Lorentz Transformation

The equations of the Standard Model must be consistent with Einstein’s principle of relativity, which states that the laws of Nature take the same form in every inertial frame of reference. An inertial frame is one in which a free body moves without acceleration. An earth-bound frame approximates to an inertial frame if the gravitational field of the earth is introduced as an external field. We shall assume that the reader is familiar with rotations, and with proper Lorentz transformations and the relativistic mechanics of particle collisions. This chapter is very largely about notation, which may make for dry reading; however an appropriate notation is crucial to the exposition of any theory, and particularly so to a relativistic theory, such as the Standard Model.

Rotations, boosts and proper Lorentz transformations
The time and space coordinates of an event measured in different inertial frames of reference are related by a Lorentz transformation. A rotation is a special case of a Lorentz transformation. Consider, for example, a frame K that is rotated about the z-axis with respect to a frame K, by an angle θ. If (t, r) are the time and space coordinates of an event observed in K, then in K the event is observed at (t, r_)
and

t = t
x = x cos θ + y sin θ
y = −x sin θ + y cos θ
z = z.

Lorentz transformations also relate events observed in frames of reference that are moving with constant velocity, one with respect to the other. Consider, for example, an inertial frame K moving in the z- direction in a frame K with velocity v, the spatial axes of K and K being coincident at t = 0. If (t, r) are the time and space coordinates of an event observed in K, and (t, r_) are the coordinates of the same event observed in K, the transformation takes the form