The Theory of General Relativity

Einstein's Theory of General Relativity encompasses Newton's theory of gravitation for weak fields, but as for Special Relativity it gives rise to a dramatic revision of our understanding of space and time. Next to the formulation of the theory, also applications important for (astro-)physical research will be discussed in some detail. Topics to be covered will include the principle of equivalence, the field equations, the experimental tests of the theory using the advance of the perihelion of Mercury, bending of light and the more recent verification on the basis gravitational radiation from binary pulsar systems. Of course we will also entertain you with black holes and cosmological consequence of the theory. Typically the last lecture will give a glossary of the more modern developments, like Hawking radiation.

Required reading: The course will follow the book " A short course in General Relativity" by J. Foster and J.D. Nightingale (Springer Verlag 1995 (2nd edition), corrected 3rd printing 2001), ISBN 0-387-94295-5 (pbk). Students are required to have the book when the course starts.

Recommend reading, but not required, is the book "Introducing Einstein's Relativity" by Ray d'Inverno (Oxford Univ. Press UK, 1992, reprinted with corrections 1995), ISBN 978-0-19-859686-8 (pbk). This book covers also the more recent (sometimes more formal) developments. In addition there are of course many other textbooks, including the one by Einstein himself, "The meaning of relativity" (his semi-popular book of 1916 is even available online in an English translation).

Required Level: Only a good and thorough knowledge of Special Relativity (and Classical Mechanics) is really necessary. Some background (from mathematics courses) in geometry will be very helpful though. The course is intended for 4th/5th year students.

Material and exercises covered per session - 2004
FN=Foster and Nightingale

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• 23 Jan: General introduction, emphasis on physical aspects.
  FN Introduction, Appendix A, sect. A.1-5.
  Equivalence principle, geometric formulation in terms of the existence of a local inertial frame. Bending of light (classical derivation based on the particle nature of light and/or the equivalence principle, see sect. 4.6-7 and problem 4.6). Curvature of space time in a central force field (free fall in a system with a finite extent). Summary of the principle of special relativity (short derivation of the Lorentz transformation).
  

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• 30 Jan: General coordinates, vectors and tensors.
  FN chapter 1
  An example of the 1st law of Newton in a uniform accelerated coordinate systeem. Interpretation of the Christoffel symbol, partly in terms of fictitious forces.
  You should carefully read through this chapter a few times.
  In particular read, check and make all the Examples and Exercises.

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• 6 Feb: metric properties of curved spaces.
  FN chapter 1 and 2.1 (the extra exercises 1 to 9 were handed out).
  Rest of chapter 1 and the first part of chapter 2: derivation of the equation for geodesics, first using embedding and then using variational calculus, is defined without reference to an embedding. During the lecture ex. 2.1.1-3 will be covered (part of extra exercise 8 returns to this issue).
  After having carefully studied chapter 1, the following set of exercises is relevant for gaining additional understanding of the material we covered:
  Problems: 1.3, 1.4, 1.6, 1.9, 1.10

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• 13 Feb: parallel transport, covariant derivative.
  FN par. 2.2 to 2.5 (thanks to Timon Idema, a translation of handwritten notes was handed out).
  Cyclic coordinates and conservation laws for geodesics. Example 2.1.2 of the Robertson-Walker metric (extra: for positive k, at fixed t the substitution k.r²=sin²(psi) gives a 3-dimensional sphere, i.e. a homogeneous and isotropic space of constant positive curvature; for negative k, at fixed t the substitution k.r²=-sinh²(psi) gives a hyperbolic space, i.e. a homogeneous and isotropic space with constant negative curvature; for k=0, at fixed t, one of course finds a flat space). Parallel transport of vectors, and related to this the definition of the covariant derivative. Introduction of geodesic coordinates.
  Problems: 2.3 (see also extra exercise 5), 2.7 and extra exercises 1, 2, 3

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• 20 Feb: The geodesic equation and Newtonian gravity.
  FN par. 2.5-9, Appendix A.6-8 (the rest of the extra exercises were handed out, as well as a paper on the binary pulsar).
  The interrelation between geodesics and Newton's equations of motion and in particular the relation beteen g₀₀ and the gravitational potential V. Covariantizing equations, the geodesic postulate, equivalence priniciple. Electromagnetism in general coordinates. Rotating coordinate frame and the relation to gravitational time dilatation and redshift.
  Problems: 2.4, 2.5 and extra exercise 4.

