Perturbative Methods in Gravity Theory (LTFY.04.011)

General information

The course will be conducted in a hybrid format. The provided digital course materials are lecture notes and recorded video lectures (see below). These will be accompanied by classes taking place Mondays at 16:15 and Thursdays at 12:15 in Physicum room A111. In order to pass the course, please do the following for each of the 16 lectures:

• Read the questions and the course materials, watch the video lecture before the class. Prepare yourself to answer the questions and explain your answer. Make yourself additional notes if you consider it helpful. Some questions will involve deriving formulas which are not explicitly derived in the course materials.
• During the class, you will be asked to present your answers to the questions. Some will require argumentation, others to show an explicit calculation. You may also use the lecture notes during the class.

The dates, lecture topics and questions are listed below.

Questions

1. 1. Derive the linear perturbation of the Christoffel symbols, the Riemann tensor and the Ricci tensor in terms of the covariant derivative of the metric perturbation for a general background metric.
   2. What changes if the background is the Minkowski metric?
   3. How would you calculate higher order perturbations of these objects?
2. 1. Derive Newton's equation of motion in a weak gravitational field. Why is only the time-time component of the metric perturbation relevant? What would change for a test mass moving at relativistic speed?
   2. Argue how the pressure of a gas is related to its kinetic energy density.
   3. What is meant by the "time derivative along flow lines"? Which kind of observer could find this notion useful?
   4. Derive the Poisson equation from Einstein's equations.
3. 1. What is the harmonic gauge?
   2. Why is it always possible to choose the harmonic gauge?
   3. Given a metric perturbation h' in an arbitrary gauge, derive an equation for the gauge transformation ξ such that h will be in the harmonic gauge.
   4. Apply the harmonic gauge condition to the plane wave solutions of the wave equation. What is the relation between the metric perturbation and the wave covector?
   5. Are plane waves the only solutions to the linearized field equations in the harmonic gauge? What is the most general solution?
4. 1. Why does one need to consider only the time-time component of the energy-momentum tensor to calculate the gravitational wave emission?
   2. What is the quadrupole tensor?
   3. How is the frequency of the observed gravitational wave related to the orbital frequency Ω of a binary system and why?
   4. Are there any gravitational waves emitted from a spherical collapse in general relativity? Why? Hint: what does the quadrupole tensor look like if you express the Cartesian coordinates under the integral through spherical coordinates, and assume that the energy-momentum tensor does not depend on the angular coordinates?
5. 1. Calculate the "electric" components of the Riemann tensor (with lower indices) in terms of the metric perturbation around a Minkowski background.
   2. Explain the shape of the gravitational wave polarization diagrams by the fact which components of the metric perturbation are non-vanishing.
   3. Show that the Lorentz transform Λ(φ, α) is indeed a Lorentz transform which preserves the wave covector and prove the formula for the product of two such transforms.
   4. What are E₂ classes?
   5. Are there any gravity theories which allow for vector modes, but not for tensor modes? Why?
6. 1. What is meant by conformal time and conformal Hubble parameter?
   2. Derive the Christoffel symbols for the homogeneous and isotropic background metric.
   3. Follow the steps of the derivation of the tensor equations and convince yourself that you could reproduce each step.
7. 1. Calculate the 3+1 decomposition of the covariant derivative of the gauge transformation ξ.
   2. Derive explicitly the gauge transformation of the irreducible components of the metric perturbation.
   3. Show that the gauge-invariant components are indeed gauge-invariant.
   4. How many independent components does the metric have, and how many gauge-invariant components? Why?
