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COURSE DATA Data Subject Código Name Relativity and cosmology Cycle Grade ECTS Credits 4.5 Curso académico Study (s) Degree Center Acad. Period year Grado en Física FACULTY OF PHYSICS

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COURSE DATA Data Subject Código Name Relativity and cosmology Cycle Grade ECTS Credits 4.5 Curso académico Study (s) Degree Center Acad. Period year Grado en Física FACULTY OF PHYSICS 4 Second term Subject-matter Degree Subject-matter Character Grado en Física 16 - Complements of Physics Optional Coordination Name PORTILLA MOLL, MIGUEL Department 16 - ASTRONOMÍA Y ASTROFÍSICA SUMMARY The subject Relativity and Cosmology, quarterly optional fourth year of the Degree in Physics, has been assigned 4.5 credits (30 hours of lectures and 15 hours of work sessions). 1 The course is an introduction to the theory of space-time in the presence of gravitation, namely Einstein's theory of gravitation, also known as the theory of general relativity (GR). The basic language of this theory is the Riemannian geometry (curved space), so this subject will also be an introduction to basic notions of curved space. RG studies can be continued in the master's program in advanced physics. The RG is in force in a very wide range of spatial scales: - Engineering related to GPS (dol) or Galileo (EU) needs to take into account relativistic corrections as are needed to explain the advancement of the perihelion of Mercury. - The evolution of massive objects, the stellar collapse, and the formation of black holes and energy processes for them produced are usual topics of relativistic astrophysics where the RG is necessary. - Gravitational lensing is an unavoidable consequence of RG that has found an important application in the detection of dark matter in remote areas. - The rapid expansion of the Universe detected in the observation of distant supernovae have revealed a mysterious energy component, unknown at the moment (dark energy) that is stimulating many studies and speculations. - In the other direction, the study of quantum gravity scale has happened to other important theoretical topic. - Finally, note that as the LHC may open perspectives in particle physics, gravitational wave detectors already built, they can also be done to the physics of gravitation. PREVIOUS KNOWLEDGE Relationship to other subjects of the same degree There are no specified enrollment restrictions with other subjects of the curriculum. Other requirements It is recommended to course RIC after having completed the basic subjects of Physics and Mathematics . 2 OUTCOMES Grado en Física - Knowledge and understanding of the fundamentals of physics in theoretical and experimental aspects, and the mathematical background needed for its formulation. - Saber aplicar los conocimientos adquiridos a la actividad profesional, saber resolver problemas y elaborar y defender argumentos, apoyándose en dichos conocimientos. - Ability to collect and interpret relevant data in order to make judgements. - Capacity to communicate information, ideas, problems and solutions to a specialist and a general audience. - Developing learning skills so as to undertake further studies with a high degree of autonomy. - Problem solving: be able to evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems. - Modelling & Problem solving skills: be able to identify the essentials of a process / situation and to set up a working model of the same; be able to perform the required approximations so as to reduce a problem to an approachable one. Critical thinking to construct physical models. - Basic & applied Research: acquire an understanding of the nature and ways of physics research and of how physics research is applicable to many fields other than physics, e.g. engineering; be able to design experimental and/or theoretical procedures for: (i) solving current problems in academic or industrial research; (ii) improving the existing results. - Foreign Language skills: Have improved command of English (or other foreign languages of interest) through: use of the basic literature, written and oral communication (scientific and technical English), participation in courses, study abroad via exchange programmes, and recognition of credits at foreign universities or research centres. - Literature Search: be able to search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development. - Learning ability: be able to enter new fields through independent study, in physics and science and technology in general. - Communication Skills (written and oral): Being able to communicate information, ideas, problems and solutions through argumentation and reasoning which are characteristic of the scientific activity, using basic concepts and tools of physics. - Cultura General en Física: Haberse familiarizado con los aspectos más importantes de la materia, y con enfoques que abarcan y relacionan diferentes áreas de la física. LEARNING OUTCOMES We expect to achieve a fundamental level in the use of Einstein's theory of gravitation. To this end, will acquire a basic level in the highlights of the geometry of curved spaces, and their relationship to the physics of gravitation: 3 Tensor-calculus: algebraic and differential. -The geometrical properties and physical implications of three basic metric spaces: Schwarzschild, Friedmann-Lemaitre-Robertson-Walker, and linearized gravitational waves, which will lead to: the phenomenon of black hole, cosmological models and the generation and analysis of radiation gravitational respectively. -The temporal and null geodesics and the problem of motion of test particles and light rays. Families of geodesic deviation equation and the corresponding physical meaning. -At the end of the skill must provide a geometric vision, which