Gravitational waves

FSU Jena, SS 2022

Lecture period: 11.04.2022 - 15.07.2022

Last updated: 29.02.2022

Moodle link

Moodle link for tutorials

Topics

• Weak field limit of GR
  • Weak-field equations
  • Solution for static sources
  • Solution for stationary sources
• Waves on flat background
  • Propagation of GWs on flat background
  • Interaction with test-masses
  • Sources of GW: quadrupole formula
• GW energy
  • Energy og GW in lienarized GR
  • Short-wave approximation
  • Isaacson tensor
• Multipolar expansion
  • Beyond the quadrupole formula
  • Multipolar expansion for scalar fields
  • STF expansion
  • Tensor spherical harmonics
• The post-Newtonian approximation
  • PN expansion of Einstein equations
  • Conservative dynamics
  • PN approach to radiation reaction
• Linearization around black holes
  • Regge-Wheeler-Zerilli equation
  • RWZ potential and black hole quasi-normal modes (ringdown)
  • Radial infall of David-Ruffini-Tiomno
• Effective-one-body description of the general-relativistic 2-body problem
  • Relative motion and Schwarzschild Hamiltonian
  • Mapping the conservative dynamics
  • Resummation aspects and waveform generation
• Strong, self-gravity and high-velocities source
  • The binary-black hole merger waveform
  • Kicks (Fouchet formula, and numerical relativity predictions)
  • Gravitational collapse and signals from CCSN
• Detection and data-analysis
  • Noise spectral density
  • Antenna pattern
  • Match-filtering & Bayesian methods

Books

• M.Maggiore, “Gravitational Waves, Vol.1: Theory and Experiments”, Oxford University Press, 2008.
• M.Maggiore, “Gravitational Waves, Vol.2: Astrophysics and Cosmology”, Oxford University Press, 2018.
• S.Chandrasekhar, “The Mathematical Theory of Black Holes”, Oxford University Press, 1983.
• R.Wald “General Relativity” University of Chicago Press (1984)
• Y.Choquet-Bruhat, “General Relativity and the Einstein Equations”, Oxford Mathematical Monographs, 2009.
• Y.Choquet-Bruhat, “Introduction to General Relativity, Black Holes, and Cosmology”, Oxford Mathematical Monographs, 2015.

Reviews

• L.Blanchet Living review on post-Newtonian methods
• E.Berti et al Review on quasi-normal modes of black holes
• T.Damour 1974: the discovery of the first binary pulsar
• T.Futamase The Post-Newtonian Approximation for Relativistic Compact Binaries
• K.Kokkotas and B.G.Schmidt Living review on quasi-normal modes of stars and black holes
• P.Jaranowski and A.Krolak Living review on GW data analysis
• C.Will The Confrontation between General Relativity and Experiment
• T.Mädler and J.Winicour Bondi-Sachs Formalism
• P.Nollert Quasinormal modes: the characteristic `sound’ of black holes and neutron stars
• M.Pitkin et al living review on GW interferometers
• D.Radice et al. The Dynamics of Binary Neutron Star Mergers and of GW170817
• B.S.Sathyaprakash and B.F.Schutz living review on GW
• G.Schäfer, P.Jaranowski Living review on Hamiltonian formulation
• S.Zschocke A detailed proof of the fundamental theorem of STF multipole expansion in linearized gravity
• Saffer et al., Sec.II on the GW stress-energy tensor

Lecture notes

• A.Buonanno Lectures on GW
• W.Del Pozzo Lectures on GW data analysis
• E.Flanagan & Hughes The basics of gravitational wave theory
• ICTS Summer School on Gravitational-Wave Astronomy
• M.Hendry Lectures on GW
• J.Novak Lectures on GR and GW
• P.Pani Lectures on black hole perturbation theory
• E.Poisson Lectures on post-Newtonian methods
• G.Schäfer, P.Jaranowski Living review on Hamiltonian formulation
• F.D’ambrosio et al. Gravitational Waves in Full, Non-Linear General Relativity (Based on Ashtekar’s lectures)

