Physics 530 Spring 2019

General Relativity

Notes:

Text:

• Wald-- General Relativity--Probably the best book from the mathematical point of view. He gets things right.
• Misner, Thorne, Wheeler-- Gravitation. Huge and discursive. It covers a lot of material
• Carroll-- Lecture Notes on Relativity -- http://preposterousuniverse.com/grnotes/ Or also in book form Spacetime and Geometry: An Introduction to General Relativity

Lecture cancellation Jan8 and 10

Final Exam

Notes

• Vectors, Abstract indicees.
• 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 Note that the "parallalogram" procedure in these notes is slightly different from the one I used in class. In these notes the parallelogram is used to displace the curve through the point at gamma(lambda) defining the vector at lambda, to the a curve through gamma(0) instead of the other way around as in the lecture. This makes no difference to the result.
• Lectures on Lie derivatives.
• 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.
• Linearization of the curvature tensor (altered Feb 27/2019)
• Perihelion advance and Einstei's luck (New versin Mar 12/2019
• Kennefink's 2009 analysis of the Eddington/Dyson experiment
• Schwartzschild solution including Eddington Finkelstein coordinates, and Kruskal
• Orbits in Schwartzschild including perihelion advance, light deflection, Shapiro time delay, Nordtvedt effect. (Some Misprints corrected and small additions Mar2/19).
• Gravitational waves solutions Including full non-linear plane wave solution.
• 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
• ADM formalism for equations
• Penrose conformal transformation for Vaidya collapse metric.
• Measurements on the double pulsar
• 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.
• "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.
• 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."

Assignments

• Assignment 1 --problem 5 added and problem 1 fixed typo. 5PM Jan3) Due Jan 15
• Assignment 2 Due Jan31
• Assignment 3 Due Feb 28 Note that the derivations of various tests are in the notes titled Orbits in Schwartschild above.
  Hint. Problem 2 and 4 Recall the Flamm paraboloid- See https://en.wikipedia.org/wiki/Schwarzschild_metric.
  Hint for problem 5 in the new problem set.
  3PM Feb 27-- problem 3 clarified.
  Having had so many requests for extention, I will now make the deadline Mar 5
• Assignment 4 Due Mar 14.(Mar 13-- Question 5 clarified as to which "acceleration" components is meant).

Solutions

• Solution 1
• Solution 2
• Solution 3
• Solution 4