Physics 617   General Relativity    (Spring 2022) 

Course info   | Prerequisites  | Plan of lectures  | Homeworks and Solutions   | Useful Links   | E-mail


Room: ARC-203
Time:   Monday,        8:30-9:50 am
             Thursday,      8:30-9:50 am
In-person classes will temporarily convert to remote classes until further notice.

Instructor: Sergei Lukyanov
         office: Serin E364
         office phone: (848) 445-9060 
         e-mail: (preferred)

Office hours: Friday 10:00 am -12:00 pm
                      Extra meetings can be held by appointment.

The main reference text: Sean Carrol, ``Spacetime and Geometry''.
Online reference material can be found at
Sean Carrol MIT lectures on General Relativity

Additional text: D. Lovelock and H. Rund, ``Tensors, Differential Forms, and Variational Principles'', Dover Publications, Inc, New York

Homework: There will one homework per 1-2 weeks.
                      Late homework will not be accepted. .
                      The absolute cutoff time for homework is 7 pm due date.
                      Ideally, solutions should be typed (in LaTeX), but
                      handwritten solutions are acceptable as long as they are clearly written. I'll not accept sloppy solutions. .
                      Homeworks will be graded and give 100% of final grade.

No exams will be given.

Students with Disabilities:

If you have a disability, it is essential that you speak to the course supervisor early in the semester to make the necessary arrangements to support a successful learning experience. Also, you must arrange for the course supervisor to receive a Letter of Accommodation from the Office of Disability Services.

Download the course info in PDF format



I will assume that you are familiar with

(I)  Graduate Classical Mechanics at the level Physics 507 or Rutgers challenge exam program:

  •   Basic: Lagrangian mechanics, invariance under point transformations, generalized coordinates and momenta, curved configuration space, phase space, dynamical systems, orbits in phase space, phase space flows, fixed points, stable and unstable, canonical transformations, Poisson brackets, differential forms, Liouville's theorem, the natural symplectic 2-form and generating functions, Hamilton-Jacobi theory, integrable systems, adiabatic invariants.

  •  Continuum mechanics: Taut string and lattice of point masses. 1-D wave equation. boundary conditions, 3-D wave equation, Laplacian, plane waves, spherical waves, volume and surface forces, stress and strain, elastic moduli (bulk, shear, Young) stress tensor. Strain tensor. longitudinal and transverse waves in solid. Fluids. "material derivative", inviscid fluid, Bernoulli, eq of continuity. Waves.

  •  Field theory: Lagrangian density, Hamilton's principle for fields, cyclic coordinates, Noether's theorem. Lagrangian formulation of electromagnetism.

    (II)  Graduate E&M at the level Physics 503 or Rutgers challenge exam program:

  •   Basic: Gauss law, differential and integral form Poisson and Laplace equations, Green's theorem, Dirichlet and Neumann boundary conditions, boundary value problems with cylindrical and spherical symmetry, Laplace equation in cylindrical and spherical coordinates, magnetostatics, vector and scalar potentials, Maxwell's equations, plane electromagnetic waves, linear and circular polarization.
  •          Download prerequisites


           This is a tentative schedule of what we will cover in the course. It is a subject to change, often without notice.
           These will occur in response to the speed with which we cover material, individual class interests, and possible changes
           in the topics covered. Use this plan to read ahead from the textbooks,

           [1] Sean Carrol, ``Spacetime and Geometry'' (2003, Pearson ISBN 978-0805387322).

           [2] D. Lovelock and H. Rund, ``Tensors, Differential Forms, and Variational Principles'', Dover Publications, Inc, New York

           [3] H.Goldstein, C. Poole and J. Safko, ``Classical Mechanics'', Third edition, Addison Wesley





    •  Space-Time in Classical Physics:  Euclidean structure. Causal structure. Inertial frames. Gravity vs Inertia. Equivalence Principle.
      Suggested literature: Lecture notes     
                                        Secs.1.1-1.3, 2.1-2.3 in [2]
                                        Secs.4.1-4.8 in [3]    (kinematics of rigid body motion)
    • Space-Time in Special Relativity:  Causal structure in SR. Poincare group. Pseudo-Euclidean (Mikowskian) space. Vectors and tensors.
      Suggested literature: Lecture notes     
                                        Secs.1.1-1.7 in [1]
                                        Secs.7.1-7.5 in [3]  (classical mechanics of special theory of relativity)
    • Basic facts from CM and EM (self study):   Maxwell's equations. Energy and momentum. Classical field theory.

