Proper-time normalization of an affine parameter ($g_{ab}{\dot{x}}^a{\dot{x}}^b=c^2$)

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.