In static spherically symmetric spacetime with metric $$ds^2=e^{2\nu}c^2dt^2-e^{2\lambda} dr^2-r^2 (d\theta^2+\sin{\theta}~d\phi^2)\equiv \delta_1~c^2 d\tau^2 \tag{1}$$ the geodesic equation in equatorial plane ($\theta=\pi/2$) reads $${e}^{2\nu+2\lambda}~\frac{\dot{r}^2}{c^2}=k^2-{e}^{2\nu}~(\delta_{1}+\frac{l^2}{c^2 r^2})\equiv k^2-V(r) \tag{2}$$ where the constants of motion are defined as $$k\equiv {e}^{2\nu}~\dot{t},~~~l\equiv r^2~\dot{\phi}~~~.$$ Correspondingly to space-, null- and timelike geodesics $\delta_{1}$ can take values $~-1,0,1$ . For circular geodesic it applies $\dot{r}=\ddot{r}=0$. Hence, a geodesic of curvature radius $r_{0}$ has constants $$k^2=\delta_{1}\frac{e^{3\nu}_{0}}{e^{\nu}_{0}-r_{0}~\partial_{r}{e^{\nu}_{0}}},~~~l^2=\delta_{1} c^2 \frac{r_{0}^3~\partial_{r}{e^{\nu}_{0}}}{e^{\nu}_{0}-r_{0}~\partial_{r}{e^{\nu}_{0}}}~~~. \tag{3}$$ For the special geodesic defined through condition $e^\nu(r_0)~=~0$ they result in $$k^2=0,~~~l^2=-\delta_{1}~ c^2 r_0^2~~~. \tag{4}$$ The second equation seems to indicate that such a geodesic can be followed only by spacelike particle ($\delta_1=-1$) with zero rest energy ($k^2=0$) and non-vanishing momentum ($l^2\geq 0$) or timelike particle $(\delta_1=1)$ with imaginary angular momentum ($l^2\leq 0$).
Assuming the first case the effective Potential reads $$V=e^{2\nu}~(-1+\frac{r_0^2}{r^2}) \tag{5}$$ and satisfy $$V(r_0)=\frac{dV}{dr}(r_0)=\frac{d^2V}{dr^2}(r_0)=0. \tag{6}$$ Thus, that special geodesic represents the innermost circular stable orbit (ICSO).
However, there only known particles in General Relativity satisfying theses conditions are transcendent tachyons. While at the same time the equation $e^{\nu}(r_0)=0$ defines event horizon of the curvature radius $r_0$, the question arises if one could interpret the event horizon as a transcendent tachyon.
Although such interpretation would contradict the prevailing understanding of event horizon as an imaginary 2-sphere it would have some other merits. Particles colliding with transcendent tachyon reflect without energy change but with charge switch, see Fig.5 in “Particles beyond the light barrier”. That has some resemblance to so called "firewall" theory. Another interesting point is that transcendent tachyon geodesic defined by equation (4) is stable. The corresponding effective potential has minimum there and the whole process of black hole formation could be understood as spontaneous symmetry breaking phenomenon, see Fig. 4 in “Tachyons and Solitons in Spontaneous Symmetry Breaking in the Frame of Field Theory”.
Appendix
I wanted to examine parametric stability of perfect fluid (sphere) spacetime using geodesic equation and effective potential $V(r)$. The parameter is $\alpha\equiv r_S/R$. For the congruence of radial geodesics $V(r)=e^{2\nu}$. In case of Schwarzschild constant density star the potential for $\alpha<8/9$ (green line) has a regular ($\sim r^2$) minimum at the center that for $\alpha=8/9$ (red line) degenerates ($\sim r^4$) and transforms to local maximum for $\alpha>8/9$ (blue line). Quasi-statically, the black hole formation process looks like classical "pitchfork" catastrophe. The former central minimum moves outwards until $\alpha=1$ is reached. In terms of radial geodesics the local minima ($r_0$) are stable and the local maximum unstable. I wanted to see the corresponding stability of circular orbits starting at different $r_0$ (minimums) and their possible physical interpretation. After all, the minimums at $r_0(\alpha)$ represent moving event horizon of curvature radius $r_0(\alpha)$.
