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Fundamental physics, year 1 (Teknisk fysik, årskurs 4)

Gravitation and Cosmology — Lecture notes

by Christian von Schultz

Lecturer: Martin Cederwall
Assistant teacher: Fredrik Ohlsson (exercise classes)

This is an introductory course in general relativity. The course home page can be found at

Monday 2008-10-27
Introduction to the course. Newton's description of gravitation — the analogy with electrostatics. Galilean transformations. Lorentz transformations. Tensor notation. Lorentz transformation as a linear transformation of the coordinates. The postulates of special relativity. Light rays. The Minkowski metric η. The transformation of vectors, scalars, tensors. Raising and lowering indices with η.
Tuesday 2008-10-28
Special relativity: Velocity and four-velocity. Proper time. Time dilation. Four-acceleration. Four-momentum. Taylor expansion of E = γ m (or, restoring the speed of light, E = γ m c2). Currents and charges. The equation of continuity. Analogy between electromagnetic charges/current and four-momentum/stress-energy tensor (also known as the energy-momentum tensor). Maxwell's equations [in relativistic (c = 1) Lorentz-Heaviside units]. The electromagnetic field tensor. Maxwell's equations in tensor notation. The four-vector potential Aα. Curl using index notation.
Thursday 2008-10-30
Exercise problems: Exercise class 1, problems 1–5.
Monday 2008-11-03
The equivalence principle. Dropping apples in accelerated elevators. Inertial mass and gravitational mass. Physics in Australia. Local inertial frames, local tangent planes, local Minkowski space. General coordinate transformations ξαxμ, ηαβgμν. The affine connection Γμνλ. The geodesic equation. Expressing Γμνλ in terms of the metric tensor gμν. The inverse of the metric g. The three assumptions of the Newtonian limit. The connection to the Newtonian gravitational potential Φ.
Tuesday 2008-11-04
Time dilation (that gives rise to the phenomenon of gravitational redshift, as opposed to the speed-induced time dilation of special relativity). The relation between proper time and coordinate time. Gravitational redshift in the Newtonian limit. Vectors and tensors. The transformation of vectors and tensors under change of coordinate system. Scalar products. 1-forms and vector fields. The affine connection Γμνλ in terms of xμ and ξα, and in terms of the metric. Γμνλ is not a tensor. Derivatives of scalars are vectors. The derivatives of vectors are problematic. The covariant derivative Dμ. The metric is covariantly constant (Dμ gνλ = 0).
Thursday 2008-11-06
Exercise problems: Exercise class 2, problems 1, 2, 4. Exercise class 3, problems 1, 2. Tensor densities. The Levi-Civita tensor density and its weight.
Monday 2008-11-10
More about the covariant derivative. Two derivatives after each other. Parallel transport. The commutator of covariant derivatives. The Riemann curvature tensor Rμνλκ. Sign convention for the curvature tensor. The curvature tensor for flat space, calculated in polar coordinates. The fully covariant curvature tensor Rμνλκ. The symmetries of the curvature tensor. The Ricci tensor Rμν. The curvature scalar R. The number of independent components in the Riemann and Ricci tensors. Gravitational waves require at least four dimensions.
Tuesday 2008-11-11
Electromagnetic analogy: covariant derivative with gauge potential or connection, field strength. The Bianchi identity. The Jacobi identity. What we want from Einstein's equations. Conservation of current. The Einstein tensor. Einstein's equations. Dimensional analysis of Einstein's equations. Calculating the constant in Einstein's equations from the Newtonian limit. The cosmological constant. Interesting solutions: Schwartzschild, weak field with gravitational waves, solutions for the universe itself. No superposition principle in general relativity. The gauge symmetry of general relativity.
Thursday 2008-11-13
Exercise problems: Exercise class 3, problems 3, 4. Exercise class 4, problems 1, 3. Why we can replace the Lagrangian of the variational problem of finding geodesics, with the square of said Lagrangian (removing a pesky square root).
Monday 2008-11-17
The Schwarzschild solution. Solving Einstein's equations in a static, spherically symmetric situation. Choosing an ansatz (use symmetry, make a gauge choice). Imposing boundary conditions: Minkowski at infinity. The Newtonian limit: identifying the integration constant as mass (-2MG to be precise). The Schwarzschild solution. The Schwarzschild radius. Schwarzschild black holes: The horizon is just a coordinate singularity. There is a real singularity in the middle. Coordinates t and r inside and outside of the horizon: space and time.
Tuesday 2008-11-18
The action principle. The action is a functional on the space of field configurations. The functional derivative. An action for the field gμν. The variation of the Riemann curvature tensor, the inverse metric, and the square root of the determinant of the metric. Integrating a covariant divergence. The Einstein-Hilbert action leads to Einstein's vacuum equations. Adding a matter action and finding the stress-energy tensor Tμν. Diffeomorphisms. The variation of the metric under an infinitesimal coordinate transformation xμxμ + εμ(x). Conservation of energy-momentum. The action for electromagnetism.
Thursday 2008-11-20
Exercise problems: Exercise class 5, problems 1, 2, 3.
