Special Relativity

1. Mathematical Formulation
   1. Lorentz Transformations
      1. Derivation and Explanation
         1. Historical Development from Classical Concepts
            1. Analysis of Galilean Transformations
            2. Introduction of Lorentz Factor (γ)
         2. Mathematical Derivation
            1. Starting from Invariance of Light Speed
            2. Detailed Step-by-Step Transformation Derivation
         3. Physical Interpretation
            1. Concepts of Relative Velocity and Simultaneity
            2. Interpretation in Various Reference Frames
      2. Effects on Time and Space Measurements
         1. Time Dilation
            1. Mathematical Expression and Derivation
            2. Analysis of Clocks’ Behavior in Motion
         2. Length Contraction
            1. Derivation from Lorentz Transformations
            2. Examples and Problem-Solving Scenarios
         3. Simultaneity
            1. Impact on Event Timing Across Different Inertial Frames
            2. Illustration through Real-World Applications
   2. Minkowski Spacetime
      1. 4-Dimensional Space and Time
         1. Definition and Structure
            1. Historical Origins by Hermann Minkowski
            2. Integration of Space and Time Dimensions
         2. Mathematical Representation
            1. Use of Tensor Calculus
            2. Space-time Interval and Its Invariance
         3. Applications to Physical Phenomena
            1. Explanation of Worldlines and World Events
            2. Importance in Predicting Physical Outcomes
      2. Light Cones and Causality
         1. Concept and Construction of Light Cones
            1. Visual Representations and Their Significance
            2. Division into Future, Past, and Elsewhere Regions
         2. Causality and Relativistic Constraints
            1. Notion of Cause and Effect in Spacetime
            2. Causal Structure and Its Implications for Physics
         3. Practical Implications in Astrophysics and Beyond
            1. Observational Impact on Cosmological Events
            2. Interaction of Light Cones with Gravitational Fields
   3. Velocity Addition Formula
      1. Composition of Velocities at Relativistic Speeds
         1. Derivation of Velocity Addition Rule
            1. Contrast with Classical Additive Velocity Rule
            2. Simplified Mathematical Approach for Relativistic Scenarios
         2. Applications and Limitations
            1. Use in High-Speed Particles and Astrophysics
            2. Discussion of Paradoxes and Resolutions
         3. Examples in Astronomy and Particle Physics
            1. Analytical Problems and Solutions
            2. Interpretations of Measurements in High-Speed Contexts
      2. Effects and Observations
         1. Practical Examples of Relativistic Velocity Transformation
         2. Considerations in Navigating High-Speed Objects
         3. Implementation in Technology and Experimental Physics