Affine parameter

An affine parameter on a geodesic is one for which the coordinate equation has the form ${\ddot{x}}^k+{\Gamma }_{ij}^k{\dot{x}}^i{\dot{x}}^j=0$. Its affine changes $\lambda \mapsto A\lambda +B$, with $A\ne 0$, preserve this form.

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.

A change $\lambda \mapsto A\lambda +B$ rescales a geodesic tangent and all its linear Killing constants by $1/A$. A null norm remains zero and cannot fix $A$, unlike proper-time normalization of a material particle. Ratios of the constants, such as the photon impact parameter $b=|h|/(ck)$ in Schwarzschild spacetime, are independent of the affine scale. A physically normalized four-momentum fixes the photon energy and hence selects a tangent scale.

For physical four-momentum $P^{\mu }=md{x}^{\mu }/d\tau$, the covariant-component equation is $d{P}_{\mu }/d\tau =({\partial }_{\mu }{g}_{\nu \sigma })P^{\nu }{P}^{\sigma }/(2m)$. The equation without $1/m$ instead uses a tangent parametrization, or the affine parameter $\lambda =\tau /m$ when the tangent is the physical momentum.