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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 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⌫ = δσ . 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 φW φ · 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 Φ 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 = Φ 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 Φ . − r with Φ 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 = Φ . 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: ↵ β ↵ ( v) = t @β v rt Thus, ↵ ↵ β v = @β v r Notice: The above expression is not valid in curvilinear coordinates. In general, ↵ δ ↵ δ δ ↵ γ δ β v (x )=v (x + ✏t )+Γβγv (x )(✏t ) k component changes basis vector changes Therefore ↵ ↵ ↵ γ β v = @β v +Γ v r βγ Lecture 1: General Relativity Consequently: Covariant differentiation of Vectors ↵ ↵ ↵ γ β v = @β v +Γ v r βγ Covariant differentiation of 1-forms λ µ!⌫ = @µ!⌫ Γ !λ r − µ⌫ Covariant differentiation of general Tensors µ1µ2 µk µ1µ2 µk σT ··· ⌫ ⌫ ⌫ = @σT ··· ⌫ ⌫ ⌫ 1 2··· l 1 2··· l r µ1 λµ2 µk µ2 µ1λ µk +Γ T ··· ⌫ ⌫ ⌫ +Γ T ··· ⌫ ⌫ ⌫ + σλ 1 2··· l σλ 1 2··· l λ µ1µ2 µk λ µ1µ2 µk ··· Γ⌫ T ··· ⌫ ⌫ Γ⌫ T ··· ⌫ λ ⌫ − 1 2··· l − 2 1 ··· l −··· Lecture 1: General Relativity Christoffel symbols From metric compatibility: λ λ ⇢gµ⌫ = @⇢gµ⌫ Γ⇢µg⌫ Γ⇢⌫ gµλ = 0 r − λ − λ µg⌫⇢ = @µg⌫⇢ Γµ⌫ g⇢ Γµ⇢g⌫ = 0 r − λ − λ ⌫ g⇢µ = @⌫ g⇢µ Γ gλµ Γ g⇢ = 0 r − ⌫⇢ − ⌫µ Subtract the second and third from the first, λ @⇢gµ⌫ @µg⌫⇢ @⌫ g⇢µ + 2Γ g⇢ = 0 .
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