The Invariant: A Direct Measure of Orbital Energy

The core discovery of this simulation is demonstrated in the fourth box: the difference between the magnitudes of velocity and gravitational time dilation is an invariant quantity for any stable orbit. While the individual dilation effects are in constant flux, this specific combination remains fixed, revealing a deeper law at work.

This is a direct consequence of the conservation of energy. The derivation involves combining the relativistic time dilation formulas with the Vis-Viva equation—the law governing orbital velocity.

When the math is performed, the terms for the object's instantaneous radius (r) and velocity (v) completely cancel each other out.

The final result depends only on the semi-major axis (a), which defines the orbit's size and total energy. This proves that the invariant term is a direct measure of the total specific energy of the system, scaled by a constant (1/c²).

System vs. Local Time

This reveals two perspectives on time:

• Local Time (The 3rd Box): The sum of both effects is the constantly changing time rate an observer on the orbiting body would actually experience.
• System Time (The 4th Box): The invariant difference represents a single, unchanging number that characterizes the entire orbit's energy. It is a property of the system as a whole.

Therefore, what you are witnessing is that the conserved energy of an orbital system is encoded directly into the fabric of its local time dilation.