Physics 530 Spring 2026

General Relativity

Blurb:

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Hennings 301 Tues and Thurs. 2:00 -3:20 Jam 6-Apr 8

Lectures

Notes

• Nordstrom's theory of gravity
• See Norton for a detailed discussion of the Hole argument (by John Norton, philosopher who is well versed in the mathematics of General Relativity).
• Vectors, Abstract indicees. (ed Feb 1 2023)
• Some examples of vectors and geodesics (ed Jan 16, 2023)
• Tensors (ed Jan 17 2023)
• Christophel Symbols Derivation of the covariant derivative by following the more usual procedure of making certain demands on the definition of derivative. This makes it seem far more arbitrary and mysterious than it is.
• Derivation of Parallel transport from metric (ed Jan 23 2023)
• Lectures on Lie derivatives.(ed Jan 25 2023)
• Curvature These notes are very similar to what Wald does in his book Ch. 3. The definition of the index order in the curvature tensor is different but due to the symmeteries of the curvature, my tensor is equal to his. (ed Feb 2 2023. Removed section on Riemann normal coordinates as for in indefinite metric the proper treatment is tricky, and since I already show by a different route that ${\Gamma^i}_{jk}=0$ at a point in an appropriate coordinate system, I do not need the Riemann normal coordinates. )
• Einstein's abortive attempts to find his equations
• Einstein's Hole argument against coordinate covariant theory.
• Energy Momentum tensors of Electromagnetism, Scalar field, and Prefect irrotational fluid. (Some edits Mar 5 2023)
  • S. Matarrese "On the Classical and Quantum Irrotational Motions of a Relativistic Perfect Fluid. I. Classical Theory" Proc Roy Soc A401, 53-66 (1985)
• A list of some equations especially coordinate equations
• Linearization of the curvature tensor (altered Feb 4/2026)
• Kennefink's 2009 analysis of the Eddington/Dyson experiment
• Flamm metric Here is also the original paper translated into English
• Schwartzschild solution including Eddington Finkelstein coordinates, and Kruskal
• Paper on the history of the various coordinates for Schwarzschild solution, especially the ones that resolve the horizon "problem".
• Orbits in Schwartzschild including perihelion advance, light deflection, Shapiro time delay, Nordtvedt effect. (Some Misprints corrected and small additions Feb 2021).
• Measurements on the double pulsar
• 16 years later See especially p20, 24, 30
• Lunar Laser ranging experiments and the weight of gravity.

  Gravitational Waves

  • Gravitational waves solutions Including full non-linear plane wave solution. [Changed Fri Mar 20 2026]
  • Period evolution of Hulse Taylor Pulsar system showing loss of energy due to grav. radiation.
  • Einsten-Rosen paper on non-linear gravitational waves in Journal Franklin Inst. 223, Pages 43-54 (1937). A previous version of this paper had been sent to Phys Rev, where the referee HP Robertson had suggested that the paper was wrong in that it argued that gravitational radiation did not exist because of the singularity of the solution. Given the first section they probably found the Plane wave solutions where the spatial components of the metric go to zero. They did not recognize that these were purely coordinate singularities. (That was only done by Bondi, Pirani and Robinson after the war. See Dan Kennefink in Physics Today Physics Today 58, 9, 43 (2005)
    Einstein was so upset that the journal had sent his paper to be refereed that he never again published in Phys Rev.
  • Bondi, Pirani, Robinson paper on plane gravitational waves.
  • Talk I gave on Ligo Grav. Wave detection
    PDF
    Openoffice ODT
    

    Livingston Gravitational Wave The Livingston gravity wave detector being hit by a gravitational wave with amplitude of about .05 (about 10,000,000,000,000,000,000 times stronger than the one that hit in Sept 2015, the first earth detection of a graviational wave)

  • 
    J McIver's talk on Grav waves Mar 24 2026
  Penrose confomal diagrams.
  • Derivations of Penrose diagrams by Yacine Ali-Ha¨ımou https://cosmo.nyu.edu/yacine/teaching/GR_2018/lectures/lecture24.pdf
  • Penrose conformal transformation of the eternal BH spacetime
  • Penrose diagram of charged black hole with outer and inner horizons.
  • Penrose conformal transformation for Vaidya collapse metric.
  • ADM formalism for equations

    conservation of Energy Momentum tensor and geodesic equations

    • Papapetrou's paper on deriving geodesic equations from conservation of Energy Momentum tensor. Note that the script T tensor is the tensor density sqrt(|g|) times the energy momentum tensor.
    • Geroch and Jang's paper making the previous one a bit more rigorous.
    • Dixon extended Papapetrou's treatment to higher moments.

    Cosmology

    • Intro t Cosmology
    • "Swiss cheese" model for cosmolgy Model in which a spherical void in the dust cosmology is replaced by a Schwartzschild solution with continuity across the boundary. This could also be done for pressure cosmologies, but a shell of pressure (negative surface tension) would be needed to hold back the pressure of the cosmology.
    • Quantum cosmology

    Black Hole evaporation

    • PhysRevD.14.870 Notes on Black Hole evaporation-- "original" paper with acceleration radiation and expansion on black hole evaporation.
    • Fulling ,Ruijsenaars Temperature, periodicity and horizons Proof of periodicity in imaginary time.
    • Bogoliubov -- [Corrections Mar 25]paper describing the generic definition of creation and annihilation operators independently of the Hamiltonian. The special case where they are related to the Hamiltonian (Hamiltonian diagonalization) are also described.
    • Field Theory The application of the previous work to field theory, and its application to Cosmological particle creation of a scalar field, and the problem with "Hamiltonian diagonalization" as a reasonable definition of "particles."
    • Accelerated Detector and excitation in vacuum
    • Toy model of black hole evaporation
    • Toy black hole and entropy flux

    Assignments

    • Assignment 1 Due Jan 22
    • Assignment 2 Due Feb 7
    • Assignment 3 Due Feb 19 (Problem 1 replaced Feb13
    • Assignment 4 Due Mar 25
    • Assignment 5 Due Apr7

    Solutions

    • Assignment 1 Solutions (A variety of misprints fixed- Feb 23-2026
    • Assignment 2 Solutions. (a misprint fixed Feb 23-2026
    • Assignment 3 Solutions.
    • Assignment 4 Soln
    • Assignment 5 Soln