Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• (a) Consider two frames S and S′ that differ only by a rotation in which the x and y axes were rotated clockwise through an angle θ to become x′ and y′. Prove that

 x′ = x cos θ − y sin θ and y′ = y cos θ + x sin θ

(and z′ = z and t′ = t). (b) Prove that the standard Lorentz transformation can be written as

 x′ = x cosh ϕ − ct sinh ϕ

and

 ct′ = ct cosh ϕ − x sinh ϕ

(and y′ − y and z′ = z) where ϕ = tanh−1(v/c). Except that the trig functions cos and sin are replaced by the hyperbolic functions cosh and sinh (and that one term has changed sign), the Lorentz transformation does to x and ct just what a rotation does to x and y. This is our first indication that x, y, z, ct should be regarded as the four coordinates in some kind of four-dimensional space-time.