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Esta es una presentacion que hice con motivo de los requisitos que exige la maestria en fisica en la Unviersidad de Bishops, en Quebec, Canada. Durante mi presentacion, hicieron incapie en un error de subindices durante el desarrollo de las ecuaciones de las ondas gravitacionales. Lamentablemente no recuerdo en que diapositiva me marcaron el error, asi que es un desafio para cualquiera que encuentre mi presentacion interesante para ser utilizada en algun proyecto. Gracias.

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- An Introduction to Gravitational Waves Adriana Marina Cardini November 24, 2017 Bishop’s University
- A Little General Relativity Gravitational Waves LIGO and Virgo Sources of Gravitational Waves
- A Little General Relativity Manifold is one of the most fundamental concepts in mathematics and physics. We are all used to the properties of n-dimensional Euclidean space, 𝑹 𝒏 = 𝑥1 , 𝑥2 , … … . , 𝑥 𝑛 , often equipped with a flat positive-definite metric. Mathematicians have developed calculus in 𝑹 𝒏 (differentiation, integration, etc.). However, there are other spaces which we think of as “curved” or topologically complicated on which we would like to perform analogous operations. To address this problem we invent the notion of a manifold, which corresponds to a space that may be curved and has a complicated topology, but in local regions looks like 𝑹 𝒏. – What is a Manifold?
- The Metric The metric tensor is an important object in curved space that it is given by the symbol 𝒈 𝝁𝝂 • This determines the “shortest distance” between two points. • This is a symmetric tensor of order 2, that is 𝑔 𝜇𝜈 = 𝑔 𝜈𝜇 • This is taken to be nondegenerate, meaning that the determinant 𝑔 = 𝒈 𝛍𝝂 does not vanish, so this allows us to define the inverse metric 𝑔 𝜇𝜈 as 𝑔 𝜇𝜈 𝑔 𝜈𝜎 = 𝑔 𝜆𝜎 𝑔 𝜆𝜇=𝛿 𝜎 𝜇 • The symmetry of 𝑔μ𝜈 implies that 𝑔 𝜇𝜈 is also symmetric.
- Line Element 𝑑𝑠2 = 𝑔 𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 Where 𝑑𝑥 𝜇 is an infinitesimal coordinate displacement This is an invariant, for example in R3 o Cartesian coordinates 𝑑𝑠2 = 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2 o Cylindrical coordinates 𝑑𝑠2 = 𝑑𝜌2 + 𝜌2 𝑑𝜑2 + 𝑑𝑧2 o Spherical coordinates 𝑑𝑠2 = 𝑑𝑟2 + 𝑟2 𝑑𝜃2 + 𝑟2 𝑠𝑖𝑛𝜃2 𝑑𝜙2 o Toroidal coordinates 𝑑𝑠2 = (𝑎 + 𝑏𝑐𝑜𝑠𝜑)2 𝑑𝜗2 + 𝑏2 𝑑𝜑2
- Curvature It depends on the metric, which defines the geometry of a manifold. All the ways in which curvature manifests itself depend on something called a connection which gives us a way of relating vectors in the tangent space of nearby points. There is a unique connection that we can construct from the metric called the Christoffel symbol, given by Γ𝜇𝜈 𝜆 = 1 2 𝑔 𝜆𝜎 𝜕𝜇 𝑔 𝜈𝜎 + 𝜕𝜈 𝑔 𝜎𝜇 − 𝜕 𝜎 𝑔 𝜇𝜈 where: 𝜕 𝛼 𝑔 𝛽𝛾 = 𝜕𝑔 𝛽𝛾 𝜕𝑥 𝛼 The notation makes Γ𝜇𝜈 𝜆 look like a tensor, but it is not; this is an “object” or “symbol”. And this is symmetric ⇒ Γ𝜇𝜈 𝜆 = Γ𝜈𝜇 𝜆
- Riemann Tensor This technical expression of curvature is contained in the Riemann tensor R 𝜌 𝜎𝜇𝜈 = 𝜕𝜇Γ𝜈𝜎 𝜌 − 𝜕𝜈Γ𝜇𝜎 𝜌 + Γ𝜇𝜆 𝜌 Γ𝜈𝜎 𝜆 − Γ𝜈𝜆 𝜌 Γ𝜇𝜎 𝜆 R 𝜌𝜎𝜇𝜈 = 𝑔 𝜌𝜆 R 𝜆 𝜎𝜇𝜈 R 𝜌𝜎𝜇𝜈 = 1 2 𝜕𝜇 𝜕 𝜎 𝑔 𝜌𝜈 − 𝜕𝜌 𝜕𝜇 𝑔 𝜎𝜈 − 𝜕𝜈 𝜕 𝜎 𝑔 𝜌𝜇 + 𝜕𝜌 𝜕𝜈 𝑔 𝜎𝜇 where: 𝜕𝜇 𝜕 𝜎 𝑔 𝜌𝜈 = 𝜕2 𝑔 𝜌𝜈 𝜕𝑥 𝜎 𝜕𝑥 𝜇 Everything we want to know about the curvature of a manifold is given to us by the Riemann tensor; and it will vanish if and only if the metric is flat.
