Lorentz Transformation Equations

Lorentz transformations are the coordinate-change rules that replace Galilean transformations when the speed of light is invariant. They let you convert (x,t) in one inertial frame into (x',t') in another moving at constant speed v along the x-axis.

Treat this maintained page as the mathematical anchor for the whole cluster; revisit it whenever sign conventions or frame conditions get fuzzy.

1. Definitions (Must Know)

• Standard configuration: frame S' moves at speed v along + x relative to frame S; axes are parallel; origins coincide at t = t' = 0.

• Lorentz factor: γ = 1/(square root of (1-v²/c²))

• Lorentz transformations (1D motion along x): x' = γ(x-vt)

  t' = γ(t-vx/c²)

  with y' = y, z' = z.

• Inverse transformations: x = γ(x' + vt')

  t = γ(t' + vx'/c²)

• Symbols used in this lesson: x,x' (m), t,t' (s), v (m s⁻¹), c (m s⁻¹), γ (dimensionless).

2. Key Ideas (What Earns Marks)

• Space and time mix: t' depends on both t and x.
• The transformations reduce to Galilean ones in the low-speed limit: v≪ c ⇒ γ ≈ 1 and t' ≈ t.
• Use the difference form for two events: it’s often cleaner for time dilation, length contraction, and simultaneity.

3. Detailed Explanations

A. The forward and inverse transformations

Forward (from S to S'): x' = γ(x-vt)

t' = γ(t-vx/c²)

Inverse (from S' to S) comes from swapping primes and replacing v → -v: x = γ(x' + vt')

t = γ(t' + vx'/c²)

B. The “difference form” for two events

For two events, define Δ x = x₂-x₁ and similarly for time. Then: Δ x' = γ(Δ x-vΔ t)

Δ t' = γ(Δ t-(vΔ x)/c²)

and inversely: Δ x = γ(Δ x' + vΔ t')

Δ t = γ(Δ t' + (vΔ x')/c²)

These are the forms you’ll use most in H3 problem-solving.

C. Classical limit (sanity check)

If c → ∞, then γ → 1 and vx/c² → 0, so: x' → x-vt, t' → t which matches Galilean transformations.

4. Common Mistakes

• Swapping the sign of v (define clearly whether S' moves at + v relative to S).
• Forgetting the vx/c² term in t' (that term is the source of relativity of simultaneity).
• Mixing coordinates from different frames in the same equation.
• Using degrees for v/c (it’s a pure ratio; no angles involved).

5. Exam Tips

• Start by stating the standard setup: “S' moves at speed v along + x relative to S.”
• Decide whether you should use the direct form (single event) or the difference form (two events).
• Quick plausibility checks:
  • if v = 0, then x' = x and t' = t.
  • if Δ x = 0 (events at same place in S), the time relation simplifies.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Relativity Results as Corollaries

Time dilation, length contraction, and relativity of simultaneity can all be derived cleanly from the difference form of Lorentz transformations:

• Simultaneity
• Time Dilation
• Length Contraction

Forward difference form:

Δ x' = γ(Δ x-vΔ t)

Δ t' = γ(Δ t-(vΔ x)/c²)

For the inverse form, replace v with -v and interchange primed and unprimed quantities.

Next in the maintained sequence: Relativistic Velocity Addition.