Can Lorentz Transformations Be Derived from General Relativity?

Lorentz transformations appear in general relativity because each tangent space carries a Lorentzian metric. Spacetime need not be flat for this local statement to hold.

This note revises my original Chinese answer.

At an event $p$ , choose an orthonormal frame for the metric, so its components are

\eta=\operatorname{diag}(-1,1,1,1).

If another frame at the same event is orthonormal, its change-of-frame matrix $\Lambda$ satisfies

\Lambda^{\mathsf T}\eta\Lambda=\eta.

That is the defining condition for a Lorentz transformation. The relation concerns frames in the tangent space at an event; arbitrary coordinate transformations on a curved region need not be Lorentz transformations.

What a local inertial coordinate system removes

For a sufficiently smooth Lorentzian metric, normal coordinates centered at $p$ can be chosen so that

g_{\mu\nu}(p)=\eta_{\mu\nu},\qquad \partial_\rho g_{\mu\nu}(p)=0.

The Christoffel symbols then vanish at $p$ . Curvature generally remains in second-order terms, so the metric is not necessarily equal to $\eta$ throughout a neighborhood. Tong's discussion of normal coordinates develops this distinction.

In flat spacetime, inertial Cartesian coordinate systems can instead describe a whole suitable region with constant Minkowski metric. Their transformations are affine:

x'=\Lambda x+a.

They are Poincaré transformations, becoming linear Lorentz transformations when the origins coincide. Even vanishing curvature everywhere does not by itself guarantee that a spacetime has the global topology of Minkowski space.

Thus general relativity contains local Lorentz geometry from the outset. Flatness is required for extending the Minkowski description across a region, not for relating orthonormal frames at a point.