Special Relativity, Tensors, And Energy Tensor: With Worked Problems

✍ Scribed by Somnath Datta

• Publisher: World Scientific Publishing
• Year: 2021
• Tongue: English
• Leaves: 384
• Edition: 2
• Category: Library

✦ Synopsis

This book takes the reader from the preliminary ideas of the Special Theory of Relativity (STR) to the doorsteps of the General Theory of Relativity (GTR).

The first part explains the main concepts in a layman's language, including STR, the Lorentz transformation, relativistic mechanics. Thereafter the concept of tensors is built up in detail, especially Maxwell's stress tensor with illustrative examples, culminating in the energy-momentum conservation in electromagnetic fields. Mathematical structure of Minkowski's space-time is constructed and explained graphically. The equation of motion is formulated and then illustrated by the example of relativistic rocket. The principle of covariance is explained with the covariant equations of classical electrodynamics. Finally, the book constructs the energy tensor which constitutes the source term in Einstein's field equation, which clears the passage to the GTR.

In the book, the concepts of tensors are developed carefully and a large number of numerical examples taken from atomic and nuclear physics. The graphs of important equations are included. This is suitable for studies in classical electrodynamics, modern physics, and relativity.

✦ Table of Contents

Contents
Preface
Symbols Used in This Book
Part I Einsteinian Relativity
1 What Is Relativity?
1.1. Influence of the Human Value System on the Evolution of Physical Theories
1.2. Rejection of Absolute Frame
1.3. Relativity Principle: A Rudimentary Form
1.4. Inertial Forces
1.5. Principle of Equivalence
1.6. Tidal Forces
1.7. The Scheme of General Relativity
1.8. Conclusion
2 Einstein’s Postulates, Their Paradoxes, and How to Resolve Them
2.1. Event Point in Space–Time
2.2. Inertial Frames
2.2.1. Newton’s equation of motion
2.2.2. Conservation of energy and momentum
2.2.3. Equivalence of inertial frames
2.3. Historical Background
2.3.1. Search for the absolute frame
2.3.2. Michelson–Morley experiment
2.4. Postulates of Special Relativity
2.5. Relativity of Simultaneity
2.6. Time Dilation
2.7. Length Contraction
3 Lorentz Transformation
3.1. Lorentz Transformation I: Special Case
3.2. Lorentz Transformation II: General Case
3.3. Simple Applications of Lorentz Transformation
3.4. Time-Like, Light-Like, Space-Like Intervals
3.5. Relativistic Doppler Formula
3.6. Worked Out Problems I
3.7. Illustrative Numerical Examples I
4 Relativistic Mechanics
4.1. Relativistic Form of Velocity Transformation
4.2. Relativistic Form of Acceleration Transformation
4.3. A New Definition of Momentum
4.4. Force
4.5. Energy
4.6. Energy–Momentum Conservation Law
4.7. The Centre of Mass of a System of Particles: The Zero Momentum Frame
4.8. The Twin Paradox
4.9. Compton Scattering
4.10. Summary of Important Formulas
4.11. Worked Out Problems II
4.12. Illustrative Numerical Examples II
4.13. Exercises for the Reader I
Part II Amazing Power of Tensors
5 Let Us Know Tensors
5.1. Introduction to Tensor
5.1.1. Vector–tensor analogy
5.1.2. Linear operator in a vector space
5.1.3. Tensor as a dyadic
5.1.4. Identity tensor, completeness relation, components of a tensor in the spherical coordinate system
5.1.5. Example: Inertia tensor
5.2. Stress in a Medium
5.2.1. Stress vector
5.2.2. Stress tensor
5.2.3. Diagonalization of a symmetric tensor
5.2.4. Gauss’s divergence theorem for a tensor field
5.2.5. Volume force density in a stress tensor field
6 Maxwell’s Stress Tensor
6.1. Introduction
6.2. Maxwell’s Stress Tensor for the Electrostatic Field
6.2.1. Volume force density in terms of the field
6.2.2. Example 1: Stress vector on a plane as a function of the angle of inclination
6.2.3. Example 2: Force transmitted between two charged particles across a spherical boundary