      Suggested literature: Secs.1.8-1.10 in [1]
                                        Secs.13.1-13.7 in [3]   (Lagrangian formulations for continuous systems and fields)
    • Curvilinear coordinates:  Equivalence Principle and Special Relativity.Transformations of coordinates. Vectors and tensor fields. Parallel vector fields. Covariant (absolute) differential. Covariant derivative. Christoffel symbols. Ellipsoidal coordinates in R^3.

      Suggested literature: Lecture notes     
                                        Secs.2.4-2.6 in [2]




    •   Manifolds:   Topological space. Definition of the manifold. Examples: S^n, O(N), SO(N), RP^n, T^n, ...  .

      Suggested literature: Lecture notes     
                                        Secs.2.1, 2.2 in [1]
                                        Sec.3.1 in [2]
    • Tensor fields on manifolds:  Tangent vector and tangent space. T_P M as space of derivations. Tangent vector fields and tangent bundle. Cotangent space and cotangent fields. Tensor fields. Tensor densities. Levi-Civita symbols.

      Suggested literature: Lecture notes     
                                        Secs.2.3, 2.4, 2.8 in [1]
                                        Secs.3.2, 3.3, 4.1, 4.2 in [2]
    • Affinely connected manifolds:   Absolute differential and covariant derivative. Affine connection. Torsion. Parallel transport. Geodesics. Parallel transport along closed curves. Curvature (Riemann) tensor.

      Suggested literature: Lecture notes     
                                        Secs.3.3-3.7 in [2]
    • Basic facts on differential forms (self study):   Differential form. Exterior derivative. Closed and exact forms. Wedge product. Integration. Stokes's theorem. Hodge star operator.

      Suggested literature: Secs.2.9, 2.10, Appendix E in [1]
                                        Secs.5.1-5.3, 5.5 in [2]

                         SPACE - TIME IN GENERAL RELATIVITY

    •  (pseudo-)Riemannian manifolds:   Metric. Physical coordinates. Geodesics in a (pseudo-)Riemannian manifold. Locally geodesic coordinates.

      Suggested literature: Lecture notes     
                                        Secs.2.5, 3.1-3.4 in [1]
                                        Secs.7.1, 7.2 in [2]
    • Levi-Civita connection:  Particle in the gravitational field (Free motion. Newtonian limit). Absence of torsion in General Relativity. Connection vs metric.

      Suggested literature: Lecture notes     
                                        Secs.3.1-3.3 in [1]
    • Curvature tensor in (pseudo)Riemannian space:   Curvature vs metric. Flatness condition. Properties of the Riemann tensor: Symmetries, number of independent components, Bianchi identity. Free fall and Fermi normal coordinates (self study). Geodesic deviation equation.

      Suggested literature: Lecture notes     
                                        Secs.3.6, 3.7, 3.10 in [1]
                                        Secs.7.3 in [2]
                                      For Fermi normal coordinates, see
                                      F.K. Manasse and C.W. Misner, ``Fermi normal coordinates and some basic concepts in                                         differential geometry",Journal of Mathematical Physics 4, no. 6, p. 735 (1963).

                                      For physical meaning of the geodesic deviation equation, see sec.1.6 in
                                      C.W. Misner, K.S. Thorne, J.A. Wheeler, ``Gravitation''.

                         GRAVITATIONAL FIELD EQUATIONS

    •  Einstein's equation:   Energy-Momentum tensor. Einstein's equation. Coordinate conditions. Harmonic coordinates.

      Suggested literature: Lecture notes     
                                        Secs.4.1, 4.2 in [1]
    • Weak gravitational field:  Linear approximation. Non-relativistic matter. Propagation of light in a weak gravitational field. Hamilton-Jacobi method in Classical Mechanics. Frequency shift in weak gravitational field. Deflection of light ray in the gravitational field of Sun. Gravitational lenses.

      Suggested literature: Lecture notes     
                                        Secs.10.1-10.5 in [3]    (Hamilton-Jacobi theory)
                                        Secs.7.1-7.3 in [1]
    • Variational principle:   Lagrangian formulation. Derivation of Einstein's equation from variational principle.

      Suggested literature: Lecture notes     
                                        Secs.1.10, 4.3-4.5 in [1]
                                        Secs.8.1-8.5 in [2]


    •  Schwarzschild solution:   Static and stationary solutions. Rotationally invariant metrics. Schwarzschild metric.

      Suggested literature: Lecture notes     
                                        Secs.5.1,5.3 in [1]
    •  Rotational symmetry and relativistic stellar structure:   Gravitating spherical body. Tolman-Oppenheimer-Volkoff equation. Incompressible matter. Gravitational stability condition for a static star. Buchdahl's theorem.

