Geodesics in Curved Spacetime: Schwarzschild, Kerr, FLRW

An interactive tour of geodesics across three spacetimes. Schwarzschild: the equatorial null-geodesic fan from the orbit equation $u' + u = 3 M u^2$ (RK4), with the conserved first integral $(u')^2 + u^2 - 2u^3 = 1/b^2$ ($b = L/E$) conserved along each ray, photons captured iff $b < b_c = 3 \sqrt{3} M$, the photon sphere at $3 M$, the ISCO at $6 M$, and the null effective potential $V(r) = (1 - 2M/r)/r^2$ peaking at $r = 3 M$. Kerr: a smaller horizon $r = M + \sqrt{M^2 - a^2}$, the ergosphere, and a perturbative frame-drag twist with the prograde/retrograde ISCO from the exact Kerr formula. FLRW (Friedmann-Lemaitre-Robertson-Walker): the comoving lattice with the Hubble flow $v = H_0 d$ (exactly linear, superluminal beyond the Hubble radius $c/H_0$, which is allowed), the particle horizon, the redshift $1 + z = 1/a$, and the monotone-expanding scale factor $a(t)$ from the Friedmann equation. Photons are captured exactly at $b_c$ with the weak-field $4M/b$ deflection tail, the Schwarzschild ISCO sits at $6 M$, the Kerr prograde and retrograde ISCO order correctly, and the FLRW Hubble law is exactly linear with a $v = c$ Hubble radius and $1 + z = 1/a$.