# Introduction into General Relativity Lecture Notes and Problem Sheets Summer 2020

## Lecture Notes (handwritten)

• Lecture 1: Recap: Newtonian concepts of machanics and gravity.
• Lecture 2: Equivalence principle and some immediate consequences.
• Lecture 3: Energy-momentum tensors.
• Lecture 4: What's wrong with a special-relativistic theory of gravity?
• Lecture 5: Affine/Inertial structure in Minkowski space, geodesic equation, affine parametrisations.
• Lecture 6: One more argument against a special-relativistic theory of gravity. A first look at Einstein's equations. Lovelock's theorem. How mass-desity curves space.
• Lecture 7: Transformation properties of tensors and connections. Riemann normal coordinates. Levi-Civita covariant derivative, its coordinate expressions, and comparison with Lie derivative of tensors of arbitrary rank.
• Lecture 8: More on the curvature tensor: Symmetries, number of independent components, 1. and 2. Bianchi identities, consequences of 2. Bianchi identity, commutator of covariant derivatives on arbitrary tensors. Further: Killing fields and their Lie algebra, finite upper bound on their dimensionality. Further: Definition of the Weyl tensor, numer of independent components, conformal invariance, definition of constant-curvature metrics.
• Lecture 9: Action principle in GR. Definition of energy-momentum tensor through metric coupling.
• Lecture 10: The linearised Einstein equations and its gauge invariance.
• Lecture 11: The linearised Einstein equations: Newtonian limit and next-to-leading-order approximation. The gravitomagnetic field. effect.
• Lecture 12: The linearised Einstein equations: Gauge fixing and polarisation states of gravitational waves.
• Lecture 13: The linearised Einstein equations: Action, energy-momentum, and quadrupole formula.
• Lecture 14: The linearised Einstein equations: Gravitational radiation emitted by a rigidly rotating body; amplitide, luminosity, polarisation.
• Lecture 15: The linearised Einstein equations: Gravitational radiation emitted by binary star-system on circular orbits. Discussion of Hulse-Taylor-Pulsar (PSR B1913+16).
• Lecture 16: Gravitationa redshift in stationary spacetimes. Gravitational lensing in static, spatially conformally flat spacetimes. Shapiro time-delay and some of its applications.
• Lecture 17: Spherically symmetric metrics. Calculation of Riemann-, Ricci-, and Einstein-Tensors using the Cartan structure equations.
• Lecture 18: The exterior Schwarzschild solution I.
• Lecture 19: Timelike and lightlike Geodesic motion in the exterior Schwarzschild geometry.
• Lecture 20: Static and spherically-symmetric perfect-fluid stars. The Tolman-Oppenheimer-Volkov equation. The interior Schwarzschild solution for incompressible fluids. The "Buchdahl Limit" and other limits from energy dominance. Gravitational binding energy with examples.
• Lecture 21: Speherically symmetric and static solutions to the Einstein-Maxwell equations with cosmological constant. Schwarzschild-DeSitter and Reissner-Nordström solutions as special cases. Penrose-diagramms of the latter.
• Lecture 22: The Kerr-Newman-family of solutions. (Stationary axisymmetric solutions to Einstein-Maxwell-eqns.). Metric in Boyer-Lindquist coordinates. Reminder on confocal elliptic coordinates. Electromagnetic field and gyromagnetic ratio (g=2). Determination of angular-momentum parameter. Limits of statioinarity, horizon and ergoregion. Geometry of spacelike 2-surface Σ = {r=r+ at t=const.}. Area A of Σ as function of (m,l,q), conversely m as function of (A,l,q). Smarr formula. Gauß curvature of Σ and application of Gauß-Bonnet theorem to show that Σ is a 2-sphere. Discussion of possible negative curvatures around poles.

## Problem sheets and solutions

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

• Sheet 1 (18 KW): problems and solutions

1. Galilei's thought experiment concerning the universality if free fall.
2. Inertial-, active-, and passive gravitational mass in Newtonian gravity. Motion of pair of points, one of which has negative gravitational and inertial masses.
3. The stress-tensor for Newtonian gravity and the impossibility of self-acceleration.
4. Non-linear modification of Newtonian gravity.

• Sheet 2 (19 KW): problems and solutions (and more!)

