Geometry of Lorentz invariance
This chapter focuses on the geometry of spacetime implied by the Lorentz invariance of the spacetime separation. It shows that boosts can be considered rotations in spacetime. In analogy with rotations, boost transformations can be considered rotations in spacetime, even though they can not be visualized as such. Rotations in Euclidean space are the result of twisting around an axis of rotation. The invariant hyperbola can be used to illustrate time dilation and length contraction. Length contraction and time dilation are thus symmetric between inertial reference frames. The chapter discusses a Lorentz transformation to be any linear transformation that preserves the spacetime separation. A geometry involves the ability to specify the distance between points, and a natural way to do that is through the inner product between vectors—which allows one to assign a magnitude to vectors. Spacetime separations are non-intuitive.