Suggested articles

• Einstein (1916) and (1918) (See also A.Einstein in Annalen der Physik)
• Regge & Wheeler (1957) Stability of a Schwarzschild Singularity
• Brill and Hartle (1963) Method of the Self-Consistent Field in General Relativity
• Peters & Mathews (1963) Gravitational Radiation from Point Masses in a Keplerian Orbit
• Isaacson PhD thesis (1967)
• Isaacson (1968) Gravitational Radiation in the Limit of High Frequency: Nonlinear Terms and the Effective Stress Tensor
• Vishveshwara PhD thesis (1968)
• Vishveshwara (1970) Scattering of Gravitational Radiation by a Schwarzschild Black-hole
• Chandrasekhar & Esposito Farese (1970) The 2½-post-Newtonian Equations of Hydrodynamics and Radiation Reaction in General Relativity
• Press&Thorne (1972) Gravitational-wave Astronomy
• David, Ruffini & Tiomno (1972) Pulses of Gravitational Radiation of a Particle Falling Radially into a Schwarzschild Black Hole
• Chandrasekhar & Detweiler (1975) The quasi-normal modes of the Schwarzschild black hole
• Witten (1981) Positive energy theorem
• Ludvigsen & Vickers (1981), Schoen & Yau (1982), Horowitz & Perry (1982) Positive Bondi mass theorem
• Damour (1983) Gravitational Radiation Reaction in the Binary Pulsar and the Quadrupole-Formula Controversy
• Leaver (1986) Analytic representation for the quasi-normal modes of Kerr black holes
• Blanchet & Damour (1986) Radiative Gravitational Fields in General Relativity (PDF)
• Schutz (1986) Determining the Hubble constant from gravitational wave observations
• Blanchet & Damour (1989) Post-newtonian generation of gravitational waves
• Nollert & Schmidt (1992) Quasinormal modes of Schwarzschild black holes: Defined and calculated via Laplace transformation
• Cutler & Flanagan (1994) Gravitational Waves from Mergin Compact Binaries: How Accurately Can One Extract the Binary’s Parameters from the Inspiral Waveform?
• Babak & Grishchuk (1999) The Energy-Momentum Tensor for the Gravitational Field
• Pretorius (2005) First binary black hole merger in numerical relativity
• Martel & Poisson (2005) Gravitational perturbations of the Schwarzschild spacetime: A practical covariant andgauge-invariant formalism
• Maggiore (2007) Mechanical interpretation of the spectrum of black hole quasinormal modes

Papers about pulsars and LIGO-Virgo observations

• Hulse-Taylor PSR B1913+16 measurement
  • 1975
  • 1993 Nobel prize, press release
  • 2010
  • 2016
• LIGO-Virgo measurements
  • Crab pulsar spindown limit (2008)
  • Black hole merger (2015)
  • Black hole merger, three detectors observations (2017)
  • Binary neutron star inspiral (2017)
  • 2017 Nobel prize, press release

Resources on the web

Videos

• A.Ashtekar’s YouTube lectures
  • Playlist
• T.Damour’s YouTube lectures
  • Part 1/4
  • Part 2/4
  • Part 3/4
  • Part 4/4
• C.Will’s YouTube lectures/talks
  • Lectures on Newtonian, post-Newtonian, relativistic gravity
  • Was Einstein right? A centennial assessment
• E.Poisson’s YouTube lectures
  • Black Hole Perturbation Theory - Lecture 1
• M.Kramer Nearly 100 years after General Relativity: Was Einstein right? (Pulsars tests)
• Gravitational Wave first detection, Press conference on February 11, 2016
• Gravity Probe B press conference

Data/software/msc

• Berti’s ringdown page
• Gravitational Wave Open Science Center
• Guth notes on STF and multipolar expansion
• Caltech’s LIGO website
• NIST Digital Library of Mathematical Functions
• xPert Mathematica package for computer algebra for metric perturbation theory. See also this paper
• Virgo website

Course material

Lecture notes:

• Introductory slides
• Gravitational waves
• Theory of Probability
• Final projects

Exercises

• Exercise sheet n.1 - Solutions
• Exercise sheet n.2 - Solutions
• Exercise sheet n.3 - Solutions
• Exercise sheet n.4 - Solutions
• Exercise sheet n.5 - Solutions

Tutor: M.Breschi

Please see the tutorials’ Moodle page

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