The Lorentz Transformation — Question 6

Question 6

Two equal boosts combine into one stronger subluminal boost

Apply successively two collinear boosts, each with $\beta=0.60$, to the coordinate column $(ct,x)^{\mathsf T}$. Each matrix is $\Lambda=\tfrac54\begin{pmatrix}1&-3/5\\-3/5&1\end{pmatrix}$.

1. Multiply the matrices and identify the effective gamma and beta.

2. Check the effective Lorentz identity and compare with adding the two velocities classically.

Question 6 - Solution

Two equal boosts combine into one stronger subluminal boost

(a) Method and calculation

$$\Lambda^2=\begin{pmatrix}17/8&-15/8\\-15/8&17/8\end{pmatrix}.$$$$\boxed{\gamma_{\mathrm{eff}}=17/8},\qquad\boxed{\beta_{\mathrm{eff}}=15/17\simeq0.882353}.$$

(b) Interpretation and further calculation

$$\gamma_{\mathrm{eff}}^2(1-\beta_{\mathrm{eff}}^2)=\frac{289-225}{64}=1.$$$$0.60+0.60=1.20,\qquad\boxed{\beta_{\mathrm{eff}}<1}.$$

Physical check

The time-space mixing in the second transformation prevents classical velocity addition.