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An <inertial frame> is a nonrotating frame in which all force-free test bodies follow straight trajectories with constant <velocity>. The qualification “all” matters: a specially chosen coordinate system can make one accelerated trajectory look straight without making the frame inertial. In practice, approximately isolated test bodies and <gyroscopes> help identify <inertial motion>; identifying the relevant forces and calibrating instruments are part of testing the theory. The definition does not select a privileged state of uniform rest.
In its usual coordinate formulation, <Newtonian mechanics> states $m\ddot{\mathbf x}=\mathbf F$ in an <inertial frame>. A <Galilean transformation>, $t'=t+b$, $\mathbf x'=R\mathbf x+\mathbf Vt+\mathbf a$ with constant orthogonal $R$ and constant $\mathbf V$, maps the class of <inertial frames> to itself. It preserves zero <acceleration>; a time-dependent translation generally does not. In the translating frame $\mathbf x'=\mathbf x-\mathbf a(t)$, the same physical law reads $m\ddot{\mathbf x}'=\mathbf F-m\ddot{\mathbf a}$. The extra <fictitious force> records the frame's <acceleration>, rather than a new interaction.
It is possible to state <Newtonian mechanics> without mentioning an <inertial frame> explicitly. On <Galilean spacetime>, let $u$ be a trajectory tangent normalized by the absolute time covector, $t_a u^a=1$. Then
$$
m\nabla_u u=F
$$
uses an <affine connection> to define <acceleration>. The free-particle equation is $\nabla_u u=0$. This formulation removes a chosen coordinate frame but retains the geometric distinction between <inertial motion> and accelerated motion. Omitting both the <inertial frame> language and all equivalent inertial structure would leave the law unable to specify which trajectories count as unforced.
The same distinction applies to <special relativity>. Einstein's original presentation uses a class of <inertial frames> and calibrated rods, clocks and <light> signals.<On the Electrodynamics of Moving Bodies by Albert Einstein>[sections 1–3: synchronization, postulates and coordinate transformation]{ref} A geometric presentation instead uses <Minkowski spacetime>, its <Lorentzian metric> and its <Levi-Civita connection>. A free massive particle obeys $\nabla_u u=0$, now with <proper time> parametrization and $g(u,u)=-c^2$. There is no preferred inertial rest frame, but there is a distinguished family of inertial trajectories and a physical <Minkowski interval>.
Accelerating coordinates are allowed in <special relativity>. They produce nonzero connection coefficients even though the <Riemann curvature tensor> remains zero. <General relativity> is needed when spacetime curvature and dynamical gravitation matter, not merely because the chosen observer accelerates. Thus neither theory requires a particular <inertial frame> in its coordinate-free statement; each still needs the structure that determines <inertial motion>.
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