  In flat spacetime, as a simple 4-vector, a particle's 4-velocity cannot be
changed by translation. Parallel translation then produces constant velocity,
motion without force. Here we consider a richer, but less well-known,
representation of the Poincare group of symmetries of flat spacetime in which
translation adds a non-linear connection term to the 4-vector. Then 4-vectors
can be meaningfully parallel translated, curved geodesics can be developed, and
the curvature for the connection can be found. By adding an assumption, that
the arc length is calculated with a position dependent metric, it is shown that
a geodesic has a 4-acceleration which is the sum of a Christoffel connection
term and a term that is the scalar product of 4-velocity and an antisymmetric
tensor. This is just like the force law obeyed by a charged, massive particle
in general relativity. Thus the dynamical laws of electrodynamics and general
relativity can be deduced as geodesic equations from the way particle proper
time depends on position and the way 4-vectors can change upon translation in
flat spacetime.
