Inertial frame

An inertial frame is a reference frame in which a force-free particle moves with constant velocity.

The rest frame of a massive particle is an inertial frame in which its spatial momentum is zero. Its four-momentum is then $(mc,0)$ and its energy is $m{c}^2$. Such a frame exists for a future-directed timelike four-momentum: the Lorentz transformation with boost velocity $\mathbf{v}=c^2\mathbf{p}/E$ makes the spatial momentum zero, and the mass shell relation implies $|\mathbf{v}|<c$. A massless particle has no rest frame, because $E=c|\mathbf{p}|$ would require a boost with speed $c$.

A local inertial frame is a coordinate frame in which, at one chosen spacetime event, the metric tensor is Minkowskian and the Levi-Civita connection vanishes. Freely falling experiments confined to a sufficiently small neighbourhood therefore obey the laws of special relativity up to tidal corrections.