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Syllabus ( MATH 595 )


   Basic information
Course title: Tensor Analysis and General Relativity
Course code: MATH 595
Lecturer: Prof. Dr. Oğul ESEN
ECTS credits: 7.5
GTU credits: 3 (3+0+0)
Year, Semester: 2016-2017 2017-2018, Fall and Spring
Level of course: Second Cycle (Master's)
Type of course: Area Elective
Language of instruction: English
Mode of delivery: Face to face
Pre- and co-requisites: none
Professional practice: No
Purpose of the course: The target of this lecture is to give the basic knowledge of Tensors to the graduate students who wish to make research on the topics of Applied Mathematics, Theoretical Physics and Differential ;Geometry. In the meantime we would like to introduce the topic of General Relativity to the graduate students .
   Learning outcomes Up

Upon successful completion of this course, students will be able to:

  1. The mathematical structure of Newton equation

    Contribution to Program Outcomes

    1. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems

    Method of assessment

    1. Written exam
    2. Homework assignment
  2. The determination of particle orbits via Newton equation.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Work effectively in multi-disciplinary research teams
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.
    5. Effectively express his/her research ideas and findings both orally and in writing
    6. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  3. Lagrange function and the application of Euler-Lagrange equations in Classical Mechanics.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Develop mathematical, communicative, problem-solving, brainstorming skills.
    5. Effectively express his/her research ideas and findings both orally and in writing
    6. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  4. The Determination of the orbits of planets under the influence of Gravitational force.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Effectively express his/her research ideas and findings both orally and in writing
    7. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  5. The learning of the basic concepts of Special and General Relativity theory.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Effectively express his/her research ideas and findings both orally and in writing
    7. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  6. Learning the Invariance concept and its combination with Tensors.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Effectively express his/her research ideas and findings both orally and in writing
    7. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  7. Observing the natural fitting of Electromagnetic Theory with Relativity Theory.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Effectively express his/her research ideas and findings both orally and in writing
    9. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  8. The awareness of the short-coming of Special Relativity in the analysis of Graviatation.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Effectively express his/her research ideas and findings both orally and in writing

    Method of assessment

    1. Written exam
    2. Homework assignment
  9. The generalization of Special Relativity to General Relativity.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    6. Effectively express his/her research ideas and findings both orally and in writing

    Method of assessment

    1. Written exam
    2. Homework assignment
  10. The basic calculational methods of Tensor Analysis

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.

    Method of assessment

    1. Written exam
    2. Homework assignment
  11. Learning the application of Tensors in Riemannian Geometry.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Effectively express his/her research ideas and findings both orally and in writing
    9. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  12. The learning of the application of Riemannian Geometry in General Relativity.

    Contribution to Program Outcomes

    1. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    2. Acquire scientific knowledge and work independently
    3. Work effectively in multi-disciplinary research teams
    4. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    5. Develop mathematical, communicative, problem-solving, brainstorming skills.
    6. Effectively express his/her research ideas and findings both orally and in writing
    7. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  13. Learning the analysis and solution techniques of Einstein Field Equations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Effectively express his/her research ideas and findings both orally and in writing
    9. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  14. Learning the frequently coincided analytıcal solutions of Einstein Field Equations and their physical interpretations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Acquire scientific knowledge and work independently
    4. Work effectively in multi-disciplinary research teams
    5. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    6. Develop mathematical, communicative, problem-solving, brainstorming skills.
    7. Effectively express his/her research ideas and findings both orally and in writing
    8. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
  15. The learning of Metric Tensors which are solutions of Einstein Field Equations and their phsical applications via Lagrange Equations.

    Contribution to Program Outcomes

    1. Define and manipulate advanced concepts of Mathematics
    2. Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
    3. Review the literature critically pertaining to his/her research projects, and connect the earlier literature to his/her own results
    4. Acquire scientific knowledge and work independently
    5. Work effectively in multi-disciplinary research teams
    6. Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
    7. Develop mathematical, communicative, problem-solving, brainstorming skills.
    8. Effectively express his/her research ideas and findings both orally and in writing
    9. Demonstrating professional and ethical responsibility.

    Method of assessment

    1. Written exam
    2. Homework assignment
   Contents Up
Week 1: Introduction to Classical Mechanics
Week 2: Lagrange Function, Euler-Lagrange Equations and their application in Classical Mechanics
Week 3: Introduction to Special Relativity
Week 4: Lorentz Transformations and the Concept of Invariance
Week 5: Special Relativity and Electromagnetic Theory
Week 6: The Shortcomings of Special Relativity and need for a General Theory
Week 7: Invariance and Tensors
Week 8: Basic Calculations with Tensors
Week 9: Midterm
Week 10: The use of Tensors in Riemannian Geometry
Week 11: Riemann Curvature, Ricci and Scalar Ricci Tensors
Week 12: Einstein Field Equations
Week 13: The Analytical solutions of Einstein Field Equations
Week 14: The applications of the Analytical solutions of Einstein Field Equations to phsical problems via Lagrange function ve Euler-Lagrange equations
Week 15*: The Fitting of General Relativity with experimental and observational physical data and introduction to General Relativistic Cosmology and Black Holes
Week 16*: Final Exam
Textbooks and materials: • Ray D’Inverno, Introducing Einstein’s Relativity, Oxford University Press, 1992
• David C. Kay, Tensor Calculus, Schaum’s Outline Series, 1988
• S. Chandrasekkhar, The Mathematical Theory of Black Holes, Oxford University Press,1992


Recommended readings: J. Foster J. D. Nightingale, A Short Course in General Relativity, Second edition, Springer-Verlag, 1995
• Articles and Lecture Notes from Arxiv.org preprint server
  * Between 15th and 16th weeks is there a free week for students to prepare for final exam.
Assessment Up
Method of assessment Week number Weight (%)
Mid-terms: 9 30
Other in-term studies: 0
Project: 0
Homework: 5, 7, 12 30
Quiz: 0
Final exam: 16 40
  Total weight:
(%)
   Workload Up
Activity Duration (Hours per week) Total number of weeks Total hours in term
Courses (Face-to-face teaching): 3 14
Own studies outside class: 3 14
Practice, Recitation: 0 0
Homework: 15 3
Term project: 0 0
Term project presentation: 0 0
Quiz: 0 0
Own study for mid-term exam: 22.5 1
Mid-term: 3 1
Personal studies for final exam: 30 1
Final exam: 3 1
    Total workload:
    Total ECTS credits:
*
  * ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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