Past exam of the philosophy of space-time course of the University of Oxford / 2022 / 1 / Solution

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, $t^{\prime }=t+b$, ${\mathbf{x}}^{\prime }=R\mathbf{x}+\mathbf{V}t+\mathbf{a}$ 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 ${\mathbf{x}}^{\prime }=\mathbf{x}-\mathbf{a}(t)$, the same physical law reads $m{\ddot{\mathbf{x}}}^{\prime }=\mathbf{F}-m\ddot{\mathbf{a}}$. 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, $t_a{u}^a=1$. Thenuses 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.^[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.