Copilot's blunder

The change in convexity of c(t) during the spiral infall is a real signature of the nonlinear coupling between angular momentum and gravitational redshift in Schwarzschild spacetime. Let's unpack what's happening.

🔄 Why Does c(t) Change Convexity in the Spiral Case?

The key lies in the interplay between radial and angular motion:

1. In the radial case:

· The photon falls straight in.

· dφ/dt = 0, so the motion is purely radial.

· The coordinate speed c(t) = dr/dt decreases smoothly and monotonically, with a convex profile due to the redshift.

2. In the circular orbit:

· The photon is at the photon sphere (r = 3GM), moving purely tangentially.

· dr/dt = 0, so the coordinate speed is entirely angular.

· The speed is constant in r, but redshifted in t, again giving a smooth, convex decay in c(t). 

3. In the spiral case:

· The photon has both radial and angular components.

· As it spirals in, the angular velocity increases due to conservation of angular momentum (L = r² dφ/dt), while r decreases.

· But the radial component slows down due to gravitational time dilation.

This creates a competition:

· Early in the trajectory, the angular component dominates and increases rapidly.

· Closer to the horizon, the redshift dominates and suppresses both components.

This nonlinear transition causes the second derivative of c(t) to change sign — hence the inflection points and shifts in convexity

🧠 Deeper Interpretation

The spiral trajectory is the only one where the photon's motion is not separable into purely radial or purely angular components. The mixing of degrees of freedom under a strongly curved metric leads to:

· Nontrivial behavior in coordinate time.

· Apparent "acceleration" or "deceleration"  in the lab frame that reflects coordinate artifacts, not physical forces.

· This is a classic example of how coordinate choices in general relativity can produce surprising features — and why interpreting them requires care.