   5. Which quantities are used to describe the perturbation of the energy-momentum tensor and how are they defined?
8. 1. Calculate explicitly the gauge transformation of the energy-momentum tensor.
   2. Derive the relations between the vector part of Einstein's equations shown in the lecture notes.
   3. Derive the energy-momentum conservation equations for the scalar perturbations.
9. 1. What do the first few (l ≤ 2) spherical harmonics look like in the angular coordinates (χ, φ) instead of the usual (θ, φ)?
   2. Calculate the scalar, vector, tensor spherical harmonics with m = 0 and l taking the lowest value such that the result is non-vanishing.
   3. What is the difference between axial and polar modes?
   4. What is the Regge-Wheeler gauge?
   5. What is the tortoise coordinate?
   6. Why can one set m = 0 for a spherically symmetric background?
   7. Follow the steps of the simplification of the axial equations and convince yourself that you could reproduce each step.
10. 1. Which manifolds have been introduced in the lecture, how are they related and what are their dimensions?
    2. What is the metric signature on these manifolds?
    3. What is meant by a conformal symmetry of a metric?
    4. What is meant by asymptotic flatness?
    5. Show how the properties of Ξ follow from its definition and the properties of g and n.
    6. Calculate the Lie derivatives of g and n on the boundary for ten Killing vector fields of Minkowski spacetime.
11. 1. Which conventions exist to denote higher order perturbations?
    2. Prove the general formula for the perturbation of the inverse metric in terms of the metric perturbation at arbitrary order.
    3. Explain the general formula for the perturbation of the Riemann tensor in terms of the perturbation of the Christoffel symbols.
12. 1. What are the basic assumptions of the PPN formalism?
    2. Derive the components of the energy-momentum tensor with lower indices up to fourth PPN order.
    3. Derive the components of the Ricci tensor in terms of the metric perturbations up to fourth PPN order.
    4. What is meant by the standard post-Newtonian gauge?
    5. How many PPN parameters are in the post-Newtonian metric and what are their values in general relativity?
13. 1. What is the gothic metric?
    2. Show how the harmonic gauge condition can be expressed in terms of the gothic metric?
    3. What is the structure of the perturbative expansion of Einstein's equations in the harmonic gauge?
    4. Derive the lowest order (linear) perturbation of the Einstein tensor, with indices raised by the gothic metric, in the harmonic gauge, expressed through the perturbation of the inverse gothic metric.
14. 1. How are second-order perturbations of the metric and energy-momentum tensors defined?
    2. How do second-order metric perturbations transform under an infinitesimal coordinate transformation?
    3. How does one decompose second-order perturbations into irreducible components?
    4. Express the second-order perturbation of the energy-momentum tensor (with upper indices) through the perturbations of the metric, density, pressure, velocity and anisotropic stress (without decomposing into the irreducible components).
15. 1. In the general theory of gauge transformations and gauge invariance, what is meant by a gauge?
    2. What is a knight diffeomorphism?
    3. How does one separate quantities defining a gauge from gauge-invariant quantities?
    4. Convince yourself that the objects ξ appearing in the Taylor expansion of the gauge transformation Φ are indeed vector fields, based on the arguments in the lecture notes.
    5. In the formulas for the gauge transformation of A up to the second order, express the gauge-dependent through the gauge invariant A and the vector fields X and Y, and derive the relation between X, Y, ξ up to second order.
16. 1. Use the transformation law of connection coefficients under coordinate changes in order to derive their Lie derivative.
    2. What are torsion and nonmetricity?
    3. What is a Finsler function?
    4. Which connection perturbations preserve the flatness of a flat connection? Prove that this is the case.
    5. Which connection perturbations would preserve vanishing curvature and vanishing torsion (but not necessarily nonmetricity)?

