Paper 4, Section I, B

Dynamics and Relativity | Part IA, 2011

Inertial frames SS and SS^{\prime} in two-dimensional space-time have coordinates (x,t)(x, t) and (x,t)\left(x^{\prime}, t^{\prime}\right), respectively. These coordinates are related by a Lorentz transformation with vv the velocity of SS^{\prime} relative to SS. Show that if x±=x±ctx_{\pm}=x \pm c t and x±=x±ctx_{\pm}^{\prime}=x^{\prime} \pm c t^{\prime} then the Lorentz transformation can be expressed in the form

x+=λ(v)x+ and x=λ(v)x, where λ(v)=(cvc+v)1/2.x_{+}^{\prime}=\lambda(v) x_{+} \quad \text { and } \quad x_{-}^{\prime}=\lambda(-v) x_{-}, \quad \text { where } \quad \lambda(v)=\left(\frac{c-v}{c+v}\right)^{1 / 2} .

Deduce that x2c2t2=x2c2t2x^{2}-c^{2} t^{2}=x^{\prime 2}-c^{2} t^{\prime 2}.

Use the form ()(*) to verify that successive Lorentz transformations with velocities v1v_{1} and v2v_{2} result in another Lorentz transformation with velocity v3v_{3}, to be determined in terms of v1v_{1} and v2v_{2}.

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