A Photon or a Probe at a Black Hole in the Schwarzschild model

Part 1

Motion of a Photon at a Black Hole:

as a trajectory r(t) and as a path r(φ)

Part 2

Motion of a Probe at a Black Hole:

as a trajectory r(t) and as a path r(φ)

This study was possible due to the fact that the Einstein's equations of the Theory of General Relativity in partial derivatives were reduced to Ordinary Differential Equations (ODEs) in the case of the Schwarzschild special spacetime model. Thanks to this fact, we could use the TCenter software for integration and kinematic animation of the trajectories of the solutions.

All along this study, the time t in trajectory r(t) means the time at a remote inertial system of coordinates in the Schwarzschild model.

In Schwarzschild spacetime, the coordinate time t used in trajectories r(t) is the time of the asymptotic inertial frame at infinity.

Because General relativity does not allow a global notion of simultaneity, distant events cannot be assigned a common "now" or synchronized physical time.

A remote laboratory receives only delayed signals carrying the proper times of local clocks along the probe’s path. From these retarded signals, and using the known geometry of Schwarzschild spacetime, the lab can reconstruct the Schwarzschild coordinate time t assigned to each event.

Thus, the time t in r(t) is not directly measured but is a mathematically reconstructed coordinate time consistent with the Schwarzschild chart.

Acknowledgement: many formulas and Ordinary Differential Equations (ODEs) in both parts were obtained mostly as queries to  the AI system Copilot by Microsoft.