(a) Using the velocity parameter introduced in Exer. 18, show that the Lorentz transformation equations, Eq. (1.12), can be put in the form
t̅ = t cosh u − x sinh u, y̅ = y,
x̅ = −t sinh u + x cosh u, z̅ = z̅.
(b) Use the identity cosh2 u − sinh2 u = 1 to demonstrate the invariance of the interval from these equations.
(c) Draw as many parallels as you can between the geometry of space-time and ordinary two-dimensional Euclidean geometry, where the coordinate transformation analogous to the Lorentz transformation is
x̅ = x cos θ + y sin θ,
y̅ = −x sin θ + y cos θ.