is not a natural time derivative orthogonal to t (e.g. n a is not dual to a ). That is, However, the vector n a a = g ab r b tr a t = 1. (9) is dual to a for any spatial shift vector t a = n a + a (10) a. That is t a a = t a r a t = 1. (11) Lecture 2: Numerical Relativity

3+1 Foliations a t + t n a t a t ab and a determine how the coordinates evolve from one slice t to the next along the time direction t a. determines how much proper time elapses between time-slices along the normal vector n a. a determines by how much spatial coordinates are shifted with respect to the normal vector n a. Lecture 2: Numerical Relativity

ADM formulation in 3+1 Coordinates Hamiltonian constraint R + K 2 K ij K ij = 16, (12) Momentum constraint D j K j i D i K = 8 j i, (13) ab evolution equation: @ t ij L ab = 2 K ij (14) K ab evolution equation: @ t K ij L K ab = D i D j + (R ij 2K ik K k j + KK ij ) 8 (S ij 1 2 ij (S )) (15) Lecture 2: Numerical Relativity

Analogy with Electrodynamics Constraint equation: Evolution equations: D i E i = 4 e (16) The gauge quantity shift i. @ t A i = E i D i (17) @ t E i = D j D j A i + D i D j A j 4 J i (18) is the analogue of the lapse and The vector potential A i is the analogue of the spatial metric ij The electric field E i is the analogue of the extrinsic curvature K ij 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 and choose ij =1. Split the extrinsic curvature as ij = e 4 ij, (19) K ij = A ij + 1 3 ij K (20) with A ij = 0 and choose the following conformal rescaling à ij = e 4 A ij. (21) Lecture 2: Numerical Relativity

BSSN formulation From @ t ln = ij @ t ij and the trace of the ij evolution equation: @ t ln 1/2 = K + D i i, (22) Substitution of =(ln )/12 yields the evolution equation: @ t = 1 6 K + i @ i + 1 6 @ i i (23) Lecture 2: Numerical Relativity

BSSN formulation Similarly, combining the trace of the K ij evolution equations with the Hamiltonian constraint gives h i @ t K = D 2 + K ij K ij + 4 ( + S) + i D i K, (24) where D 2 ij D i D j. Substitution of Ãij = e 4 A ij in this equations yields the K evolution equation: @ t K = D 2 + apple à ij à ij + 1 3 K 2 + 4 ( + S) + i @ i K. (25) Lecture 2: Numerical Relativity

BSSN formulation Subtracting the @ t and @ t K evolution equations from the @ t ij and @ t K ij yields and @ t ij = 2 Ãij + k @ k ij + ik @ j k + kj @ i k 2 3 ij@ k k. (26) @ t à ij = e 4 h (D i D j ) TF + (R TF ij i 8 Sij TF ) + (K Ãij 2ÃilÃl j ) + k @ k à ij + Ãik@ j k + Ãkj@ i k 2 3Ãij@ k k. where TF denotes B TF ij = B ij ij B/3. (27) Lecture 2: Numerical Relativity

BSSN conformal connection Define the conformal connection functions i jk i jk = ij,j, (28) The Ricci tensor can be written as 1 R ij = 2 lm ij,lm + k(i @ j) k + k (ij)k + lm 2 k l(i j)km + k im klj. (29) Notice: The only second derivatives of ij left over in this operator is the Laplace operator lm ij,lm all others have been absorbed in first derivatives of i. Lecture 2: Numerical Relativity

Why bother introducing ei? Recall the wave equation @ tt = (30) or @ t = (31) @ t = (32) With e i the BSSN equations have the structure of @ t ij / Ãij (33) @ t à ij / ij (34) Lecture 2: Numerical Relativity

BSSN formulation: i evolution equation From the time derivative of i = equation for ij one gets @ t i ij,j and the evolution h = @ j 2 Ãij 2 m(j i),m + 2 i ij l,l + l ij 3,l. (35) The divergence of the Ãij can be eliminated with the help of the momentum constraint yielding the i evolution equation: @ i t = 2Ãij @ j + 2 i jk à kj 2 3 ij @ j K 8 ij S j + 6Ãij @ j + j @ j i j @ j i + 2 3 i @ j j + 1 3 li j,jl + lj i,lj. (36) Lecture 2: Numerical Relativity

