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Introduction. Special relativity. Relativistic kinematics.
(27/02/24)
Keywords:
Inertial frame of reference;
Lorentz transformation;
Spacetime;
Space-time interval;
Metric;
Four-vectors;
Special principle of relativity;
Least action principle and Lagrangian;
Canonical momentum and energy;
Relativistic action of a massive body;
Relativistic momentum and energy of a massive body.
Recommended exercises:
Coordinate transformations between inertial frames form a Lie group;
Galilean transformations as a Lie group, galilean law of velocity addition;
Lorentz transformations as a Lie group, relativistic law of velocity addition;
Relativistic action of a massive body, relativistic momentum and energy;
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Equivalence principle.
(27/02/24)
Inertial forces;
Equivalence of inertial and gravitational masses;
Equivalence principle;
Gravitation as geometry of spacetime;
Curved spacetime;
General principle of relativity;
Recommended exercises:
Pendulum with mg≠m;
Potential formulation of Newtonian gravitation;
Gravitation as geometry of space;
Newtonean limit of Nordström's theory;
Equivalence principle implies bending of light and gravitational redshift.
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Vectors in curvillinear coordinates.
Covariant differentials of tensors in metric spaces.
(27/02/24)
Covariant transformaion of physical quantities under a (general)
coordinate transformation; Principle of covariance;
Covariant objects: scalars, vectors, tensors;
Covariant differentials of tensors;
Christoffel symbols in metric spaces;
Recommended exercises:
All except for transformation formula for Christoffel symbols ;)
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Motion of free bodies in curved spaces. Geodesics.
(27/02/24)
General covariance; Geodesics;
Null geodesics;
Gravitational potential.
Recommended exercises:
All but "geodesics on a sphere".
[pdf]
Electrodynamics in curved spaces.
(27/02/24)
Principle of stationary action; Action of charges in electromagentic
fields; Covariant volume element; Action of the electromagentic field;
Lorentz force equation in gravitational fields; Maxwell equations in
gravitational fields.
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The source of gravity: the stress-energy-momentum tensor of matter.
(27/02/24)
Action of matter in general relativity;
Stress-energy-momentum tensor of matter as the source of gravitation;
Stress-energy-momentum conservation law;
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Action of gravitation. Gravitational field equation (Einstein equation).
(27/02/24)
Riemann tensor;
Ricci tensor;
Ricci scalar;
Einstein equation;
Hilbert-Einstein action;
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Newtonian limit of general relativity.
Gravitational waves.
(27/02/24)
Newtonian limit; Linearized gravity; Gravitational waves in vacuum.
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Schwarzschild solution. Motion in the Schwarzschild metric.
(27/02/24)
Schwarzschild coordinates; Schwarzschild solution; Schwarzschild radius.
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Classical tests of GTR
(27/02/24)
Anomalous perihelion shift; Bending of light; Gravitational red-shift.
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Radial fall in the Schwarzschild metric.
(27/02/24)
Lemaitre coordinates for Schwarzschild solution;
Coordinate singularity;
Gravitational singularity;
Event horizons; Black holes.
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Geometry in a homogeneous and isotropic universe. Friedmann equation.
(27/02/24)
Friedmann universe;
Friedmann-(Lemaitre-Robertson-Walker) metric;
Friedmann equations.
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Solutions to Friedmann equation.
Big Bang. Cosmological red-shift. Hubble-Lemaitre constant.
(27/02/24)
Big Bang; Cosmological red-shift; Hubble (-Lemaitre) law;
Cosmological constant;
Radiation-, matter-, and dark-energy-dominated universes.
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[pdf]
Proper time and length intervals in general relativity.
(27/02/24)