• Timetable:
    • Lectures: Friday — 08:00 AM – 11:30 PM — Room: TBA
    • Tutorial: Monday — 06:30 PM – 07:30 PM — Room: Google Meet
  • TA: Muhammad Zeeshan Amir
  • Course Outline: The outline can be downloaded from here.
  • Another Course: A crash course on Special Relativity can also be found here. I gave these lectures online in during COVID-19 lockdown and they contain some extra material as well which I will go through in this course. However, I will go into details of some relevant topics in tutorials.
  • Note about Figures: In first two lectures, you will find “PH” is written in purple color almost all figures. Those figures are taken from Dr. Pervez Hoodbhoy’s video lectures “Teach Yourself Special Relativity“. It is highly recommended to watch those lectures for a deeper understanding of special relativity.
  • Errors and Omissions: In case of any errors and omissions in lectures notes, please email me at bilalazam31@gmail.com.
  • Another Note: Solutions of in-class quizzes and homeworks are not made public. If interested, instructors can ask for solutions by sending an email from their institutional email address.
  • Lecture 1: Space and Time (Lecture Notes)
    • Spacetime diagrams
    • Inertial frames of reference
    • Galilean transformations between two inertial frames
    • Galilean transformations to sound
    • Galilean transformations to light
    • Michelson-Morley experiment
  • Lecture 2: Einstein’s Postulates (Lecture Notes)
  • Lecture 3: The Geometry of Special Relativity-I (Lecture Notes)
    • Definition of an inertial observer in SR
    • New units
    • Spacetime diagrams
    • Construction of the coordinates used by another observer
  • Lecture 4: The Geometry of Special Relativity-II (Lecture Notes)
  • Lecture 5: Spacetime Interval (Lecture Notes)
    • Invariance of the Interval
  • Lecture 6: Calibration of Coordinates (Lecture Notes)
  • Lecture 7: Vectors in Relativity-I (Lecture Notes)
    • Definition of a Vector
    • Transformation of Components of a Vector
    • Transformation of Basis Vectors
    • Inverse Transformation
  • Lecture 8: Vectors in Relativity-II (Lecture Notes)
    • Four-Velocity
    • Momentarily Comoving Reference Frame (MCRF)
    • Four-Momentum
    • In-Class Quiz 4
    • Homework 4
    • Tutorial 4: Problems from vectors and metric tensor (PDF)
  • Lecture 9: Four Velocity and Four Acceleration (Lecture Notes)
    • Four-Velocity
    • Four-Acceleration
  • Lecture 10: Four-Momentum and Discussion on Photons (Lecture Notes)
  • Lecture 11: One-Forms (Lecture Notes)
    • Recap of Metric Tensor
    • Definition of Tensors
    • One-Forms
    • How do the components of one-form transform?
    • Contraction is Lorentz invariant.
    • Basis 1-Forms
    • How to picture a One-Form?
  • Lecture 12: Gradient One-Form (Lecture Notes)
  • Lecture 13: (0 2) Tensors (Lecture Notes)
    • Tensor Product (Outer Product)
    • (0 2) Tensors
    • How the components of (0 2) tensors transform?
    • What are the basis elements of (0 2) tensors?
    • Symmetric (0 2) Tensor
  • Lecture 14: Metric Tensor (Lecture Notes)
    • Decomposition of Tensor into Symmetric and Anti-Symmetric Parts
    • Metric
    • Distance in 2d Euclidean Space with Cartesian Coordinates
    • Distance in 2d Euclidean Space with Polar Coordinates
    • Length of a Curve (a quarter circle) in 2d Euclidean Space with Cartesian Coordinates
    • Length of a Curve (a quarter circle) in 2d Euclidean Space with Polar Coordinates
    • In-Class Quiz 7
    • Homework 7
  • Lecture 15: Mixed Tensors (Lecture Notes)
    • Lowering and Raising the Indices
    • (0 m) tensors
    • Number of Components of Tensors
    • (1 0) tensors
    • (2 0) tensors
    • (n 0) tensors
    • (1 1) tensors
    • (1 2) tensors
    • (2 1) tensors
    • (n m) tensors
  • Lecture 16: Inverse Metric and Accelerated Observers (Lecture Notes)
    • Inverse Metric
    • Accelerated Observers
    • Uniform Acceleration
    • In-Class Quiz 8
  • Midterm
  • Lecture 17: Accelerated Observers and Communication (Lecture Notes)
    • Homework 8
    • Accelerating Observers
    • Causally-Disconnected Regions
    • Communication between Different Observers
  • Lecture 18: Coordinates of an Accelerated Observer (Lecture Notes)
  • Lecture 19: Einstein’s Equivalence Principle (Lecture Notes)
    • Weak Equivalence Principle
    • Einstein’s Equivalence Principle
    • Meaning of “Local”
  • Lecture 20: Manifolds-I (Lecture Notes)
    • Map
    • Composition
    • One-to-One/Injective Map
    • Onto/Surjective Map
    • Invertible/Bijective Map
    • Continuity
    • Diffeomorphism
    • Open Ball
    • Open Set
    • Chart/Coordinate System
    • Atlas
  • Lecture 21: Manifolds-II 
    • More about Manifolds
    • Stereographic Projection
  • Lecture 22: Manifolds-III
    • Vectors
    • Directional Derivatives
    • Tangent Spaces and Tangent Bundles
    • One-Forms
    • Cotangent Spaces and Cotangent Bundles
    • Tensors
    • Vector Fields
    • One-Form Fields
    • Tensor Fields
    • Homework 10
  • Lecture 23: Metric on Manifold
    • Metric and the Causal Structure of Spacetime
    • Computation of Path Lengths and Proper Time from Metric
    • Metric tells about Geodesics
    • Metric is the GR analog of Newtonian Gravitational Field
    • Metric and the notion of Curvature
    • Local Inertial Frames
  • Lecture 24: Connections and Parallel Transport (Lecture Notes)
    • Transformations of Partial Derivatives under Arbitrary Coordinate Transformation
    • Connections/Covariant Derivatives
    • Levi-Civita Connections
    • Christoffel Symbols
    • Brief Introduction of Riemannian Geometry and Affine Geometry
    • Torsion Tensor
    • Parallel Transport
  • Lecture 25: Parallel Transport and Geodesics (Lecture Notes)
    • Directional Covariant Derivative
    • Parallel Transport
    • Geodesic Equation
    • Affine Parameters
    • “Spacetime tells matter how to move […].” (John Wheeler)
  • Lecture 26: Curvature (Lecture Notes)
    • Notion of Curvature
    • Reimann Curvature Tensor
    • Symmetries of Reimann Curvature Tensor
  • Lecture 27: Einstein Field Equations (Lecture Notes)
    • A Very Brief Summary of Lectures 1 to 26
    • Minimal Coupling
    • Generalization of Integrals on Manifolds
    • Ricci Tensor
    • Ricci Scalar
    • Bianchi Identities
    • Comparison of Newton’s and Einstein’s Gravity
    • Stress-Energy Tensor
    • Einstein Field Equations
    • Homework 11
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