1. Periastron precession in perturbed Kepler potentials.
2. Newtonian gravitational potential in quadrupole aproximation: General case.
3. Newtonian gravitational potential in quadrupole aproximation: Special case of axisymmetric mass distributions; the J2 parameter.
4. Application to homogeneous mass distribution inside a spheroid. Determination of J2 as function of "flattening" and estimation of J2 for the Sun.
5. Estimation of the Sun's quadrupole contribition to the perihelion precession of Mercury.

• Sheet 3 (20 KW): problems and solutions

1. Energy-momentum tensor of perfect fluid: General discussion.
2. Energy-momentum tensor of perfect fluid: Imposing vanishing divergence and work out consequences.
3. Current density of electric point charge and energy-momentum tensor of point mass (as distributions). Testing divergencelessness.
4. Necessary and sufficient condition for symmetric energy-momentum tensor to be diagonalisable.
5. Energy conditions (weak, strong, energy-dominance) on energy-momentum tensors. Interpretation as image restrictions. Consequences for perfect fluid.
6. Alternative form of dominant-energy condition.

• Sheet 4 (21 KW): problems and solutions

1. Newtonian equations of motion as geodesic equation
2. Eigentime of clocks in homogeneous static gravitational field. Comparison of eigentimes of a freely falling and stationary clock in a homogeneous static gravitational field.
3. General stationary metric. Necessary and sufficient Condition for it to be static.
4. Vorticity, shear, and expansion of a timelike vector field.
5. The Raychaudhuri equation.
6. Points of infinite convergence on irrotational timelike geodesic congruences.

• Sheet 5 (22 KW): problems and solutions

1. Another, index-free expression for the vorticity of a four-velocity field.
2. Rotating four-velocity fields in Minkowski space.
3. Geometric aspects of "space" in rotating reference frames in Minkowski space.
4. Geometric aspects of "simultaneity" in rotating reference frames in Minkowski space.
5. Einstein's equations for static metrics in vacuum; the "no -soliton"-theorem.
6. Timelike and lightlike geodesics in static metrics. Proof that spatial projections of lightlike geodesics are geodesics in the "optical metric" with static time as affine parameter.

• Sheet 6 (24 KW): problems and solutions

1. Gravitational field inside a hollow, non rotating, homogeneous spherical shell in linearised approximation.
2. Gravitational field inside a hollow, slowly rotating, homogeneous spherical shell in linearised approximation. Coriolis-like terms in the geodesic equation.
3. Possibility to detect the Earth's gravitomagnetic field through Sagnac effect.
4. Thomas precession in Minkowski space as a result of Fermi-parallel-transport.
5. Relation between Fermi parallel-transport along a worldline and Levi-Civita parallel-transport along the hodograph.

• Sheet 7 (25 KW): problems and solutions

1. Once more: Timelike and lightlike geodesics in static metrics. Proof that spatial projections of lightlike geodesics are geodesics in the "optical metric" with static time as affine parameter.
2. Representation of spatial rotations on gravitational-plane-wave amplitudes.
3. Gesodesics under gravitational-plane-waves and the spatial "motion" of test particles.
4. Fermi parallel transport along worldlines of stationarity; Thomas precession and its relation to vorticity of statiorary frames.

• Sheet 8 (26 KW): problems and solutions

1. Energy-current density of a monochromatic plane gravitationa wave.
2. Rotating rod: Upper bound on gravitational-wave luminosity by breaking-stress of material.
3. Rotating rod: Upper bound on gravitational-wave amplitide by breaking-stress of material.
4. Rotating rod: Polarisation of gravitational-wave as function of angle between line of sight and rotation axis.

• Sheet 9 (27 KW): problems and solutions

1. Energy loss of the Earth-Sun system due to gravitational wave emission.
2. Energy loss of th Crab Pulsar. How does it compare to gravitational-wave energy loss of a slightly defomed neutron star of same mass, size, and pulse-period?
3. Distinguishing energy-loss due to gravitational quadrupolar and magnetic dipolar radiation by the braking index.
4. Derivation of the lensing map.
5. Specialisation of the lensing map to a rotationally symmetric point lens.
6. Derivation of geometric optics in general space-times from asymptotic expansion ("short-wavelength limit") short-wavelength limit arbitrary of Maxwell's equations.

• Sheet 10 (28 KW): problems and solutions

1. Coordinate-free derivation of the Jacobi Equation of Geodesic Deviation (JEGD).
2. JEGD in adapted frame components.
3. JEGD for static weak gravitational fields.
4. JEGD for linearised plane gravitational waves in plus-mode.
5. Riemann, Ricci, and Einstein-Tensor components for the most general spherically symmetric metric.
6. Derivation of geometric optics in general space-times from asymptotic expansion ("short-wavelength limit") short-wavelength limit arbitrary of Maxwell's equations.