6.3. Maxwell’s Stress Tensor for the Magnetostatic Field
6.3.1. Volume force density in terms of the field
6.3.2. Example 3: Force transmitted between two magnetic dipoles across a spherical boundary
6.4. Example 4: The Force Between Two Hemispheres of a Charged Sphere
6.5. Maxwell’s Stress Tensor for the Electromagnetic Field and Momentum Conservation
Part III Physics in Four Dimensions
7 Space–Time and Its Inhabitants
7.1. World Line in Space–Time
7.2. Hyperbolic World Line of a Particle Moving Under a Constant Force
7.3. Lorentz Transformation in Space–Time
7.3.1. Graphical procedure
7.3.2. Graphical construction of length contraction
7.3.3. Graphical construction of time dilation
7.3.4. Simultaneity, or absence of it
7.4. Minkowski Space–Time
7.5. 3-Vectors, Contravariant and Covariant Families
7.6. 3-Tensors, Contravariant and Covariant Families
7.7. Transformation of the Metric 3-Tensor from Cartesian to Spherical
7.8. 4-Vectors in Relativity
7.9. 4-Tensors in Relativity
7.9.1. Contravariant, covariant and mixed tensors
7.9.2. Equality of two tensors
7.9.3. The metric tensor of the Minkowski space–time
7.9.4. New tensors from old ones, index gymnastics
8 Four Vectors of Relativistic Mechanics
8.1. 4-Displacement
8.2. 4-Velocity
8.3. 4-Acceleration
8.4. 4-Momentum, or En-Mentum
8.5. 4-Force, or Pow-Force and Minkowski’s Equation of Motion of a Point Particle
8.6. Force on a Particle with a Variable Rest Mass
8.7. Lorentz Transformation of the 4-Vectors of Relativistic Mechanics
8.7.1. LT of 4-velocity
8.7.2. LT of 4-acceleration
8.7.3. LT of 4-momentum
8.7.4. LT of 4-force
8.8. Conservation of 4-Momentum of a System of Particles
8.8.1. Zero momentum frame, equivalence of E and mass
8.9. Illustrative Numerical Examples III
8.9.1. Example 1: Relativistic billiard balls
8.9.2. Example 2: Threshold energy for a p + p collision resulting in the production of a π0 particle
8.9.3. Example 3: Threshold energy for a photon hitting an electron to produce an electron–positron pair
8.9.4. Example 4: Compton scattering and inverse compton scattering
8.9.5. Example 5: Doppler effect
8.10. Exercises for the Reader II
9 Relativistic Rocket
9.1. Introduction
9.2. The Rocket, Its Specifications
9.3. Review of the Non-relativistic Results
9.4. Relativistic Mass Equation
9.5. The Thrust 4-Force
9.6. The Equation of Motion
9.7. Solution of the EoM for Two Special Cases
10 Magnetism as a Relativistic Effect
10.1. Velocity-Dependent Force from a Velocity-Independent One under a Lorentz Transformation
10.2. How Magnetic Force Originates from Lorentz Transformation
11 Principle of Covariance with Application in Classical Electrodynamics
11.1. The Principle
11.2. The Flux of a Vector Field in E3
11.2.1. 2D surface embedded in 3D space, and the outward normal
11.2.2. Surface integral, flux of a vector field
11.2.3. Continuity equation
11.3. Conservation of Electric Charge
11.4. The Electromagnetic Field Tensor
11.5. The Field Equations of Electrodynamics in the Covariant Language
11.6. EM Field of a Charged Particle in Uniform Motion
11.6.1. Transformation from the rest frame to Lab frame
11.6.2. Pictorial interpretation of the fields
11.6.3. Ionizing effect of a heavy ion in its passage through matter
11.7. Exercises for the Reader III
Part IV 4-Momentum Conservation in Continuous Media
12 The Energy Tensor
12.1. Why Energy Tensor?
12.2. Minkowski Volume Force Density
12.3. Energy and Momentum Conservation in One Voice
12.4. Euler’s (Non-relativistic) Equation of Motion for a Perfect Fluid
12.5. Relativistic Equation of Motion for a Continuous Incoherent Media
12.6. Energy Tensor for a System of Charged Incoherent Fluid
12.7. Energy Tensor of a Closed System
12.8. Energy Tensor of a Perfect Fluid
Appendices
A.1. Energy Conservation in Electromagnetic Field
A.2. Examples of Lowering and Raising an Index
A.3. Components of Maxwell’s Stress 3-Tensor and Maxwell’s 4-Tensor, and Their Traces
B.1. Useful Integrals
Epilogue
Bibliography
Index