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Physics
617 General Relativity
(Spring 2022)
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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:
sergei@physics.rutgers.edu (preferred)
Office hours: Friday
10:00 am -12:00 pm
Extra meetings can be held by appointment.
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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:
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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.
http://www.physics.rutgers.edu/ugrad/disabilities.html |
Download the course
info in PDF format
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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.
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Download
prerequisites
PLAN OF LECTURES
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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
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PRELIMINARIES
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ELEMENTS OF DIFFERENTIAL GEOMETRY
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- 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]
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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]
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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]
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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]
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SPACE - TIME IN GENERAL RELATIVITY
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- (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]
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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]
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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''.
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GRAVITATIONAL FIELD EQUATIONS
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- Einstein's equation:
Energy-Momentum tensor. Einstein's equation. Coordinate conditions. Harmonic coordinates.
Suggested literature:
Lecture notes
Secs.4.1, 4.2 in
[1]
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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]
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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]
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EXACT SOLUTIONS OF EINSTEIN'S EQUATION
Homeworks and
Solutions
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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. |
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Assigned
on
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Assignment
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Due
Date
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Solutions
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1. |
Jan. 20,
2022 |
pdf
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Jan. 31,
2022 |
pdf
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2. |
Jan. 20,
2022
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pdf
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Feb. 3,
2022 |
pdf
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3. |
Jan. 20,
2022 |
pdf
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Feb. 10,
2022 |
pdf
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4. |
Jan. 20,
2022 |
pdf
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Feb. 17, 2022 |
pdf
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5. |
Jan. 20,
2022 |
pdf
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Feb. 28, 2022 |
pdf
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6. |
Jan. 20,
2022 |
pdf
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Mar. 10, 2022 |
pdf
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Spring Recess:
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March 12-20, 2022
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7. |
Feb. 10, 2022
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pdf
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Mar. 28, 2022
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pdf
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8. |
Feb. 24, 2022
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pdf
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Apr. 4, 2022
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pdf
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9. |
Mar. 30, 2022
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pdf
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Apr. 11, 2022
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pdf
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10. |
Apr. 6, 2022
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pdf
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Apr. 18, 2022
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pdf
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11. |
Apr. 11, 2022
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pdf
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Apr. 25, 2022
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pdf
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12. |
Apr. 25, 2022
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pdf
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May 2, 2022
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pdf
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Grades:
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May 5, 2022
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