Why is the Lorentz Transformation Linear, and How is its Linearity Derived?

Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.

If physical and historical factors are disregarded and the question is considered from a purely mathematical standpoint, the linearity of the Lorentz transformation can be established as follows:

Assume the physical universe is a flat, complete, and simply connected pseudo-Riemannian manifold. Such a manifold must be isometric to a pseudo-Euclidean space. Consequently, any isometry of this manifold must be an affine transformation.

If this pseudo-Riemannian manifold is endowed with the Minkowski metric, its isometries are precisely the Poincaré transformations. The linear part of any Poincaré transformation is a Lorentz transformation.

The selection of the Minkowski metric is predicated on its mathematical convenience. From a historical perspective, however, the formalization of Minkowski space occurred after Lorentz had proposed his transformations and Einstein had published the theory of Special Relativity. The impetus for Minkowski’s formulation of spacetime was his discovery that the quantity known as the spacetime interval remains invariant under the Lorentz transformation. Subsequently, to systematically handle transformations between arbitrary inertial reference frames within Special Relativity, Minkowski defined Minkowski spacetime based on this principle of “spacetime interval invariance.”

The requirement for isometries, subsequent to adopting the Minkowski metric, stems from the fact that transformations preserving this metric inherently satisfy the two fundamental postulates of Special Relativity: “The laws of physics are the same in all inertial frames of reference” and “The speed of light in a vacuum has the same value in all inertial frames of reference.”

As to why the Lorentz transformation (a linear transformation) is more widely recognized and emphasized in the context of Special Relativity than the Poincaré transformation (an affine transformation), it is most likely because the Lorentz component directly leads to the derivation of canonical relativistic phenomena such as time dilation and length contraction. In contrast, the translational component of the Poincaré transformation can appear trivial. Nevertheless, the Poincaré transformation represents the most general isometry within Minkowski spacetime.