      Suggested literature: Lecture notes     
                                        Sec.5.8 in [1]
    • Geodesics of Schwarzschild:   Kepler problem. Fall to the center.

      Suggested literature: Lecture notes     
                                        Secs.5.4, 5.5 in [1]
                                        Secs.3.7,3.8 in [3]    (Kepler in Classical Mechanics problem)
    • The maximally extended Schwarzschild solution:   Rindler space-time. Kruskal coordinates.

      Suggested literature: Lecture notes     
                                        Secs.5.6,5.7 in [1]
                                        Sec.7.3 in [2]
    • Isometries:   Killing vectors. Lie bracket. Birkhoff's theorem.

      Suggested literature: Lecture notes     
                                        Secs.3.8, 5.2, Appendix B in [1]
                                        Sec.4.4 in [2]

                         GRAVITATIONAL RADIATION

    •  Gravitational waves:   Gravitational wave solutions. Production of gravitational waves.
      Suggested literature: Lecture notes     
                                        Secs.7.4,7.5 in [1]
    •   Energy-momentum of gravitational field:   Gravitational energy-momentum pseudo-tensor. Gravitational energy-momentum in linear approximation.
      Suggested literature: Lecture notes     
                                        Secs.7.4,7.5 in [1]
    • Energy loss due to gravitational radiation:   Energy loss. Quadrupole radiation. Gravitational radiation of a double star.
      Suggested literature: Lecture notes     
                                        Secs.7.6,7.7 in [1]

                         UNIFORM ROTATION

    •  Rotating star:   Weak field approximation for uniform rotation. Orbit precession due to rotation of central body. Kerr metric.
      Suggested literature: Lecture notes     
                                        Secs.6.1, 6.6 in [1]
                                        Secs.3.7-3.9 in [3]   (Kepler in Classical Mechanics problem,
                                        Laplace-Runge-Lenz vector)
                                        Lense and H. Thirring, Phys. Z.,19, p.156 (1918)


    •   Geometry of isotropic spaces:   Isotropic and homogenous manifolds. Spatially isotropic Space-Time.
      Suggested literature: Lecture notes     
                                        Sec.8.1 in [1]
    • Cosmological models:   Robertson-Walker metric. Friedman equations. Role of the cosmological constant. Hubble's law. Big Bang. FRW cosmology. Vacuum-dominated universe: de Sitter model.

      Suggested literature: Lecture notes     
                                        Secs.8.2-8.4 in [1]
    • Isotropic models and observations (self study):  Cosmological redshift and distances. Our universe. Inflation.

      Suggested literature: Secs.8.5-8.8 in [1]

    Download syllabus


    Homeworks and Solutions  

    The assignments and solutions are stored in PDF format. The absolute cutoff time for homework is 7pm due date. I'll not accept sloppy solutions.

    Assigned on
    Due Date
    1. Jan. 20, 2022 pdf   Jan. 31, 2022 pdf  
    2. Jan. 20, 2022 pdf   Feb. 3, 2022 pdf  
    3. Jan. 20, 2022 pdf   Feb. 10, 2022 pdf  
    4. Jan. 20, 2022 pdf   Feb. 17, 2022 pdf  
    5. Jan. 20, 2022 pdf   Feb. 28, 2022 pdf  
    6. Jan. 20, 2022 pdf   Mar. 10, 2022 pdf  

    Spring Recess:

    March 12-20, 2022
    7. Feb. 10, 2022 pdf   Mar. 28, 2022 pdf  
    8. Feb. 24, 2022 pdf   Apr. 4, 2022 pdf  
    9. Mar. 30, 2022 pdf   Apr. 11, 2022 pdf  
    10. Apr. 6, 2022 pdf   Apr. 18, 2022 pdf  
    11. Apr. 11, 2022 pdf   Apr. 25, 2022 pdf  
    12. Apr. 25, 2022 pdf   May 2, 2022 pdf  


    May 5, 2022

    Useful Links  

    1. "Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables",
    M. Abramowitz and I. Stegun
    2. Sean Carrol MIT lectures on General Relativity
    3 F.K. Manasse and C.W. Misner,
    ``Fermi normal coordinates and some basic concepts in differential geometry", Journal of Mathematical Physics 4, no. 6, p. 735 (1963).

    Course info  |  Prerequisites  |   Plan of lectures  |   Homeworks and Solutions  |   Useful Links  |  E-mail