- Ricci Tensor and Ricci Scalar o If we make a contraction of indices, 𝜌 = 𝜈, we obtain 𝑹 𝝆 𝝈𝝁𝝆 = 𝑹 𝝈𝝁 which is called Ricci tensor. o If we rise an index, 𝒈 𝝈𝜶 𝑹 𝜶𝝁 = 𝑹 𝝁, 𝝈 we obtain a mixed system. o We again make other contraction of indices, = , we have 𝑹 𝝁, 𝝁 = 𝑹 which is called Ricci scalar (or scalar curvature) and it is invariant. Finally, with all these elements, we can present the Einstein equation.
- Einstein’ Equation where 𝑮 𝝁𝝂 is the Einstein tensor 𝑹 𝝁𝝂 is the Ricci tensor R is the Ricci scalar 𝒈 𝝁𝝂 is the metric 𝑮 is the gravitational constant of Newton c is the speed of light 𝑻 𝝁𝝂 is the energy-momentum tensor The term 8𝜋GT𝜇𝜆 generalizes the density term in the Poisson equation for the Newtonian potential Φ = − GM 𝑟 . 𝛻2 Φ = 4πG𝜌 𝑮 𝝁𝝂 = 𝑹 𝝁𝝂 − 𝟏 𝟐 𝒈 𝝁𝝂 𝑹 = 𝟖𝝅𝑮𝑻 𝝁𝝂 𝒄 𝟒
- Minkowski Space 𝑑𝑠2 = 𝝁𝝂 𝑑𝑥 𝜇 𝑑𝑥 𝜈 where 𝑥 𝜇 : 𝑥0 = 𝑐𝑡 𝑥1 = 𝑥 𝑥2 = 𝑦 𝑥3 = 𝑧 𝜇𝜈 is the Minkowski metric 𝜇𝜈 = −1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 So, we have a spatiotemporal metric 𝑑𝑠2 = −𝑐2 𝑑𝑡2 + 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2 this is flat space, so 𝒈 𝝁𝝂 𝝁𝝂
- Gravitational Wave This is a ripple in the curvature of spacetime, which propagates with the speed of light c. In the real universe, gravitational waves propagate on the background of a large-scale, slowly-changing spacetime curvature, created by the universe’s lumpy cosmological distribution of matter (stars, galaxies, clusters, …). What is this?
