Inertial frame

An inertial frame is a reference frame in which a free particle follows a straight path at constant velocity. This is the setting in which the usual form of Newton's second law applies without additional forces caused by acceleration of the frame itself.

Frames moving at constant relative velocity can both be inertial. In Newtonian mechanics, a Galilean transformation relates their spatial velocities while preserving absolute time; in special relativity, the corresponding relation is a Lorentz transformation.

A freely moving object viewed from an accelerating vehicle can acquire coordinate acceleration even when no new interaction acts on it. Apparent-force terms account for that frame motion. An inertial description is an idealization whose accuracy depends on the spatial and temporal scales of the experiment.

Motion with zero acceleration relative to the theory's inertial affine connection. In an inertial frame, its spatial trajectory is uniform and straight. The geometric statement refers to the connection, not to a chosen coordinate system; force-free motion supplies its physical interpretation. In general relativity the corresponding freely falling trajectories are timelike geodesics, with local inertial coordinates rather than a global inertial frame in a curved region.

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 $c$.

An inertial frame in which a selected body's velocity vanishes at one event on its worldline. For an accelerated body it is a local construction: rest frames at different events need not coincide, and simultaneity slices chosen at one event do not equal laboratory equal-time slices.

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.