Why Is the Lorentz Transformation Linear?

The transformation between inertial Cartesian coordinates in Minkowski spacetime is affine. It becomes linear when the coordinate origins coincide. The qualification “inertial Cartesian” matters: accelerated or curvilinear coordinates can have nonlinear transformations even in flat spacetime.

This note revises my original Chinese answer.

A geometric argument

Use coordinates $x^\mu$ and $x'^\mu$ in which the metric is the same constant Minkowski matrix $\eta$. In both systems, the Levi-Civita connection coefficients vanish. The connection transformation law then implies

On a connected coordinate domain, all first derivatives are constant, so the inverse coordinate map is affine. Its inverse therefore has the form

with constant invertible $\Lambda$ and constant translation $a$. Preserving the metric gives

Consequently $\Lambda$ is a Lorentz matrix. When both coordinate systems assign zero to the same event, $a=0$, yielding a linear Lorentz transformation. The general affine transformation is called a Poincaré transformation.

What this assumes

The argument starts from Minkowski geometry and inertial coordinates. It is not a derivation of special relativity from pure mathematics: that geometry encodes physical assumptions about spacetime, supported by experiment. Another familiar route assumes spacetime homogeneity to obtain an affine transformation, then imposes the relativistic interval and a common origin.

Metric preservation is also not sufficient to establish the relativity principle for every imaginable law. A physical theory must have dynamics compatible with the transformations. The geometric condition identifies the transformations; covariance of the equations establishes the corresponding symmetry of that theory.

For the distinction between flat-space inertial coordinates and local frames in curved spacetime, see Lorentz transformations and general relativity and Tong's account of Lorentzian geometry.