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• 27 Fed: Energy-momentum and curvature tensors.
  FN paragraaf 3.1-3
  Energy-momentum tensor, conservation laws (see handout); definition of the curvature tensor. Its symmetry properties will be derived somewhat different from what is done in the book. For FN.3.14 we use problems 2.4 and 2.5. For FN.3.16 we can repeat step by step the derivation on pg. 104 with a contravariant vector (see exercise 3.2.1.a) as compared to FN.3.12. To derive FN.3.18 we make (a few times) use of FN.3.14, FN.3.16 and FN.3.17. Par. 3.3 concerns the relation between the path dependence of parallel transport of a vector and the Riemann curvature tensor, derived using problem 3.5. Study example 3.3.1 and extra exercise 6 to obtain more insight in these matters.
  Problems: 3.1, 3.3, 3.4 + extra exercises 5 and 6

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• 6 Mar: Einstein equations and the Schwarzschild solution.
  FN paragraaf 3.4-7 and 4.1 (thanks to Timon Idema, a translation of the rest of the handwritten notes was handed out).
  Formulation of the Einstein equations. Geodesic deviation (tidal forces) derived by replacing the difference of two geodesics by a family of geodesics, x(u,v), with u the affine parameter for the geodesics and replacing xi by dx/dv, where xi in FN par. 3.4 is defined by the difference of the positions on the two geodesics with the same u. From the geodesic equation for x(u,v) we then find after some work FN.3.35 (see the handout). In the classical limit the Einstein equations reduce to the Poisson equation. Outside a mass distribution this can in part be seen by inspecting the expression for the geodesic deviation. We discuss the Schwarzschild metric as a solution of Einstein's equations. Extra: Description of the geometry of the Schwarzschild solution by viewing the spatial part (dt=0) as a 3-dimensional space embeded in a 4-dimensional euclidian space, through the equation (v(r)/4m)²=r/2m-1 (see the handout). Problem 3.7 will be discussed in class.
  Problems: extra exercises 7, 8 and 9

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• 13 Mar: Properties of the Schwarzschild solution I.
  FN paragraaf 4.1-4
  Description of the radar-delay experiment (Shapiro) [see also this paper (pdf)] and the gravitational redshift (Pound/Rebka) as tests of the theory. Description of the geodesics in the Schwarzschild metric for test particles as well as photons, with the special case of free fall (radial) and circular orbits (in terms of resp. the proper and the coordinate time this remarkably gives exactly Newton's result). For radial free fall we use r=r₀(1+cos(eta))/2 to integrate FN.4.29 (it can also be used to explicitly integrate FN.4.30), giving a so-called cycloid (see also exercise 4.8.2, which uses r=r₀sin²(psi). Substituting psi=eta/2 this amounts, however, to the same result). The proper time for free fall is finite and from the point of view of the falling observer nothing special happens when passing the horizon at r=2m. However, an observer at a large distance, using coordinate time, sees the free fall to come to a "halt" ("freezing") on the horizon of a black hole. At the same time the signal extinguishes due to the gravitation redshift (with a characteristic time of 2m/c). Stable circular orbits are only possible up to r=3m (so up to half the Schwarzschild radius from the horizon, which is therefore the shortest distance to which an accretion disc can extend). At r=3m on such a circular orbit the geodesic becomes lightlike (the orbital velocity approaches the speed of light when r approaches 3m). Photons can exist in a circular orbit only for r=3m (since, in a sense, their orbital velocity is fixed), but this orbit is not stable under small perturbations.
  Problems: 4.2, 4.3, 4.7 and 4.8.

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• 20 Mar: Properties of the Schwarzschild solution II.
  FN par. 4.5-9
  Bending of light, first restricted to the curvature of the spatial part (see the handout). This gives the same contribution as due to the Newtonian equations of motion, with the photon treated as a classical particle (see the first part of the handout). Adding these two contributions together gives Einstein's result, as a test for his theory, but now derived with the exact relativistic equations of motion, FN par. 4.6. Planetary orbits (also as preparation for studying the binary pulsar) and the perihelion advance are discussed, as well as the geodesic effect -- precession of a giroscope. After many delays the launch of the Gravity Probe satellite took place on 20 April 2004. We will briefly discuss the results. Black holes: Eddington-Finkelstein and Kruskal coordinaten, which are better behaved in the neighborhood of the horizon.
  Alternative derivation for the perihelion advance by using (as with the bending of light) u=u₀+w, where u₀ is given by FN.4.42. Substituting this in FN.4.43 (where the expression for E should be modified to
  E=(GM/h²)² [e²-1-2(GM/hc)²(1+3e²)],
  which amounts to redefining e), or by first differentiating with respect to the angle, gives
  w=[3(GM)³/(c²h⁴)](1+e²/2+e phi sin(phi)-e²cos(2 phi)/6).
  To first order in the perturbation we thus find:
  u=GMh^-2[1+e cos(q phi)+3(GM/ch)²(1+e ²/2-e²cos(2 phi)/6)],
  where q=1-3(GM/ch)², such that the perihelion advance per completed orbit is given by -2pi(1-1/q)=6pi(GM/ch)². When we finally use u₁+u₂=2GM/h², the result in FN.4.45 is found (see the handout).
  For those interested: The perihelion advance can even be calculated exactly(!).
  Problems: 4.4, 4.5 + extra exercise 10.

