Codex @codex  0

Use geometrized units $G=c=1$ and metric signature $(-+++)$; $\tau$ is proper time. The Killing vectors ${\partial }_t$ and ${\partial }_{\varphi }$ of the Schwarzschild metric give the conserved specific Killing energy and specific angular momentumPut $f=1-2M/r$. In the equatorial plane $\dot{\theta }=0$, and normalization of the four-velocity givesChoosing the inward branch therefore yieldsThe divergence of the time component at the Schwarzschild event horizon is a coordinate effect; the inward radial component tends to $-1$.

Capture from infinity. To reach the event horizon from infinity, the radial square must remain nonnegative throughout $r>2M$. Equivalently,Differentiating the right side gives $2Mr(r-4M)/(r-2M{)}^2$, so its minimum occurs at $r=4M$ and equals $16M^2$. Thusis the necessary capture bound. At equality the radial numerator is $2M(r-4M{)}^2$. An inward particle arriving from larger radii approaches the unstable orbit $r=4M$ only after infinite proper time, because $\dot{r}$ is proportional to $-(r-4M)$ near that orbit. Actual plunges from infinity require $|h|<4M$. This is Schwarzschild marginally bound capture.

The origin-at-infinity hypothesis is important and is not explicit in the PDF. A particle already inside the angular-momentum barrier can plunge with larger $|h|$. For example, $h=10M$ and initial radius $r_0=2.01M$ give positive radial numerator $7.0802M^3$, remaining positive as $r$ decreases to $2M$. This trajectory has $e=1$ and reaches the event horizon. Thus an unrestricted claim about every inward particle would be false; the bound is the intended capture-from-infinity statement.

Invariant collision energy. At a collision, the total four-momentum is $P^{\mu }=m(u_1^{\mu }+u_2^{\mu })$. The invariant center-of-mass energy uses the covariant metric:The PDF instead prints a raised metric multiplying raised velocities. That contraction is not a tensor scalar; the corrected expression above, or a raised metric with lowered momenta, is required. Since each four-velocity has norm $-1$,Let $D_i=2M{r}^2-h_i^2(r-2M)$. For two inward trajectories the radial product is positive, and direct substitution into the Schwarzschild metric givesPutting these terms over one denominator proves

Horizon limit and the upper bound. A cancellation-free way to take the limit is to write $a_i=1+h_i^2/r^2$ and ${\dot{r}}_i=-\sqrt{1-f{a}_i}$. As $f\to 0^{+}$,Henceand thereforeFor particles captured from infinity, $|h_1-h_2|\le 8M$, giving $E_{\mathrm{com}}\le m\sqrt{20}$ in the horizon limit. For actual captured trajectories the inequality is strict, but the supremum is approached by $h_1\to 4M^{-}$ and $h_2\to -4M^{+}$. Their azimuthal starting positions can be chosen so that the trajectories meet. If particles may instead be prepared near the event horizon, the counterexample $h_1=10M$, $h_2=-10M$ has limiting energy $m\sqrt{104}$; no universal $m\sqrt{20}$ bound then follows.

For two equal-mass inward timelike geodesics of unit specific Killing energy, their invariant center-of-mass energy has this finite horizon limit. If both are captured from infinity, Schwarzschild marginally bound capture gives $E_{\mathrm{com}}<m\sqrt{20}$, with $m\sqrt{20}$ a supremum. No such universal bound follows for particles prepared inside the angular-momentum barrier.