Fall 2024
Shadi Tahvildar-Zadeh
Subtitle:
Einstein's Relativistic Theory of Gravitation
Course Description:
This is an introductory course on Einstein's theory of general relativity and gravitation, emphasizing the geometric-PDEs point of view of the equations. Although welcomed and very much appreciated, no previous knowledge of physics is assumed. Some knowledge of differential geometry and PDEs is helpful. The following is an outline of the course:
I. The Geometry of Space-time:
Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit, The Penrose singularity theorem. Black holes and cosmic censorship conjectures. Homogeneous and isotropic solutions.
II. The Cauchy problem for Einstein Vacuum Equations (EVE):
The symbol and the characteristics of EVE. The local existence theorem in wave coordinates.
III. Conservation Laws and Noether's Theorem:
Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps, and for sections of vector bundles. Asymptotic flatness. The definition of global energy, momentum and angular momentum. The Positive Energy Theorem.
IV. Reduction under Symmetry:
One- and Two-Killing field reductions of Einstein's Equations. Kerr and Kerr-Newman solutions. Cosmological solutions and asymptotically flat solutions. Spherically symmetric solutions of vacuum Einstein and Einstein-Maxwell equations. Complete Integrability and the Vesture Method for generating new solutions (if time permits.)
Text:
none
Prerequisites:
permission of instructor
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Fall 2023
Shadi Tahvildar-Zadeh
Subtitle:
Introduction to General Relativity
Course Description:
Same as above Fall 2024
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Fall 2022
Shadi Tahvildar-Zadeh
Subtitle:
Introduction to Mathematical Relativity
Course Description:
This is an introductory course on Einstein's theory of general relativity and gravitation, emphasizing the geometric-PDEs point of view of the equations and the recent advances in the field. Although welcomed and very much appreciated, no previous knowledge of physics is assumed. Some knowledge of differential geometry and PDEs is helpful. The following is an outline of the course:
I. The Geometry of Space-time:
Causal structure, curvature and gravitation, the energy tensor and the matter equations of motion, the Einstein equations: variational formulation, derivation of the constraints and the evolution equations, maximal hypersurfaces and the Newtonian limit, The Penrose singularity theorem. Black holes and cosmic censorship conjectures. Homogeneous and isotropic solutions.
II. The Cauchy Problem for Einstein Vacuum Equations (EVE):
The symbol and the characteristics of EVE. The local existence theorem in wave coordinates. Local existence using maximal hypersurfaces.
III. Conservation Laws and Noether's Theorem:
Lagrangian and Hamiltonian formulations. The Noether current in the theory of maps, and for sections of vector bundles. Asymptotic flatness. The definition of global energy, momentum and angular momentum. The Positive Energy Theorem.
IV. Reduction under Symmetry:
Kerr and Kerr-Newman solutions. Cosmological solutions and asymptotically flat solutions. Spherically symmetric solutions of vacuum Einstein and Einstein-Maxwell equations.
Text:
none
Prerequisites:
517 and 532, or permission of instructor
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Fall 2021-Cancelled 8/20/21
Sheldon Goldstein
Subtitle:
Relational Mechanics, Classical and Quantum: Physics and Dynamics on Shape Space
Course Description:
Relational mechanics is based on two ideas:
(i) Physical (3-dimensional) space should be regarded relationally, so that configurations that differ by a uniform overall translation, rotation, or rescaling of distances should not be regarded as physically different.
(ii) The appropriate notion of time is that of qualitative time and not quantitive time, so that insofar as the time evolution of a configuration is concerned, what is physically meaningful is the corresponding path in configuration space but not the speed with which the configuration moves along the path.
Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space, the space that results when the symmetries referred to above are taken into account and the quotient of the usual configuration space for a system of particles is taken with respect to the group of those symmetries. For example, for a 3-particle system the shape space is the space of all triangles, with similar triangles regarded as equivalent. In this course we shall analyze the metric structure of shape space and describe how that structure can be used to straightforwardly define both a classical and a quantum dynamics on shape space, i.e., both a classical and a quantum relational mechanics.
We will see how these motions gives rise to the more or less familiar physical laws and theories formulated in terms of absolute space and time. We will see how free motion on shape space, when lifted to configuration space, becomes an interacting theory. And since many different lifts are possible--- corresponding to different choices of "gauges"---we will see that much of what is regarded as fundamental in physics corresponds merely to a choice of gauge and is thus a reflection of somewhat arbitrary choices that we make in forming physical theories.
Possible topics:
Emergence of Absolute Space and Time
Emergence of Interaction
Emergence of Probability
Geometrodynamics and the Emergence of Spacetime
Relativity Without Relativity
Text:
none
Prerequisites:
none
Schedule of Sections:
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Previous Semesters:
FALL 2019:
Sheldon Goldstein
Subtitle:
Introduction to Mathematical Relativity I
Course Description:
Relational mechanics is based on two ideas:
(i) Physical (3-dimensional) space should be regarded relationally, so that configurations that differ by a uniform overall translation, rotation, or rescaling of distances should not be regarded as physically different.
(ii) The appropriate notion of time is that of qualitative time and not quantitive time, so that insofar as the time evolution of a configuration is concerned, what is physically meaningful is the corresponding path in configuration space but not the speed with which the configuration moves along the path.
Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space, the space that results when the symmetries referred to above are taken into account and the quotient of the usual configuration space for a system of particles is taken with respect to the group of those symmetries. For example, for a 3-particle system the shape space is the space of all triangles, with similar triangles regarded as equivalent. In this course we shall analyze the metric structure of shape space and describe how that structure can be used to straightforwardly define both a classical and a quantum dynamics on shape space, i.e., both a classical and a quantum relational mechanics.
We will see how these motions gives rise to the more or less familiar physical laws and theories formulated in terms of absolute space and time. We will see how free motion on shape space, when lifted to configuration space, becomes an interacting theory. And since many different lifts are possible--- corresponding to different choices of "gauges"---we will see that much of what is regarded as fundamental in physics corresponds merely to a choice of gauge and is thus a reflection of somewhat arbitrary choices that we make in forming physical theories.
Possible topics:
Emergence of Absolute Space and Time
Emergence of Interaction
Emergence of Probability
Geometrodynamics and the Emergence of Spacetime
Relativity Without Relativity
Text:
None
Prereq:
None