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  Note the gap in the schedule -- use it wisely: to catch up with the exercises!

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• 8 May: Gravitational Radiation.
  FN chapter 5, upto Eq. FN.5.41
  A graviton (the analogue of what the photon is for electromagnetic radiation) has spin 2, but (like the photon) has only two polarisations. This is related on the one hand to the invariance under general coordinate transformations (as compared to the gauge invariance for electromagnetisme), and on the other hand to the fact that the graviton can not stand still (like the photon it moves with the speed of light). For detection one can use resonant detectors or laser interferometers. Here you can read about MiniGRAIL, the spherical detector for gravitational radiation in Leiden. For laser interferometers it is essential to note that in the the TT-gauge only the spatial components of the metric (transverse to the gravitational wave) change, which implies that there is no gravitational redshift that can undo the phase-difference due to the displacement of the mirrors in the arms of the interferometer. For details, see the recent review of Alberto Lobo (in particular chapter 4).
  Problems: 5.1, 5.2, 5.3 and 5.4.
  Chapter 5 of FN (for earlier printings) has some annoying typos:
  Pg. 167, 4th line below Eq. FN.5.29 and the 2nd line from the bottom the d'Alembertiaan is written as 2², which should of course be as in Eq. FN.5.17. The same correction holds for Eq. FN.A.50-51 and the footnote on pg.208.
  Pg. 168, an i for a circularly polarized light wave (7th line below Eq. FN.5.30) is missing.
  Pg. 169, Table 5.1, the horizontal component of course should indicate the x-direction in stead of the z-direction.

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• 15 May: The binary pulsar.
  FN rest of par. 5.3 and material handed out on 24 Feb.
  We use the paper by J. Weisberg e.a., Scientific American, October 1981 (study this paper in detail) to analyse the binary pulsar, where the extra exercises 11 and 12 will, step by step, show how to compute the energy loss of the binary pulsar due to gravitational radiation. This energy loss can be deduced from the change in the orbital elements. For this test of general relativity Hulse and Taylor in 1993 received the Nobel prize. [Note: Up to now only one of the neutron stars in such a binary system turned out to be a pulsar. In 2004 (pdf) a binary system was found (pdf) where both neutron stars are pulsars. With an orbital period of only 2.4h this gives a unique opportunity to test the theory even more accurately!] Extra exercise 11 is partly meant as preparation for exercise 12, and partly to derive FN.5.44, which gives the radiated power of a rotating dumbbell. Exercises 11 and 12 will discussed in class to prepare you for the homework; in particular for exercise 11 some of the details will be worked out.
  Extra exercises 11 and 12

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• 22 May: Cosmology.
  FN chapter 6
  The field equations will be used to determine the scale factor R(t) in the Robertson-Walker metric. The Friedmann model is found by neglecting the pressure p. From this follows the well-known Big Bang and the Hubble-expansion. The cosmological constant is an extra free parameter which can be added to the Einstein equations. This gives rise to the so-called De Sitter model (see extra exercise 13 -- for historical background see this pdf as well as the Studium Generale presentation (in Dutch, 3.5 Mb pdf) of Michel Janssen).
  Problems: 6.2, 6.3, 6.4, 6.5, 6.6 and 6.7 + extra exercise 13.

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• 29 May: Modern developments (last class)
  We discuss modern developments in a casual way, to wet your appetite. Rotating black holes (the Kerr solution): where you can reach another universe by moving through a ring, in which you can travel backwards in time, but cannot tell those you left behind. Thermodynamics of black holes: black holes cannot split and their horizon has an area that has to increase always (the entropy of a black hole). Hawking radiation: Classically nothing can escape from a black hole, but through quantum effects a black hole emits thermal radiation (for astrophysical black holes the temperature is far too small to be measurable). We also briefly discuss how gravity in 5 dimensions (the extra dimension being a small circle) can give standard gravitaty in 4 dimensions, electromagnetisme and a scalarair field (this is called Kaluza-Klein compactification).
  A nice book which covers many of these things in a lucid, but accurate way is ``Black Holes'', by J.-P. Luminet, Cambridge University Press, 1995 - translated from French, ``Les Trois Noirs'', Belfond, Paris, 1987.

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• Examination rules
  The course is offered on average every 2nd year. Each time there are two opportunities to take the exam. To get access to the exam you are required to hand in the last 5 homework exercises, 9-13 (PS (237kb), PDF (136kb)). They will be graded and count for one quarter of the final score. During the exam you will be allowed to consult the book by Foster and Nightingale, as well as your notes.

  Results: fall 1999, spring 2002 and spring 2004.