- Linearized Einstein Equations for Weak Gravitational Fields We assume weak gravitational field The weakness of the gravitational field allows us to decompose the metric into Minkowski metric plus a small perturbation in Cartesian coordinates: 𝑔 𝜇𝜈 = 𝜂 𝜇𝜈 + ℎ 𝜇𝜈 ℎ 𝜇𝜈 ≪ 1 This ℎ 𝜇𝜈 being so small, it allows us to ignore anything that is higher than first order in this quantity. Far from sources, and if r is the radial coordinate, the space must be asymptotically flat, that is lim 𝑟→∞ ℎ 𝜇𝜈 = 0 ∀𝜇, 𝜈
- We want to find the equation of motion obeyed by the perturbations ℎ 𝜇𝜈, which comes by examining Einstein’s equation to first order. o We begin with the Christoffel symbols Γ𝜇𝜈 𝜆 = 1 2 𝑔 𝜆𝜎 𝜕𝜇 𝑔 𝜈𝜎 + 𝜕𝜈 𝑔 𝜎𝜇 − 𝜕 𝜎 𝑔 𝜇𝜈 Γ𝜇𝜈 𝜆 = 1 2 (𝜂 𝜆𝜎 −ℎ 𝜆𝜎 ) 𝜕𝜇(𝜂 𝜈𝜎+ℎ 𝜈𝜎) + 𝜕𝜈 𝜂 𝜎𝜇 + ℎ 𝜎𝜇 − 𝜕 𝜎(𝜂 𝜇𝜈+ℎ 𝜇𝜈) But, 𝜂 𝜇𝜈 are all constants, and to first order in h, we have Γ𝜇𝜈 𝜆 ≈ 1 2 𝜂 𝜆𝜎 𝜕𝜇ℎ 𝜈𝜎 + 𝜕𝜈ℎ 𝜎𝜇 − 𝜕 𝜎ℎ 𝜇𝜈 o The Riemann tensor becomes R 𝜇𝜈𝜌𝜎 = 1 2 𝜕𝜌 𝜕𝜈ℎ 𝜇𝜎 + 𝜕 𝜎 𝜕𝜇ℎ 𝜈𝜌 − 𝜕 𝜎 𝜕𝜈ℎ 𝜇𝜌 − 𝜕𝜌 𝜕𝜇ℎ 𝜈𝜎
- R 𝜆 𝜈𝜌𝜎 = 𝑔 𝜆𝜇 R 𝜇𝜈𝜌𝜎 ≈ 𝜂 𝜆𝜇 R 𝜇𝜈𝜌𝜎 R 𝜆 𝜈𝜌𝜎 ≈ 1 2 𝜂 𝜆𝜇 𝜕𝜈 𝜕 𝜌ℎ 𝜇𝜎 − 𝜕𝜇 𝜕𝜌ℎ 𝜈𝜎 − 𝜕𝜈 𝜕 𝜎ℎ 𝜇𝜌 + 𝜕𝜇 𝜕 𝜎ℎ 𝜈𝜌 R 𝜆 𝜈𝜌𝜎 ≈ 1 2 𝜂 𝜆𝜇 𝜕𝜈 𝜕𝜌ℎ 𝜇𝜎 − 𝜂 𝜆𝜇 𝜕𝜇 𝜕𝜌ℎ 𝜈𝜎 − 𝜂 𝜆𝜇 𝜕𝜈 𝜕 𝜎ℎ 𝜇𝜌 + 𝜂 𝜆𝜇 𝜕𝜇 𝜕 𝜎ℎ 𝜈𝜌 Doing the contraction = R 𝜆 𝜈𝜌𝜆 = R 𝜈𝜌 ≈ 1 2 𝜂 𝜆𝜇 𝜕𝜆 𝜕𝜌ℎ 𝜇𝜆 − 𝜂 𝜆𝜇 𝜕𝜇 𝜕𝜌ℎ 𝜈𝜆 − 𝜂 𝜆𝜇 𝜕𝜈 𝜕𝜆ℎ 𝜇𝜌 + 𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 R 𝜈𝜌 ≈ 1 2 𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 + 𝜕𝜆 𝜕𝜌ℎ 𝜇 𝜇 − 𝜕𝜇 𝜕𝜌ℎ 𝜈 𝜇 − 𝜕𝜈 𝜕𝜆ℎ 𝜌 𝜆 R 𝜈𝜌 ≈ 𝟏 𝟐 𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 + 1 2 𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜇 𝜕𝜌ℎ 𝜈 𝜇 − 𝜕𝜈 𝜕𝜆ℎ 𝜌 𝜆 where ℎ = ℎ 𝜇 𝜇 = 𝑖=1 4 ℎ𝑖 𝑖
- 2R 𝜈𝜌 ≈ 𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 + 1 2 𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜇 𝜕𝜌ℎ 𝜈 𝜇 + 1 2 𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜈 𝜕𝜆ℎ 𝜌 𝜆 2R 𝜈𝜌 ≈ 𝜼 𝝂𝝁 𝝏 𝝁 𝝏 𝝂 𝒉 𝝂𝝆 + 𝜕𝜌 1 2 𝜕𝜈ℎ − 𝜕𝜇ℎ 𝜈 𝜇 + 𝜕𝜈 1 2 𝜕𝜌ℎ − 𝜕𝜆ℎ 𝜌 𝜆 2R 𝜈𝜌 ≈ 𝜼 𝝂𝝁 𝝏 𝝁 𝝏 𝝂 𝒉 𝝂𝝆 + 𝜕𝜌 𝜕𝜇 1 2 𝛿 𝜈 𝜇 ℎ − ℎ 𝜈 𝜇 + 𝜕𝜈 𝜕𝜆 1 2 𝛿 𝜌 𝜆 ℎ − ℎ 𝜌 𝜆 Imposing 𝜕𝜇 1 2 𝛿 𝜈 𝜇 ℎ − ℎ 𝜈 𝜇 ≡ 0 𝜕𝜆 1 2 𝛿 𝜌 𝜆 ℎ − ℎ 𝜌 𝜆 ≡ 0 Hilbert – Einstein gauge Doing =