Monday 2008-11-24
Moving around in the Schwarzschild solution. Staying in the θ = π/2 plane. Radial motion. Radial motion close to the event horizon. The precession of Mercury's orbit; calculating such precessions. The eccentricity of an ellipse. Isometries (symmetries of the metric). Isometries of the two-sphere, isometries of flat D dimensional space. The Killing equation, Killing vector fields ξμ(x).
Tuesday 2008-11-25
Gravitational radiation. The graviton. Analogy with the photon. Gravity will act as its own source, unlike the electromagnetic case (the photon has no charge). The weak field approximation. Fix the gauge. The harmonic gauge. The linearised Einstein's equations in harmonic gauge. The particular solution using a Green's function. The plane wave solution. The polarisation tensor. The gravitational wave moves with the speed of light. (The graviton is massless.) Maxwell's equations in Lorentz gauge. Two physical modes in electromagnetism. Two in general relativity. Helicity and the spin of the graviton.
Wednesday 2008-11-26
Exercise problems: Exercise class 6, problems 1, 2, 3.
Monday 2008-12-01
Isometry. Form invariant metric. Killing vectors of flat, two-dimensional space. In D dimensions we get D (D + 1)/2 equations for D components ξμ. The maximal number of isometries is D (D + 1)/2. The second covariant derivative of a Killing vector relates to the Riemann tensor and the Killing vector itself: Dμ Dν ξλ = - Rνλμσ ξσ. The Killing vector and its covariant derivative at one point determines the entire function ξμ(x), if it exists. The covariant derivatives Dμ ξν of a Killing vector are antisymmetric in μ and ν. Interpretation of Killing vector and its derivatives as translation and rotation, respectively. Symmetries of the Universe: homogeneous and isotropic. That implies six Killing vectors, and that implies constant curvature. Given dimensionality, curvature and signature, a maximally symmetric space is unique. Embed D-dimensional space (-time) in a flat space (-time) of dimension D + 1.
Tuesday 2008-12-02
Starting with cosmology. Einstein's equations for the universe: ansatz for the metric + models for matter and energy. Metric by embedding ensures that we have all of the symmetry. The art of embedding. The curvature sign k. de Sitter and anti-de Sitter. The Robertson-Walker metric. Proper time t and conformal time τ. Energy-momentum tensors: Dust (cold matter). Radiation (and hot, relativistic matter). Tλλ = 0 in electromagnetism (this means something). The equation of state parameter w. The acceleration equation and the Friedmann equation. Dust scales as ρ = ρ0 a-3. Radiation (w = 1/3) scales as ρ = ρ0 a-4, since the wavelength grows as the universe expands.
Thursday 2008-12-04
Exercise problems: Exercise class 7, problems 1, 2.
Monday 2008-12-08
Cosmology. Ansatz for the metric with maximally symmetric space part. Proper time for observers at rest. Nice parametrisation of the spatial part, using curvature sign k. The metric with conformal time τ. Scale factor as a function of time: the differential equations. Energy density in terms of the scale factor. Radiation is essentially negligible today. The Hubble parameter, or Hubble constant (which is not a constant). The Friedmann equation. Critical energy density ρc, relates to curvature sign k. Density parameter Ω(i). Flat universes: Scale factor as a function of time: the solution (for w ≠ 1). For w > -1 the universe exhibits a Big Bang at t = 0. This is a singularity. Estimating the age of the universe. Non-flat universes: Conformal time τ is a suitable parameter. The scale factor as a function of conformal time, for general w, for dust and for radiation. The connection between proper time t and conformal time τ. For k = +1, we have a Big Crunch. All common matter has a Big Bang, but only k = 1 has a Big Crunch.
Tuesday 2008-12-09
The cosmological constant Λ. Modifications to Einstein's equations. Thinking of the cosmological constant as modifying the energy-momentum tensor: Positive energy density and negative pressure (for positive Λ). Here w = -1. Λ is like a vacuum energy. Solving the Friedmann equation with cosmological constant: cosh, exp, sinh for k = +1, 0, -1. In all cases, exponential growth for large t. This is de Sitter space. Maximally symmetric, exponential expansion. Discussion about negative pressure. Negative cosmological constant (not the case in the present universe): anti-de Sitter space. The cosmological constant was not very important early in the universe. With different kinds of energy, the equations have to be solved numerically. Patching solutions from times where different types of energy was dominant. For baryonic matter ΩB ≈ 0.04 (there is not much baryonic matter). Ωdark matter ≈ 0.26. There is a lot of dark matter, and it is not any known elementary particles. If really lucky, we could detect it at the Large Hadron Collider LHC. Ωradiation ≈ 0. Ωdark energy ≈ 0.7. Dark energy is something with w = -1. This is a mystery. Ωtotal ≈ 1, the total energy density is very nearly the critical energy density. We would expect a much larger Λ.
Thursday 2008-12-11
Earlier examination problems number 37 and 38.


Photos from the class

A few smiling students More students enjoying the course The lecturer himself: Martin Cederwall Martin Cederwall again Assistant teacher: Fredrik Ohlsson

Christian von Schultz <>