2016-2107 Fall Semester

PHYS405: Theory of General Relativity

Syllabus of PHYS 405 - Theory of General Relativity

Course Coordinator: Metin Gurses

Semester: 2016-2107 Fall

Contact Hours: 4 hours of lecture per week

Textbook and Other Required Material:

Required - Textbook: General Relativity : An Introduction for Physicists, M. P. Hobson, G. Efstathiou and A.N. Lasenby, 2014/7th Printing, Cambridge Universty Press

Catalog Description:

The Spacetime of Special Relativity; Lorentz Transformations; Manifolds and Coordinates; Vector and Tensor Calculus on Manifolds; Electromagnetism; The Equivalence Principle and Spacetime Curvature; The Gravitational Field Equations; The Schwarzscild Geometry; Experimental Tests of General Relativity; Schwarzschild Black Holes; The Friedman-Robertson-Walker Geometry; Cosmological Models; Linearized General Relativity; Gravitational Waves.

Prerequisite(s): PHYS 212

Assessment Methods:

Midterm1: %20 First Midterm Exam 2016: pdf file (November 8)
Midterm2: %20 Second Midterm Exam 2016: pdf file (December 16)
Final: %30 Final Exam 2016: pdf file ((January 13, 9:00-11:00 SB-Z10)
Homework: %30

Minimum Requirements to Qualify for the Final Exam:
Midterms total must be > 50

Weekly Syllabus:

1. The Spacetime of Special Relativity: Reading Assignment
2. Lorentz transformations: Reading Assignment
3. Manifolds and Coordinates : We begin the course starting with the second Chapter of the text book. Namely we will study The concept of a manifold, Coordinates,Curves and surfaces,Coordinate transformations,Summation convention,Geometry of manifolds,Riemannian geometry,Intrinsic and extrinsic geometry
4. Vector Calculus on Manifolds: Scalar fileds on manifolds,Vector fileds on manifolds,Tangent vectors to a curve, Basis vectors, Raising and lowering vector indices,Basis vectors and coordinate transformations,Coordinate independent properties of the vectors,Derivatives of basis vectors and the affine connection,Transformation properties of the affine connection,Realtionship of the connection and metric, Local geodesics and Cartesian coordinates,covariant derivative of a vector, Vector operations on component form,Intrinsic derivative of avector along a curve,Parallel transport,Null curvesa and non-null curves and affine parameters,Geodesics,Stationary properties of non-null geodesics, Lagrangian procedure for geodesics.(Appendix 3A and 3B).
First homework (14 October): Chapter 2 of the book: Exercies 2.1, 2.2, 2.3, 2.4,2.12. Chapter 3 of the book: Exercies 3.1,3.2,3.3,3.4,3.6,3.12, 3.14
Solutions of the First Homework Problems ,
5. Tensor Calculus on Manifolds: Tensor fileds on manifolds,Components of tensors,symmetries of tensors, The metric tensor,Raising and lowering tensor indices,Mapping tensors into tensors,Elementary operations with tensors, Tensors as geomterical objects, Tensors and coordinate transformations, Tensor equations, The quotient theorem, Covariant derivative of a tensor,Intrinsic dersivative of a tensor along a curve.
First Exercise set
Second Homework (4 November): All exercises of Chapter 4 of the text book
Solutions of the Second Homework Problems ,
6. Special Thoery of Relativityy
Minkowski spacetime in Cartesian coordinates, Lorentz transformations, Cartesian basis vectors, Four-vectors and lightcone, Four vectors and Lorentz transformations, Four-velocity,Four-momentum of a massive particle, Four-momentum of a photon,Doppler effect and relativistic aberration, Relativistic mechanics, Free particles, Relativistic collisions and Compton scattering.
Third Homework Assignment (November 22): Solve the problems from 5.1 to 5.13 (13 problems) from Exercises of Chapter 5. Hw3 is due 22 November (Tuesday).
Solutions of the Third Homework Problems ,
Solutions of the First Midterm Exam Problems (2016) pdf file
7. The Equivalence Principle and Spacetime Curvature
Newtonian gravity, The equivalence principle, Gravity as spacetime curvature, Local inertial coordinates, Weak gravitational fields and the Newtonian limit, intrinsic curvature of a manifold, The curvature tensor, Properties of Curvature tensor, The Ricci tensor and curvature scalar, Curvature and parallel transport, Curvature and Geodesic deviation.
Fourth Homework set : (December 16) Chapter 7 Problems 7.1-7.10 and 7.13 (11 problems).
Second Exercise set
8. The Gravitational Field Equations
The energy momentum tensor,The energy momentum tesnor of a perfect fluid, Conservation of energy and momentum for a perfect fluid,The Einstein equations, The Einstein equations in empty space,The weak field limit of the Einstein equations, Geodesic motion from the Einstein equations (Reading assignment), Sign Conventions (Reading assignment).
9. The Schwarzschild Geometry
The general static isotropic metric, Solution of the empty-space field equations,Birkhoff's theorem, Gravitational redshift for a fixed emitter and receiver, Geodesics in the Schwarzschild geometry, Trajectories of massive particles,Radial motion of massive particles, Circular motion of massive particles,Stability of massive particle orbits, Trajectories of photons,Radial motion of photons, Circular motion of photons,Stability of photon orbits,
10. Experimental Tests of General Relativity
Precession of planetary orbits, The bending of light,Gravitational Lensing,Radar echoes.
Solutions of the Second Midterm Exam Problems (2016) pdf file
Fifth Homework set : (December 30) Chapter 10 Problems 10.1,10.2 10.3,10.4,10.5 (5 problems).
11.Schwarzschild Black Holes
Solutions of the Final Exam Problems (2016) pdf file
12.The Friedman- Robertson-Walker Geometry
13.Cosmological Models
14.Linearized General Relativity
15.Gravitational Waves

Last update November 2016

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