Special Topics in GR Lecture Notes and Problem Sheets Winter 2020/21

Lecture Notes (handwritten)

  • Lecture 1: Newtonian Cosmology I: Basics.
  • Lecture 2: Newtonian Cosmology II: The Friedmann Equations in a Newtonian Context.
  • Lecture 3: Recap of GR.
  • Lecture 4: What is a cosmological model in GR? The FLRW models. Curvature properties of FLRW metrics.
    First Addendum: Constant curvature metrics and their normal form.
    Second Addendum: Conformal flatness of FLRW metrics.
  • Lecture 5: Geodesics in FLRW-type geometries.
  • Lecture 6: The Friedmann Equations in General Relativity.
  • Lecture 7: Observable quantities: Redshift, luminsity distance, and the "Hubble Plot" as a relation between them.
  • Lecture 8: Age of universe, galaxy counts, and electromagnetic radiation.
  • Lecture 9: Horizons.
  • Lecture 10: Some typical problems in relativistic cosmology.
  • Lecture 11: Λ-dominated universes.
  • Lecture 12: Discussion of de Sitter and anti-deSitter manifolds and the cosmological models assiciated to them: De Sitter as closed, flat, and open model. Anti de Sitter as open Big-Bang/Big-Crunch cosmology (somewhat similar to Milne's model, with Minkowski space replaced by adS).

Problem sheets and solutions

The information "XX CW" in brackets refers to the XXth calendar week in which the corresponding sheet will be discussed. The solutions will be posted afterwards.

  • Sheet 1 (44 CW): problems and solutions
    1. Solutions to Poisson-Equation for constant mass-density on ℝ3 and S3.
    2. Newtonian equations for N gravitationally interacting point masses.
    3. Homothetic solutions for the Newtonian equations for N gravitationally interacting point masses and the notion of "central configurations".
    4. Integrals for homothetic solutions.
    5. Properties of homothetic solutions.
    6. Further analysis of "central configurations".
  • Sheet 2 (46 CW): problems and solutions
    1. Newtonian equation of motion in expanding universe. Integration of free motion for a(t)~t(2/3) and ~ exp(λt).
    2. Keplerian orbits in an Newtonian exponentially expanding universe. Existence of upper bound for radius of stable circular orbits.
    3. The interior future light cone in Minkowski space as an open FLRW Universe (so-called "Milne Universe").
    4. Space (constant comoving proper time) in the Milne Universe: Curvature, isometry group, isometry groups fixing a point and obeserver at point.
    5. Redshift in Milne Universe.
  • Sheet 3 (48 CW): problems and solutions
    1. Particle motion in FLRW universe.
    2. Consequences of divergence-free energy momentum tensor of a perfect fluid in FLRW spacetime.
    3. Null geodesics in conformally related spacetimes.
    4. Relation of affine parameters for null geodesics in conformally related spacetimes.
    5. Conformal time in FLRW spacetimes.
  • Sheet 4 (50 CW): problems and solutions
    1. Proof of Mattig's formula for the "open" and "flat" case.
    2. An alternative form of Mattig's formula.
    3. Derivation of an explicit formula for the angle-of-sight as function of redshift in case ΩΛ = Ωk = 0.
    4. Calculation of the redshift for Ωrad = Ωdust and comparison with redshift at "recombination".
    5. Analytic determination of all solutions for Ωrad = Ωk = 0. Derivation of formula for "age-of-the-universe" in case ΩΛ > 0 and evaluation for current cosmological parameters.
  • Sheet 5 (1 CW 2021): problems and solutions
    1. Calculation of redshift for epoch at which CMB-spectrum has its maximum at H-ionisation frequency. Comparison with redshift at recombination.
    2. Angular separation for causally disconnected regions on surface of last scattering, this time using radiation dominat dynamics.
    3. Recollapsing FLRW models with ΩΛ=0, Ωk < 0 (positive spatial curvature k=+1; i.e., "closed" models) and either Ωrad or Ωdust vanishing. Analytic solutions, lifetimes, and fronts of light flashes emerging from the "Big-Bang".
  • Sheet 6 (3 CW 2021): problems and solutions
    1. Relations linking the Ω-parameters at any two times/scale-factors.
    2. Planck-Scales and a quantitaive re-assessment of the "flatness-problem".
    3. Exponential expansion and number of e-folds needed to solve flatness-problem.

Latest science news related to Cosmology and GR

  • Once more (this happend time and again in the past) there is a controversy over the "right" value of the Hubble paramater H0, depending on whether its measured on standard candles (supernovae of type 1A) by the HST in either the SCP or SH0ES, on the CMB background via WMAP and more recently PLANCK, or other methods. CMP-based measurements systematically tend to lower values (around 67) than SN1A-based ones (around 74). A nice compilation of supernova datasets can be found here.
  • GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object; see recent paper of June 23. 2020. This observation is potentially very interesting, since the 2.6 solar-mass companion seems too heavy for a neutron star and very light indeed for a black hole. What can it be?
  • Lense-Thirring frame dragging induced by a fast-rotating white dwarf in a binary pulsar system; see recent paper. The effect was measured on a compact binary star system called PSR J1141-6545, which consists of the rare combination of a pulsar (neutron star) and a white dwarf.
  • First precise (5 to 6 σ) measurement of perihelion precession of star S2 orbiting the Galactic Black-Hole Sgr A* confirms GR; see paper and ESO science release (nice pictures!).
  • News on gravitational-wave detection from binary mergers of black-holes and neutron-stars are best drawn directly from the LSC site (LIGO Sientific Collaboration). Most recent highlights are the merger of two black-holes of fairly unequal masses (30 and 8 solar masses) on April 20. and a second neuron-star merger on January 6.
  • The first picture of a "Black-Hole-Shadow" (press release April 10. 2019) is shown on the EHT site (Event Horizon Telescope).
  • Stallite-based test of UFF (Universality of Free Fall) at the 10-15 level by ESA's MICROSCOPE mission. The Satellite was launched on April 25. 2016 and decommissioned at October 18. 2018. See paper and ESA site.
  • Earth-bound tests of UFF (Universality of Free Fall) at the 10-13 level using torion balances. See Eöt-Wash site and the 2012 paper.

Supplementary reading

  1. General background and history of modern cosmology
  2. Mathematical background
    • D. Giulini: Differentialgeometrie für Physiker (pdf)