Course materials

Lectures

Please click on 🎥 to watch the recorded lectures. To execute the Mathematica notebooks, you need the tensor algebra package suite xAct, for the cosmological perturbations also xPand and for the calculation of PPN parameters xPPN. Alternatively, there is also a PDF made from every Mathematica notebook.

Additional recommended literature

• Gravity, general relativity and linear perturbations
  • C. W. Misner, K. S. Thorne, J. A. Wheeler: Gravitation; W. H. Freeman (1973)
  • R. Wald: General Relativity; Chicago (1984)
  • S. Carroll: Spacetime and Geometry - An Introduction to General Relativity; Pearson (2004)
• Parametrized post-Newtonian formalism, Newman-Penrose formalism
  • C. M. Will: Theory and Experiment in Gravitational Physics; Cambridge University Press (1993) - recommended edition, uses same conventions as in lecture notes
  • C. M. Will: Theory and Experiment in Gravitational Physics; Cambridge University Press (2018) - newer edition, introduces parametrized post-Newtonian formalism in different convention
  • C. M. Will: The Confrontation between General Relativity and Experiment, Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377 [gr-qc]]
• Post-Newtonian expansion, compact systems, binaries, gravitational wave emission
  • E. Poisson, C. M. Will: Gravity; Newtonian, Post-Newtonian, Relativistic; Cambridge University Press (2014)
  • L. Blanchet: Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries, Living Rev. Rel. 17 (2014) 2 [arXiv:1310.1528 [gr-qc]]
  • T. Di Matteo, A. King, N. Cornish: Black Hole Formation and Growth; Springer (2019)
• Quasinormal modes, perturbations of black hole spacetimes
  • H.-P. Nollert: Quasinormal modes: the characteristic "sound" of black holes and neutron stars, Class. Quant. Grav. 16 (1999) R159
  • P. Pani: Advanced Methods in Black-Hole Perturbation Theory, Int. J. Mod. Phys. A28 (2013) 1340018 [arXiv:1305.6759 [gr-qc]]
  • C. M. Will, B. S. Sathyaprakash, E. Berti: Lecture materials from the ICTS Summer school on gravitational wave astronomy
• Asymptotically flat spacetimes, Bondi-Metzner-Sachs group
  • R. Geroch: Asymptotic Structure of Space-time; Plenum Press (1977)
  • É. É. Flanagan, K. Prabhu, I. Shehzad: Extensions of the asymptotic symmetry algebra of general relativity, JHEP 2001 (2020) 002 [arXiv:1910.04557 [gr-qc]]
  • H. Bondi, M. G. J. van der Burg, A. W. K. Metzner: Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A269 (1962) 21
  • R. K. Sachs: Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A270 (1962) 103
• Gauge-invariant higher order perturbation theory
  • M. Bruni, S. Matarrese, S. Mollerach, S. Sonego: Perturbations of space-time: Gauge transformations and gauge invariance at second order and beyond, Class. Quant. Grav. 14 (1997) 2585 [arXiv:gr-qc/9609040]
  • M. Bruni, S. Sonego: Observables and gauge invariance in the theory of nonlinear space-time perturbations, Class. Quant. Grav. 16 (1999) L29 [arXiv:gr-qc/9906017]
  • S. Sonego, M. Bruni: Gauge dependence in the theory of nonlinear space-time perturbations, Commun. Math. Phys. 193 (1998) 209 [arXiv:gr-qc/9708068]
  • K. Nakamura: Gauge invariant variables in two parameter nonlinear perturbations, Prog. Theor. Phys. 110 (2003) 723 [arXiv:gr-qc/0303090]
  • K. Nakamura: Second order gauge invariant perturbation theory, Prog. Theor. Phys. 113 (2005) 481 [arXiv:gr-qc/0410024]
• Cosmology, linear cosmological perturbation theory
  • S. Weinberg: Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity; Wiley (1972)
  • S. Weinberg: Cosmology; Oxford University Press (2008)
  • S. Dodelson: Modern Cosmology; Academic Press (2003)
  • H. Kodama, M. Sasaki: Cosmological Perturbation Theory, Prog. Theor. Phys. Suppl. 78 (1984) 1
  • V. F. Mukhanov, H. A. Feldman, R. H. Brandenberger: Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203
  • K. A. Malik, D. Wands: Cosmological perturbations, Phys.Rept. 475 (2009) 1 [arXiv:0809.4944 [astro-ph]]
• Higher order cosmological perturbation theory
  • K. Tomita: Non-Linear Theory of Gravitational Instability in the Expanding Universe I, Prog. Theor. Phys. 37 (1967) 831
  • K. Tomita: Non-Linear Theory of Gravitational Instability in the Expanding Universe II, Prog. Theor. Phys. 45 (1971) 1747
  • K. Tomita: Non-Linear Theory of Gravitational Instability in the Expanding Universe III, Prog. Theor. Phys. 47 (1972) 416
  • K. Nakamura: Second-order gauge invariant cosmological perturbation theory: Einstein equations in terms of gauge invariant variables, Prog. Theor. Phys. 117 (2007) 17 [arXiv:gr-qc/0605108]
• Perturbations of tetrad, connection, Finsler Geometry

Links to computational tools

• xAct - general tensor calculus
• xPand - cosmological perturbations
• xPPN - parametrized post-Newtonian formalism
• Black Hole Perturbation Toolkit - perturbations of black hole spacetimes