the Lorentz factor γ is defined in terms of the boost velocity to be
We recognize that (3.7) can also be written in the matrix form as
where we have defined
so that
Since the range of the boost velocity is given by −1 ≤ β ≤ 1 (namely, |v| ≤ c = 1), we conclude from (3.10) that −∞ ≤ ω ≤ ∞.
Thus, we note that a Lorentz boost along the x-direction can be written as a matrix relation
where
From this, we can obtain,
which would lead to the transformation of the covariant coordinate vector as
The matrix representing the Lorentz transformation of the coordinates,
where we have used
From (3.16), we see that the matrix Λµν has a unit determinant, much like the rotation matrix R in (3.3). (Incidentally, (3.16) also shows that the covariant vector transforms in an inverse manner compared with the contravariant vector.) Therefore, we can think of the Lorentz boost along the 1-axis (x-axis) as a rotation in the 0-1 plane with an imaginary angle (so that we have hyperbolic functions instead of ordinary trigonometric functions). (That these rotations become complex is related to the fact that the metric has opposite signature for time and space components.) Furthermore, as we have seen, the “angle” of rotation, ω, can take any real value and, as a result, Lorentz boosts correspond to noncompact transformations unlike space rotations.
Let us finally note that if we are considering an infinitesimal Lorentz boost along the 1-axis (or a rotation in the 0-1 plane), then we can write (see (3.13) with ω = ϵ = infinitesimal)
where,
It follows from this that
In other words, the matrix representing the change under an infinitesimal Lorentz boost is anti-symmetric just like the case of an infinitesimal rotation. In a general language, therefore, we note that we can combine rotations and Lorentz boosts into what are known as the homogeneous Lorentz transformations, which can be thought of as rotations in the four dimensional space-time.
General Lorentz transformations are defined as transformations
which leave the length of the vector invariant, namely,
where we have used the fact that the metric, ηµν, remains invariant under a Lorentz transformation. Equation (3.22) is, of course, what we have seen before in (3.16). Lorentz transformations define the maximal symmetry of the space-time manifold which leaves the origin invariant.
Choosing ρ = σ = 0, we can write out the relation (3.22) explicitly as
Therefore, we conclude that
If Λ00 ≥ 1, then the transformation is called orthochronous. (The Greek prefix “ortho” means straight up. Thus, orthochronous means straight up in time. Namely, such a Lorentz transformation does not change the direction of time. Incidentally, “gonia” in Greek means an angle or a corner and, therefore, orthogonal means the corner that is straight up (perpendicular). In the same spirit, an orthodontist is someone who can make your teeth straight.) Note also that since (see (3.16))
we obtain
where we have used det ΛT = det Λ. The set of homogeneous Lorentz transformations satisfying
are known as the proper orthochronous Lorentz transformations and constitute a set of continuous transformations that can be connected to the identity matrix. (Just to emphasize, we note that the set of transformations with det Λ = 1 are known as proper transformations and the set for which Λ00 ≥ 1 are called orthochronous.) In general, however, there are four kinds of Lorentz transformations, namely,
