| CODE | MAT3515 | ||||||||
| TITLE | General Relativity and Cosmology | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | Not Applicable | ||||||||
| ECTS CREDITS | 6 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | Gravity is the fundamental force that effects everything from dynamics on Earth to the evolution of the Universe. Newton's theory of gravitation explains to a very good extent the gravitational dynamics on Earth and within the solar system however it has several insurmountable inconsistencies. Moreover, it does not agree with observation when scrutinized to sufficient accuracy. For this reason, Einstein formed his theory of general relativity which is the current standard picture of gravity. The study-unit will introduce general relativity as a natural extension of Einstein's special theory of relativity. The study-unit will close with an investigation into the effects this theory has had on standard cosmology and all the new physics that has emerged as a result in recent decades. Study-unit Aims: This study-unit aims to provide a/an: - Review of Newtonian gravity and its internal inconsistencies as well as its failure to explain a number of key observations; - Introduction to flat spacetime (Minkowski metric, inertial and non-inertial observers, proper time and length, Lorentz transformations); - Introduction to Manifolds (definition, summation notation, Riemann geometry); - Introduction to tensor analysis (definition, contraction, symmetry operations, the metric tensor and its properties); - Introduction to vector calculus (covariant derivative, parallel transport); - Explanation of geodesics and geodesic deviation; - In-depth analysis of curvature (The Riemann curvature tensor, Ricci tensor, Ricci scalar, Kretschmann scalar, Bianchi Identities); - Explanation of the equivalence principle; - Derivation and analysis of Einstein's field equations (introduction to the stress-energy tensor); - Analysis of electromagnetism in tensor mechanics (Maxwell's equations, equations of motion, curved spacetime); - In-depth analysis of the Schwarzschild solution (derivation, Birkhoff's theorem, geodesics, orbits, stability); - Explanation of other solutions (black holes, Reissner-Nordstroem metric, Kerr metric); - Analysis of the classical tests of general relativity (precession of orbits, light deflection, redshift); - Introduction to the FLRW metric (derivation, introduction to cosmology); - Analysis of geodesics in the FLRW universe; - Introduction to cosmography (Hubble parameter, deceleration parameter, look-back time); - Analysis of the cosmological constant; - Introduction to the Friedmann equations; - In-depth analysis of cosmological models (flat, open, closed, dust, radiation, matter); - Analysis of density parameter evolution; - Introduction to inflation (definition, phase transition, inflaton field); Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - State the flat spacetime metric; - Identify distinction between inertial and non-inertial observers; - Explain the notion of a manifold and Riemann geometry; - Explain tensor analysis including contraction, symmetry operations; - State the definition and properties of the metric tensor; - Explain vector analysis including the covariant derivative and parallel transport; - State how to analyse geodesics and geodesic deviation; - Identify measures of curvature and general identities in Riemann geometry; - Explain the equivalence principle and simple applications; - State the Einstein field equations and some stress-energy tensors; - Explain the role of electromagnetism in Einstein's theory of gravity; - Explain how mechanical calculations in the Schwarzschild geometry can be performed; - Identify other vacuum solutions in general relativity; - Identify black hole solutions; - Explain the classical tests of general relativity; - State the FLRW metric and its role in modern cosmology; - Etate the basic cosmographic parameters; - Identify the role of the cosmological constant in modern cosmology; - State the standard cosmological models; - State what the state parameters are using different stress-energy tensors and using the Friedmann equations; - Explain density parameter evolution; - Explain inflation and the role of the inflaton. 2. Skills: By the end of the study-unit the student will be able to: - Show how manifolds differ from one another; - Perform calculations from the point of view of both inertial and non-inertial observers using manifolds; - Calculate tensor quantities using their basic properties; - Perform the covariant derivative and parallel transport; - Determine geodesics and their deviation; - Use tensor quantities to measure curvature on a manifold; - Perform simple derivations using the equivalence principle; - Perform calculations using Einstein's field equations; - Use the Maxwell tensor to perform electromagnetic calculations in curved spacetimes; - Perform analysis on the Schwarzschild solution including orbital analysis; - Perform the classical tests of general relativity; - Use the FLRW metric to calculate geodesics; - Determine cosmographic quantities; - Perform calculations in flat/open/closed cosmologies; - Perform state parameter calculations for different stress-energy tensors; - Use the cosmological constant in cosmological calculations; - Perform calculations using the Friedmann equations; - Show the evolution of density parameters; - Calculate simple outputs of inflation. Main Text/s and any supplementary readings: Main textbook: - Carroll, S., `Spacetime and Geometry: An Introduction to General Relativity', First edition, Pearson. Supplementary textbooks: - Schutz, B., `A first course in general relativity', any edition, Cambridge University Press. - Hobson, M. P., Efstathiou, G. P. and Lasenby, A. N., `General Relativity: An Introduction for Physicists', First edition, Cambridge University Press. |
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| ADDITIONAL NOTES | Pre-requisite Qualifications: A good understanding of special relativity Pre-requisite Study-unit: PHY2195 |
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| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Jackson Said Joseph Sultana (Co-ord.) |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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