- o Then the Ricci tensor reduces to where is the d’Alembertian, and for a flat space is = − 1 𝑐2 𝜕𝑡 2 + 𝜕 𝑥 2 + 𝜕 𝑦 2 + 𝜕𝑧 2 = − 1 𝑐2 𝜕𝑡 2 + 𝛻2 o We calculate the Ricci scalar from the mixed tensor R 𝜈 𝜇 = 𝑔 𝜇𝜌 R 𝜈𝜌 ≈ 𝑔 𝜇𝜌 1 2 𝜂 𝜈𝜇 𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌 ≈ 𝜂 𝜇𝜌 1 2 𝜂 𝜈𝜇 𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌 R 𝜈 𝜇 ≈ 1 2 𝜂 𝜈𝜇 𝜕𝜇 𝜕𝜈ℎ 𝜈𝜇 ≈ 1 2 𝜕𝜇 𝜕𝜈ℎ 𝜇 𝜇 ≈ 1 2 𝜕𝜇 𝜕𝜈ℎ Contracting = we obtain R 𝜇 𝜇 = R ≈ 1 2 𝜕𝜇 𝜕𝜇ℎ = 1 2 ℎ 𝑅 𝜈𝜌 ≈ 1 2 𝜂 𝜈𝜇 𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌 = 1 2 ℎ 𝜇𝜎
- o Putting it all together, we obtain the Einstein tensor in vacuo. 𝑅 𝜇𝜈 − 1 2 𝑔 𝜇𝜈 𝑅 = 0 ⇒ 𝑅 𝜇𝜈 − 1 2 𝜂 𝜇𝜈 + ℎ 𝜇𝜈 𝑅 = 0 ⇒ 1 2 ℎ 𝜇𝜈 − 1 2 𝜂 𝜇𝜈 + ℎ 𝜇𝜈 1 2 ℎ = 0 Again, to first order, 1 2 ℎ 𝜇𝜈 − 1 4 𝜂 𝜇𝜈ℎ = 0 ⇒ 1 2 ℎ 𝜇𝜈 − 1 2 𝜂 𝜇𝜈ℎ = 0 o Finally, we obtain the wave equation 𝒉 𝝁𝝂 − 𝟏 𝟐 𝜼 𝝁𝝂 𝒉 = 𝟎 Thus, the quantities 𝒉 𝝁𝝂 − 𝟏 𝟐 𝜼 𝝁𝝂 𝒉 propagate in space-time with the speed of light in vacuo.
- (a) and (b) transverse, spin 2 (tensor) (c) transverse, spin 0 (scalar) (d) purely longitudinal, spin 0 (scalar) (e) and (f) longitudinal transverse, spin 1 (vector) Polarization This describes how spacetime is distorted in three different spatial directions when a gravitational wave propagates. General Relativity predicts these waves are transversal. This means that they stretch and compress space-time in the plane perpendicular to the direction of propagation. The allowed distortions are only of two types, called polarizations (a) "+" ("more"), and (b) "x" ("cross"). The space-time distortion occurs in the plane perpendicular to the direction in which the gravitational wave propagates (axis z). (c) again shows a transverse polarization, while (d), (e) and (f) illustrate distortions that propagate in one direction (indicated by the arrow) that is in the same plane as the space- time distortion.
