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Lecturer(s)
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Richterek Lukáš, Mgr. Ph.D.
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Course content
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Brief Description The aim of this lecture is to provide a general overview of the basic ideas and principles of the general theory of relativity (GTR), which is one of the fundamental components of 20th-century physics. The lecture will cover the most essential basics of tensor calculus on manifolds, and the conclusion will focus on the astrophysical and, above all, cosmological implications of GRT (the evolution of stars and their final stages, selected aspects of the theory of black holes, gravitational waves, and cosmological models). Topics 1. Geometry and physics; the role of gravity in the physical picture of the world; the historical development of ideas about space, time, and gravity. 2. The equivalence principle, Mach's principle, the covariance principle, and the correspondence principle; gravity as a manifestation of the curvature of spacetime. 3. Coordinates, metric, light cones, worldlines, local inertial frames, and basic geometric concepts in curved spacetime. 4. Geodesics, integrals of motion, and conservation laws in curved spacetime. 5. Schwarzschild geometry and its physical implications: gravitational redshift, perihelion precession, light bending, and signal delay. Experimental tests of general relativity in the Solar System, the PPN formalism, and measurements of the parameters and . 6. Relativistic phenomena in astrophysics: gravitational lensing, accretion disks, binary pulsars, gravitational collapse, and black holes. 7. Rotating gravitational sources, dragging of inertial frames, Kerr geometry, and the ergosphere. 8. Linearized gravitational waves, their polarization, energy, and detection principles. 9. Fundamentals of the tensor description of gravity, covariant derivatives, the curvature tensor, Einstein's equations, and their Newtonian limit. The Riemann tensor and its physical significance, the metric, the Ricci, Einstein, and Weyl tensors, and the Bianchi identities. 10. Relativistic stars, hydrostatic equilibrium, stability, and critical masses of compact objects. Current observational and experimental aspects of general relativity.
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Learning activities and teaching methods
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Lecture, Work with Text (with Book, Textbook), Activating (Simulations, Games, Dramatization)
- Attendace
- 13 hours per semester
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Learning outcomes
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This course aims to provide an introductory, physics-based overview of the fundamental principles of general relativity as a modern theory of gravity. The course is based on historical and physical motivations, the principle of equivalence, the principle of covariance, and the geometric description of gravity as the curvature of spacetime. Attention is given to the basic mathematical tools necessary for understanding the theory, particularly the concepts of metric, geodesic, tensor, and covariant derivative. The course also introduces students to Schwarzschild and Kerr geometries, experimental tests of general relativity, gravitational waves, black holes, and selected relativistic phenomena in astrophysics. The course is designed as a general overview. It cannot replace a full-fledged course in differential geometry or advanced general relativity.
A course focused on acquiring knowledge. Upon completion of the course, students will be able to: - explain the basic principles of general relativity, in particular the principle of equivalence, the principle of covariance, and the geometric interpretation of gravity; - describe the significance of metrics, geodesics, light cones, and local inertial frames in the physical description of curved spacetime; - interpret the fundamental consequences of Schwarzschild geometry, particularly gravitational redshift, light bending, perihelion precession, and signal time delay; - be familiar with the fundamental experimental tests of general relativity and be able to explain their physical significance; - characterizes selected astrophysical applications of general relativity, particularly black holes, gravitational lenses, binary pulsars, and gravitational waves; - explains the fundamental significance of Einstein's equations and their relationship to the Newtonian theory of gravity; - solves simple model problems in general relativity or prepares a physically correct, annotated analysis of them; - works with specialized literature and can briefly present a selected topic in general relativity within a broader physical context.
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Prerequisites
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Recommended prerequisites include knowledge of special relativity, classical mechanics, the fundamentals of mathematical analysis, linear algebra, and vector calculus. A basic understanding of differential equations and electrodynamics is advantageous. The course is suitable for master's students or students in their final year of a bachelor's program.
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Assessment methods and criteria
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Student performance
Assessment of learning outcomes takes the form of a colloquium. The requirements for passing are: - active participation in at least 50% of class sessions; - presentation of a short academic paper on a selected topic in general relativity, its experimental tests, or astrophysical applications; - submission of annotated solutions to selected model problems; - an oral discussion/exam on the completed problems. The goal of the oral exam is to verify that the student understands the fundamental principles of general relativity, can explain the significance of the mathematical tools used, and can interpret the results in the appropriate physical context.
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Recommended literature
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d'Inverno R., Vickers J. (2022). Introducing Einstein's Relativity: A Deeper Understanding. Oxford.
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d'Inverno Ray. (1992). Introducing Einstein's Relativity. Oxford.
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Guidry M. (2019). Modern General Relativity: Black Holes, Gravitational Waves, and Cosmology.
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Hartle, J.B. (2003). Gravity: An introduction to Einstein's general relativity. San Francisco.
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Horský J., Novotný J., Štefaník M. (2001). Mechanika ve fyzice. Praha.
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Lambourne R. (2010). Relativity, Gravitation and Cosmology. Cambridge.
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Lightman A. P., Press W.H., Price R.H., Teukolsky S.A. (1975). Problem Book on Relativity and Gravitation.
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Moore T. (2013). A General Relativity Workbook.
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Rindler, W. (2006). Relativity: special, general and cosmological. New York.
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Schutz B. F. (2022). A First Course in General Relativity. Cambridge.
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Schutz B.F. (1985). A First Course in General Relativity.
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Ullman V. (1986). Gravitace, černé díry a fyzika prostoročasu. Ostrava.
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Ullmann V. (1986). Gravitace, černé díry a fyzika prostoročasu. Ostrava.
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Weinberg S. (1972). Gravitation and Cosmology. New York.
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