Chapter 1: Lorentz Transformations
The Lorentz transformations are the mathematical heart of special relativity. They describe exactly how space and time coordinates transform between different inertial reference frames, replacing the simpler Galilean transformations that fail at high velocities.
The Lorentz Transformation Equations
Consider frame S' moving at velocity v in the +x direction relative to frame S. Both frames use standard configuration (origins coincide at t = t' = 0, axes parallel).
Lorentz Transformations (S → S')
\( t' = \gamma \left( t - \frac{vx}{c^2} \right) \)
\( x' = \gamma (x - vt) \)
\( y' = y \)
\( z' = z \)
where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \)
Inverse Transformations (S' → S)
\( t = \gamma \left( t' + \frac{vx'}{c^2} \right) \)
\( x = \gamma (x' + vt') \)
\( y = y' \)
\( z = z' \)
Just replace v with −v (S moves at −v relative to S')
Derivation from Einstein's Postulates
Step 1: Linearity
The transformation must be linear (to preserve uniform motion):
\( x' = Ax + Bt \) and \( t' = Cx + Dt \)
where A, B, C, D depend only on v
Step 2: Origin of S' at x' = 0
The origin of S' moves at x = vt in frame S:
\( 0 = A(vt) + Bt \Rightarrow B = -Av \)
Step 3: Light Speed Invariance
A light pulse at x = ct must also be at x' = ct':
\( ct' = c(Ct + Dt) \) and \( x' = A(ct - vt) = ct' \)
This gives us relations between A, C, D
Step 4: Reciprocity
S sees S' moving at +v; S' sees S moving at −v. By the principle of relativity:
The inverse transformation uses −v (not different coefficients)
Final Result: Combining all constraints yields A = D = γ, B = −γv, C = −γv/c², giving us the Lorentz transformations.
Matrix Formulation
The Lorentz transformation can be written elegantly in matrix form:
\( \begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma \\ -\beta\gamma & \gamma \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} \)
where β = v/c
Using rapidity φ where tanh(φ) = β:
\( \Lambda = \begin{pmatrix} \cosh\phi & -\sinh\phi \\ -\sinh\phi & \cosh\phi \end{pmatrix} \)
Compare to rotation matrix with cos and sin!
Key Properties
1. Preserves the Spacetime Interval
\( c^2t'^2 - x'^2 = c^2t^2 - x^2 \) (invariant under Lorentz transformation)
2. Forms a Group (the Lorentz Group)
Successive transformations give another Lorentz transformation. Identity exists (v = 0). Every transformation has an inverse (v → −v).
3. Reduces to Galilean at Low Speeds
When v ≪ c, γ ≈ 1 and vx/c² ≈ 0, so t' ≈ t and x' ≈ x − vt (Galilean form).
4. Composition is Non-Linear in Velocities
Two successive boosts (v₁, then v₂) give a combined velocity less than v₁ + v₂. Rapidities add linearly: φ₁₂ = φ₁ + φ₂.
Reading Time Dilation and Length Contraction
Time Dilation
Clock at rest in S' (x' = 0). In S:
\( \Delta t = \gamma \Delta t' \) (moving clock runs slow)
Length Contraction
Rod at rest in S' (proper length L₀). In S, measured simultaneously:
\( L = L_0/\gamma \) (moving rod is shortened)