Geodesic eqn for non-affine parameterization

Question

I am confused about a point regarding parallel transport and geodesics. The basic idea of a geodesic is the unaccelerated test particle moves in a straight line, or the tangent vector of a curve $x^b(\lambda)$ will be parallelly transported along the curve. In curved spacetime or in general coordinates, the eqn looks $$ u^b\nabla_b u^c=0\tag{1} $$ where $u^b=\frac{dx^b}{d\lambda}$ tangent to the curve. The covariant derivative vanishes, means no external force acting on the particle, hence velocity is constant. But if we rewrite the same eqn with a non-affine parameter $\xi$ then $$ \exists \eta: u^b\nabla_bu^c=\eta(\xi)u^c\tag{2} $$ which means there is some acceleration working on the particle and the velocity $u^b=\frac{dx^b}{d\xi}$ is not of constant magnitude. Now this is not anymore a parallel transport of the tangent vector. My question is now can we say this is a geodesic?

My comment is I am not sure whether the appropriate definition of the geodesic is the magnitude of the tangent vector can be changed but the direction should be unchanged. If a curve is actually geodesic, we can find a unique affine parameter which will make the RHS zero. This makes sense for particle in EM field where the RHS is $F^c_du^d$ and we can not get some parameter which will make the RHS zero, hence it is not a geodesic. But this idea is not working for null geodesics at $r=2m$ sphere of the blackhole, where the RHS is $\frac{1}{2m}u^c$. Here also we can not find an affine parameter which makes the RHS zero. So why we call this as geodesic

Answer 1

Yes, OP is right: Eq. (2) is the equation for an arbitrarily parametrized geodesic, while eq. (1) is the equation for an affinely parametrized geodesic. An affine parameter $\lambda$ can locally be found as the following nested indefinite integral
$$ \eta(\xi)~=~\frac{d}{d\xi}\ln\left|\frac{d\lambda}{d\xi}\right| \qquad\Leftrightarrow\qquad \lambda ~=~\pm \int e^{\int \eta}.$$