• Sheet 11 (29 KW): problems and solutions

1. Local boost-symmetry of the curvature tensor for the exterior Schwarzschild metric.
2. Application of JEGD: Genral calculation of the tension inside an elastic body in radial free-fall in the Schwarzschild geometry.
3. Experiments we always wanted to do: 1) Dropping a railway line onto a neutron star. Into how many pieces will it break before hitting the surface? 2) Jumping feet-first across the horizon of a black hole. How massive should it be for this experience to be reasonably painless?
4. Definition, properties, and interpretation of Eddington-Finkelstein coordinates.
5. Definition, properties, and interpretation of Gullstrand-Painlevé coordinates.
6. Rindler coordinates in the x>|ct| part of 2-dimensional Minkowski space. Causal future, casual past, and causal complement of constantly accelerated observers.

• Sheet 12 (30 KW): problems and solutions

1. Newtonian misjudgements of neutron-star gravitational binding energies.
2. Curvature tensor and Kretschmann-scalar for Reissner-Nordström solution.
3. Dimensionless charge/mass ratio for the electron.
4. When does the Compton wavelength exceed the Schwarzschild radius?
5. A heuristic calculation: self-energy of a gravitating charged sphere in the limit of vanishing radius.

## Latest science news related to GR

• 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, sinde 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.

1. Mathematical background
• D. Giulini: Differentialgeometrie für Physiker (pdf)
2.
3. Discussion of the law of inertia (in german)
• D. Giulini: Das Problem der Trägheit (pdf)
4.
5. Discussion of special-relativistic generalisations of Newtonian Gravity. Do such theories exist and, if so, what's wrong with them.
• D. Giulini: What is (not) wrong with scalar gravity (pdf)
6.
7. Some semi-popular information on the experimental status of GR (as of 2005, in german).
• D. Giulini: Einsteins Kunstwerk. Die Allgemeine Relativitätstheorie -- aus mittlerer Entfernung betrachtet. (Physik Journal 10 (2005) 27-33) (pdf)
8.
9. Gravitomagnetism (in german)
• D. Giulini: Kosmische Kreisel: Inertialsysteme und Gravitomagnetismus. (Physik in unserer Zeit 35 (2004) 160-167) (pdf)
10.
11. Relativistic effects in measurements of the Earth's rotation
• N. Beverini et al. High-Accuracy Ring Laser Gyroscopes: Earth Rotation Rate and Relativistic Effects. (Journal of Physics, Conf. Series 723 (2016) 012061) (pdf)
12.
13. General Foundations
• Kip Thorne, David Lee und Alan Lightman: Foundations for a Theory of Gravitation Theories. (Physical Review D, Vol. 7, Nr. 12 (1973) 3563-3577) (pdf)
14.
15. On the many fascinating aspects concerning the perihelion-precession of Mercury.
• Anna Nobili & Clifford Will (Nature 320 (1986) 39-40): The real value of Mercury's perihelion advance. (pdf)
• Sophie Pireaux und Jean-Pierre Rozelot (Astrophysics and Space Science 284 (2003) 1159-1194): Solar quadrupole moment and purely relativistic gravitation contributions to Mercury's perihelion advance. (pdf)
• Frank Pijpers (Mon. Not. R. Astron. Soc. 297 (1998) L76-L80): Helioseismic determination of the solar gravitational quadrupole moment (pdf)

16. The first direct detection of gravitational waves
• B.P. Abbott et al.: Observation of Gravitational Waves from a Binary Black Hole Merger (Physical Review Letters Vol. 116 (2016) 061102) (pdf)
• B.P. Abbott et al.: GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence (Physical Review Letters Vol. 116 (2016) 241103) (pdf)
17.
18. Some histrory
• F.W. Dyson, A.S. Eddington und C. Davidson: A Determination of the Defelction of Light by the Sun's Gravitational Field, from Observations made at the Total Eclipse of May 29, 1919. (Philosophical Transactions of the Royal Societey A: Mathematical, Physical and Engineering Sciences, Volume 220, Issues 571-581, 1920) (pdf). A very readable modern discussion also containing supplementary information is by C. Will: The 1919 measurement of the deflection of light. (Classical and Quantum Gravity, 32 (2015) 124001). (pdf).