- LIGO and Virgo The LIGO and Virgo detectors are giant Michelson-type laser interferometers with arms of 4 and 3 kilometers respectively, both have been designed to detect gravitational waves. o LIGO Laser Interferometer Gravitational-Wave Observatory Hanford, Washington, USA
- o Virgo is named for the Virgo Cluster of about 1,500 galaxies in the Virgo constellation. Cascina, Pisa, Italy
- The Observed Signal GW170814 GW170814 was a gravitational wave signal from two merging black holes, detected by the LIGO and Virgo observatories on 14 August 2017 at 10:30:43 UTC. This was the fourth confirmed event after GW150914, GW151226 and GW170104. It was the first binary black hole merger detected by LIGO and Virgo together. The detected gravitational waves, "wrinkles in spacetime", were emitted during the final moments of the fusion of two black holes with masses of approximately 30.5 𝑴⨀ and 25.3 𝑴⨀ and located 540 Mpc (1.8x102 ly) away. The resulting spinning black hole is about 53.2 𝑴⨀. This means that approximately 2.6 𝑴⨀ were converted into energy in the form of gravitational waves during coalescence.
- The blue area corresponds to the fast localization based only on data from the two LIGO detectors. Adding Virgo leads to the orange area, which is an order of magnitude smaller, 100 𝑑𝑒𝑔2 compared to 1160 𝑑𝑒𝑔2 . The green region is the result of the complete analysis of parameter estimation using the three detectors: an area of 60 𝑑𝑒𝑔2 . The right part of the figure compares the probability distributions for the luminosity distance of the source. Location of the source of GW170814 in the sky.
- What detected each of them? Signal-to-noise ratio as a function of time. The peak occurs at different times in each detector because the gravitational wave propagates at the speed of light. GW170814 arrived first at LIGO-Livingston, 8 ms later it arrived at LIGO- Hanford, and 6 ms later it arrived at Virgo. Temporal series of amplitude with the best wave models selected by the adapted filtering technique (black curves) and search methods without models (gray bands) superimposed Time-frequency representation of the amplitude of the signal.
- Sources of Gravitational Waves In general, any acceleration that is not spherically or cylindrically symmetric (with varying 4-pole moment) will produce a gravitational wave. For example: • Supernovae This explosion will produce gravitational waves if the mass is not ejected in a spherically symmetric way. • Spinning star A perfectly spherical star will not produce a gravitational wave, but a lumpy star will. The gravitational waves that modern detectors are sensitive to would be in the audible frequency range if they were sound waves. In that sense, these detectors can be thought of as ‘gravitational wave radios.’
- Continuous Gravitational Waves They are produced by systems that have a constant and well-defined frequency. Examples of these are binary star or black hole systems orbiting each other or a single star rotating rapidly about its axis. The sound these gravitational waves would produce is a continuous tone since the frequency of the gravitational wave is nearly constant.
- Inspiral Gravitational Waves They are generated during the end-of-life stage of binary systems where the two objects merge into one. These systems are usually two neutron stars, two black holes, or a neutron star and a black hole whose orbits have degraded to the point that the two masses are about to coalesce.
- Burst Gravitational Waves They come from short-duration unknown or unanticipated sources. • Gamma ray bursts (GRBs) gamma-ray bursts (GRBs) are extremely energetic explosions that have been observed in distant galaxies. • Supernovae (SN) it is a dramatic and catastrophic destruction which is marked by one final titanic explosion.
- Stochastic Gravitational Waves They are the relic gravitational waves from the early evolution of the universe. Much like the Cosmic Micro-wave Background (CMB), which is the leftover light from the Big Bang, these gravitational waves arise from a large number of random, independent events combining to create a cosmic gravitational wave background. If these gravitational waves truly originated in the Big Bang, these waves will have been stretched as the universe expanded and they can tell us about the very beginning of the universe.
- Black Hole Hunter
- Thankyou foryourpatience

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