| week 1
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First lecture:
Special Relativity: Introduction / Einstein’s Postulates / Time dilation and Length contraction / Time Triangle / Inertial reference frames / Einstein's synchronization.
Second lecture:
Minkowsky diagrams: Light cone / Causality / Simultaneity / Change of frame.
Twin Experiment / Clock hypothesis / Equivalence principle
Exercise class: Time dilation and spacetime diagrams. worksheet.
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| week 2
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Third lecture:
Vector Space: Contravariant & covariant components and basis / metric / tensors.
Fourth lecture:
Change of coordinates: General case / Lorentz transform / Measuring length in relativity / Composition of velocities / Doppler effect.
Exercise class: Vector Space, Change of coordinates and Composition of velocities. worksheet.
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| week 3
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Fifth lecture:
Four-vectors: 4-velocity / 4-acceleration / Mass-energy equivalence / Energy and momentum / Inelastic collisions / Barycentric frame / Cosmic Rays.
Sixth lecture:
Relativistic Dynamics: Equations / Lorentz force / Relativistic motion in an electric and magnetic field / Particles trapped in a magnetosphere belt.
Exercise class: 4-vectors and energy-momentum conservation. worksheet.
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Tensor
Training
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Exercise class: Scalar and cross products, completely antisymmetric unit tensors, nabla operator, formulas, determinant, symmetric and antisymmetric tensors, electromagnetic tensor, electric and magnetic transform laws, 4-vector current, tensorial Maxwell equations, Lorentz invariants, covariant equation of motion, Lorentz covariant force. worksheet.
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| week 4
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Lecture 7:
Metrics: Euclidean / Euclid's Postulates / Metric on the sphere / Riemann sphere stereographic projection / Mercator projection / Topology.
Lecture 8:
Lagrangian formalism: Geodesics / Demonstration of the equation of motion / Equations in general relativity / Examples / Toy models.
Exercise class: Lambert cylindrical projection / Lagrangian and Maxwell's equations. worksheet.
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| week 5
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Lecture 9:
Non-inertial reference frames: Local basis and connections / Covariant derivative / Properties / Ricci Theorem / Examples.
Lecture 10:
Riemann curvature tensor: Parallel transport / properties / Ricci tensor and scalar / inertial effects versus gravitation / Flat and curve spaces and spacetimes.
Exercise class: Riemann tensor symmetries and antisymmetries / Bianchi's Identity / Calculations of Christoffel and Riemann components for different metrics. worksheet.
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| week 6
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Lecture 11:
Accelerating frame: Proper clocks / Doppler Effect / Coordinate clocks / Clock synchronization / Metric / Christoffel symbols / Riemann tensor / Coordinate velocity / Wordlines for light ans non-zero mass particles / Event horizon / Rigidity.
Lecture 12:
Rotating Disk: Porper clocks / Synchronization of coordinate clocks / Change of coordinates / Metric / Sagnac effect.
Exercise class: Ehrenfest Paradox / spatial metric / Space curvature. worksheet.
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Tensor
Training
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Exercise class: Metric determinant / Lengths, surfaces and volumes / Covariant Derivative Properties / Ricci theorem / Commutator of the covariant derivatives. worksheet.
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| week 7
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Lecture 13:
General Relativity: Equivalence Principle / Cartan's Theorem / Stress-energy tensor / Einstein's tensor / Ricci tensor and scalar / Einstein's Equation.
Lecture 14:
Schwarzschild Metric: Spherical bodies and Black Holes / Demonstration of the metric / Effet Einstein.
Exercise class: worksheet.
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| week 8
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Lecture 15:
Black Holes: Caracteristics / Event Horizon / Singularity / Worldlines / Free fall in a Black Hole / Kerr metric.
Lecture 16:
Effective Potential: Deflection of light / Perihelion Precession / Inward spiraling of matter / Accretion Disk.
Exercise class: worksheet.
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| week 9
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Lecture 17:
Gravitational Waves: Perturbation of the metric / Wave equation / Gauge invariance / Analogy with electromagnetic waves / Polarization States.
Lecture 18:
Emission of Gravitational Waves: Quadrupole Moment and radiation at large distance / Detection.
Exercise class: worksheet.
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| Examination
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Test validation for certification
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