Analogy with Electrodynamics Recall: @ t A i = E i D i (37) @ t E i = D j D j A i + D i D j A j 4 J i (38) Auxiliary variable: =D i A i. (39) New evolution equations: @ t A i = E i D i (40) @ t E i = D j D j A i + D i 4 J i (41) @ t = @ t D i A i = D i @ t A i = D i E i D i D i = D i D i 4 e. (42) 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 @ t = 2 K + i @ i i @ t B i @ t B i = @ i t B i j @ j i 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 (t, x, y, z) = 1 (r t) r n Since @ t + @ r = 0, then @ t = 1 r n @ r [r n ] Lecture 2: Numerical Relativity

Initial Data Recall the constraints: R + K 2 K ab K ab = 16 D b K b a D a K = 8 j a, Notice: 4 equations and 12 variables { ij, K ij } York et.al suggested conformal and transverse-traceless decompositions ij = 4 ij K ij = 10 Ā ij + 1 3 ij K Lecture 2: Numerical Relativity

Initial Data Ā ij = Āij TT + Āij L, where the transverse part is divergenceless D j Ā ij TT = 0 and where the longitudinal part satisfies Ā ij L = D i W j + D j W i 2 3 ij Dk W k ( LW ) ij. Lecture 2: Numerical Relativity

Initial Data Hamiltonian Constraint 8 D 2 R 2 3 5 K 2 + 7 Ā ij Ā ij = 16 5, Momentum constraint ( LW ) i 2 3 6 ij Dj K = 8 10 j i. 4-equations for 4-unknowns {, W i } Lecture 2: Numerical Relativity

Initial Data 8 Assumptions Then Hamiltonian Constraint Momentum constraint ij = ij K = 0 Ā ij TT = 0 8 D 2 + 7 LWij LW ij = 16 5, ( LW ) i = 8 10 j i. Lecture 2: Numerical Relativity

Initial Data: Binary Black Holes For black holes there are well known solutions (Bowen-York) to the momentum constraint ( LW ) i = 0. Thus, constructing initial data reduces to solving the Hamiltonian Constraint 8 D 2 + 7 LWij LW ij = 0, Lecture 2: Numerical Relativity

Gravitational Wave Extraction The Weyl tensor scalar 4 is related to the grav. wave strain polarizations: 4 = ḧ+ How does one construct 4 from the numerical relativity simulations? Start with an orthonormal tetrad {ê(n) a } and build the null-tetrad: ḧ l a = p 1 ê(0) a + êa (1) 2 k a = 1 p 2 ê a (0) ê a (1) m a = p 1 ê(2) a + iêa (1) 2 m a = p 1 ê(2) a iê(1) a 2 Lecture 2: Numerical Relativity

Gravitational Wave Extraction Then 4 = C abcd k a ˆm b k c ˆm d Lecture 2: Numerical Relativity

Spherical Harmonics rm 4(,, t) = X`,m 2Y`,m (, )C`,m (t) Lecture 2: Numerical Relativity

Higher Order Modes Lecture 2: Numerical Relativity

Conversion to Strain h(t) =h + (t) ih x (t) = Z t Z t dt 0 dt 00 4. 1 1 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 c 2 h µ @ 0h @ i h 2 Lecture 3: Waveforms

In The Know Key definitions and lingo In progress = M = m 1 + m 2 q = m 1 m 2 = 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 0.2 0.3 0.2 numerical relativity 0.1 0.1 h(t) 0 h(t) 0-0.1-0.1-0.2 inspiral-plunge merger-ring-down -0.2-200 -100 0 100 t/m -200-100 0 100 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 PN inspiral 10 4 15 SNR at 100 Mpc 10 5 SNR at 3Gpc 10 3 numerical relativity PN inspiral 0 30 60 90 120 150 180 M (M sun ) 10 2 10 5 10 6 10 7 M (M sun ) 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. Introduction to the EinsteinToolkit: https://arxiv.org/abs/1305.5299 1. Numerical Relativity Review: https://arxiv.org/abs/gr-qc/0106072 2.

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