#### Certificate Course on A Tensorial Way to Relativity & Cosmology

__Certificate Course on A Tensorial Way to Relativity & Cosmology __

**Course Type:** Online Certificate Course (It’s free, no Course Fee required)

**Certificate Issuing Authority:** Banwarilal Bhalotia College, Asansol, West Bengal, India

**Course Organizer: **Department of Mathematics, Banwarilal Bhalotia College, Asansol Under the Aegis of IQAC, Banwarilal Bhalotia College, Asansol

**Target Audience:** UG/PG Level Mathematics and Physics Students

**Pre-Requisites:** Basic Knowledge on Geometry, Differential Calculus & VectorsCourse Opening Date: 17th October, 2021 (Tentative)Course Registration: Register on/before 10th October, 2021 through the following linkhttps://forms.gle/dfQo3E9HtNfsxAEw5Course

**Duration:** 12 Weeks (Approx. 3 Classes per Week, Each of 1 Hour Duration)Course Mode: Virtual/Online through Google Meet Platform

**Class Timings:** All the Classes will be Scheduled in the Evening between 7 PM – 9 PM (IST)

**Course Exams:** Regular Weekly Assignments & Final Exam at the end of the Course

**Course Instructor: **Dr. Sudipta Das, Assistant Professor in Department of Mathematics Banwarilal Bhalotia College, Asansol - 713303, West Bengal, India

** Email id: **sudipta_jumaths@yahoo.co.in

**Course Outline & Outcome:**This course is specifically designed for advanced UG and PG level students in Mathematics and Physics. Wewill start with the basic idea of tensors as a generalization of vectors. We will then gradually move on tolearn the details of algebraic operations on the tensors along with the transformation laws. Then we willproceed to the core topic of tensor calculus and study the ideas like metric tensors, curvature and derivativesof tensors with adequate examples. Finally, we will delve deeper into the course where the applications oftensor calculus in different fields of Physics, especially in Special & General Theory of Relativity andCosmology will be discussed in details with some elaborate demonstrations. This course will enable thestudents to learn about the modern-day research topics such as Relativity and Cosmology and to acquire thenecessary tools which will surely help them in their future fields of research. Certificates will be issued at theend of the Course to the successful participants.

**Course Syllabus/ModulesWeek 1 & 2 (Module – 1): **Basics of Tensors & Tensor Algebra (5-6 Classes)Vectors, Euclidean & non-Euclidean Geometry, notations about indices & summation convention, symmetric& anti-symmetric systems, tensors as generalization of vectors, transformation properties of tensors:contravariant, covariant & mixed tensors, algebraic operations on tensors: contraction, inner product &outer product of tensors, idea of metric tensor and its applications. Assignments at the end of each week.

**Week 3, 4 & 5 (Module – 2):**

Introduction to Tensors Calculus (8 Classes)Differentiation of tensors: covariant derivative, definition and properties of affine connection & Christoffelsymbols, differential operators & their properties: gradient, curl, divergence & Laplacian, Lagrangian &geodesic equations, definition & properties of Riemann curvature tensor, Ricci tensor & Ricci scalar.Assignments at the end of each week.

**Week 6, 7 & 8 (Module – 3):** Tensorial Description of Special Theory of Relativity (9-10 Classes)4-dimensional Minkowski Spacetime & four-vectors, Principles of Relativity, interval between two events,Lorentz transformations: time dilation & length contraction, velocity & acceleration four-vectors, energymomentum four-vector, angular momentum tensor, relativistic dynamics: covariant Newton’s law, covariantlaws of electromagnetism. Assignments at the end of each week.

**Week 9, 10, 11 & 12 (Module – 4): **Applications to General Relativity & Cosmology (12 Classes)Action integral, electromagnetic field equations, action for matter fields, Einstein tensor, Introduction toTheory of General Relativity, Einstein field equations, Black Holes: Schwarzschild solution, introduction toCosmology: Robertson-Walker metric, Friedmann equations, solutions to Friedmann equations and modelsof our Universe, distance measurements, cosmological redshift, Hubble law and expanding universe, Basicidea of Cosmic Microwave Background Radiation (CMBR). Assignments at the end of each week.

**Course Materials/Reference Books:Basic Texts:**1. Tensors, Relativity and Cosmology by Mirjana Dalarsson & Nils Dalarsson, Academic Press.2. Introducing Einstein’s Relativity by Ray D’Inverno, Oxford University Press.3. Spacetime and Geometry by Sean M. Carroll, Addison Wesley.For Further Reading:4. Lecture Notes on General Relativity by Matthias Blau.http://www.blau.itp.unibe.ch/GRLecturenotes.html5. Relativity, Gravitation and Cosmology by Robert J. A. Lambourne, Cambridge Publishers.

**About Course Instructor:**Dr. Sudipta Das has joined at Banwarilal Bhalotia College, Asansol on 25th February, 2015 as an AssistantProfessor in Mathematics and working here till date. He has completed his Ph.D. from Indian StatisticalInstitute, Kolkata on 7th January, 2015. His specialization is in Mathematical Physics and his research interestlies in the fields of Relativity and Cosmology, especially on the topic of Dark Energy in Cosmology. Presentlyhe is offering numerous UG Courses for B.Sc. Mathematics Hons. students on Differential Calculus &Differential Equations, Calculus of Several Variables, Particle Dynamics, Dynamics of Rigid Bodies, Probability& Statistics, Special Theory of relativity; and one PG Course for M.Sc. Physics students on Special Theory ofRelativity. An avid reader and traveller, whenever not busy with teaching jobs or research works, Dr. Das canoften be found to be travelling in the Himalayas.

02-Jan-21