will result in the use of diagrams temporary spaces in the discussion of physical problems. Others characteristic of the degree: -Develop critical thinking skills and application of scientific method. -Be able to identify problems, including the similarities with other solution which is known, and devise strategies for their solution. -Extend the ability to plan and organize own learning, based on individual work, from the literature and other sources. -Evaluate the different causes of a phenomenon and its relative importance. -Identify the essential elements of a complex situation, make the approximations needed to construct simplified models that describe and to understand their behavior and in other situations. -Be able to perform an update of existing information on a specific problem, sort and analyze it. -Building capacity to work together in addressing the complex problems that require collaboration with others. -Promote the acquisition of resources for speaking and writing to complete a clear and coherent scientific argument. -Encourage communication skills of the physical concepts involved in a problem by way of speaking and writing. -Promote the understanding and use of new information technologies. DESCRIPTION OF CONTENTS 1. Special Relativity. Affine structure of spacetime. Inertial basis. Light cones, time and space lines. A model of geometric watch. Quadrivector momentum. Energy and 3-momentum respect an inertial observer. Minkowski tensors and volume element. 2. The principle of equivalence. The Einstein equivalence principle. Geometric interpretation of the gravitational redshift. The chronometric hypothesis. 4 3. Curved spacetime. Lorentzian 4-dimensional variety. Tangent space. Tensorials fields. Metric tensor. Coordinate basis. Orthonormal bases and locally inertial systems. Spatial hypersurfaces. Null hypersurfaces. 4. Geodesics. Geodesics as extremal curves. The geodesic equation. Null geodesics, temporary geodesics and the equivalence principle. Symmetries and first integrals of geodesics. Killing s vectors. 5. Metrical Connection. Covariant derivative of vector fields. Covariant derivative of tensorials fields. Metrical Connection. 6. Curvature and geodesics. The Riemann curvature tensor. Equation of geodesic deviation (EDG). EDG in a locally inertial system. Connection with Newtonian tidal forces. 7. The Einstein equations. The energy-momentum tensor. Local conservation of energy-momentum tensor. Einstein's equations. 8. The linearized Einstein equations. Perturbations of Minkowski in harmonic coordinates. The Newtonian limit. 9. The Schwarzschild metric. Geometry of the Schwarzschild solution. Temporary and null geodesics. Finkelstein-Lemaitre Extension. Black hole. 10. Gravitational radiation. Gravitational waves in vacuum. Polarization waves. Foundation of detectors. Emission of radiation. 11. Cosmological models. Space-times homogeneous and isotropic. Spaces with null, positive and negative curvature. Cosmological redshift. The horizon. Models with initial singularity. Evolution equations. Matter and dark energy. 5 WORKLOAD ACTIVITY Hours % To be attended Theory classes Readings supplementary material Preparing lectures Preparation of practical classes and problem TOTAL TEACHING METHODOLOGY The course will consist of two distinct types of classes: a) The mathematical development will be performed gradually, accompanied by discussion of the physical problem of gravity. We will pay little attention to the solution of the Einstein equations for spacetime metric, compared to the understanding of the consequences that flow from them. It will be studied: metrics with spherical symmetry, the maximum symmetry, and Minkowski perturbacions. There will be abundant examples, discussion of paradoxes, and experiments imaginaries to highlight the influence of gravity in space-time relations. It seems that the Hartle's book will be appropriate. b) Workshops (1 h per week). The RIC-hour seminar will be in charge to students. Individually or in small groups will present the solution to issues and problems identified during the development of the theory. He will be requested to previously discuss the problems with the teacher of the subject. EVALUATION The assesment system is as follows: 1) Written examinations: One part will assess the understanding of the theoretical-conceptual and formal nature of the subject, both through theoretical questions, conceptual questions and numerical or simple particular cases. Another part will assess the applicability of the formalism, by solving problems and critical capacity regarding the results. Proper argumentations and adequate justifications will be important in both cases. 2) Continuous assessment: assessment of exercices and problems presented by students, questions proposed and discussed in class, oral presentation of problems solved or any other method that involves an interaction with students. COMMENTS: The weight of each item is established in accordance to the agreements of the CAT. 6 REFERENCES Basic - Gravity. An introduction to Einsteins general relativity . J.B. Hartle, Addison Wesley Relativity and Cosmology . Wolfgang Rindler, Oxford General Relativity. Woodhouse N.M.J. Springer 2007 Additional - Relativity. An introduction to special and general relativity. Hans Stephany, Cambridge. U.P Spacetime and geometry. Sean M. Carroll. Addison Wesley General relativity an introduction to physicists Hobson, M.P., Efstathiou G.P., Lasenby, A.N., Cambridge A first course in general relativity Bernard Schutz General relativity . Bob. Wald La relativité générale. Une approche geometrique Malcolm Ludvigsen Dunod, Gravity from the ground up Bernard Schutz. Cambridge New perspectives in astrophysical cosmology Martin Rees. Cambridge

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