We all know how the metric of GR is a generalization of the flat Minkowski spacetime metric. But I wonder if the SR metric is generalized from the kinetic energy term of Newtonian physics. There the kinetic energy is (1/2)m*v^2=(1/2)m*dx*dx. If the mass/2 plays the role of the metric, then this looks like n_uv*dx^u*dx^v, like a differential line segment. Or what part does the kinetic term play in the SR inner product. Thanks.