Special Relativity - Morin - Chapter 4

Chapter 4: 4-vectors

Introduction to 4-vectors

  • 4-vectors are crucial in simplifying calculations and clarifying concepts in special relativity.

  • Previous discussions derived results in special relativity without 4-vectors, illustrating that while not necessary, utilizing them makes derivations easier.

  • General relativity requires a solid understanding of tensors (4-vectors generalization). Understanding 4-vectors is foundational before tackling general relativity.

  • Outline of Chapter 4:

    • 4.1 Definition of 4-vectors

    • 4.2 Examples of 4-vectors

    • 4.3 Properties of 4-vectors

    • 4.4 Energy-momentum 4-vector

    • 4.5 Force and acceleration 4-vectors

    • 4.6 Form of physical laws in special relativity

4.1 Definition of 4-vectors

  • Definition 4.1: A 4-tuple, A = (A0, A1, A2, A3), is a 4-vector if the Ai transform between frames under Lorentz transformations, specifically:

    • A0 = γ(A′0 + (v/c)A′1)

    • A1 = γ(A′1 + (v/c)A′0)

    • A2 = A′2

    • A3 = A′3

  • The Ai components must behave as standard 3-D space vectors under rotations.

  • The definition applies in scenarios where changes in coordinates result from Lorentz transformations.

  • Components:

    • A0: time component

    • A1, A2, A3: space components

4.2 Examples of 4-vectors

  • Displacement 4-vector:

    • dS = (dt, dx, dy, dz)

  • Velocity 4-vector:

    • V = (γ, γv) where v is the velocity in the direction of motion.

  • Energy-momentum 4-vector:

    • P = (E/c, p) = (γm, γmv)

  • Acceleration 4-vector:

    • A = (γ4 vv̇ , γ4 vv̇v + γ2a)

  • Force 4-vector:

    • F = (γdE/dτ, γf) where f is the 3-force.

  • Key Point: For a 4-vector, the components must transform consistently under both Lorentz and rotational transformations.

4.3 Properties of 4-vectors

  • Linearity: Any linear combination of 4-vectors remains a 4-vector.

  • Inner product invariance: The inner product of two 4-vectors A · B = A0B0 - A1B1 - A2B2 - A3B3 is invariant under Lorentz transformations.

  • Norm: The square of the norm |A| = √(A · A) is invariant, leading to useful conservation equations in relativistic physics.

4.4 Energy-momentum 4-vector

  • The norm of the energy-momentum 4-vector, P · P = E² - |p|², is invariant across frames.

  • For a single particle, it simplifies to m²c⁴ in its rest frame.

4.5 Force and acceleration 4-vectors

  • The force 4-vector relates to the relativistic form of Newton's second law, encompassing both force and acceleration in relativistic contexts. Specifically:

    • F = dP/dτ

    • Also gives the familiar mA in contexts where mass is constant.

4.6 Form of physical laws in special relativity

  • Physical laws must be expressed in terms of 4-vectors or tensors to be frame-independent.

  • Basic 3-vector forms are inadequate, as laws expressed in them may not hold true when frames change.

4.7 Summary

  • Defined 4-vectors and explained their transformation laws.

  • Provided examples of common 4-vectors (displacement, velocity, energy-momentum, etc.).

  • Asserted properties like linear combinations, invariance of inner products, and norms.

  • Connected energy-momentum 4-vectors with conservation principles.

  • Noted the necessity of using 4-vectors to express physical laws consistently across reference frames.