Relational Orbital Mechanics

Or: How to build a Universe without Mass and $G$

1. The Two Rulers of Reality

We describe the system solely through two measurable light projections:

1. Kinematic Projection ($\beta$): How fast is the relationship changing? (Doppler shift) \[ \beta = v/c \]
2. Gravitational Projection ($\kappa$): How deep is the relationship? (Gravitational Redshift) \[ \kappa \approx \sqrt{2z} \]

We don’t need Mass. We don’t need Newton’s Constant $G$. We just need these two numbers.

2. The Vector $Q$ (Relational Displacement)

Combine these two projections into a single vector $Q$ on your “Causal Radar”: \[ Q = \sqrt{\beta^2 + \kappa^2} \]

This vector defines the State of the System.

• Length of $Q$: Defines the total energy scale (Orbit size).
• Angle of $Q$: Defines the orbital shape (Eccentricity).

3. Interactive Demonstrator: The Causal Radar

Below is the WILL Relational Engine. It visualizes the “Phase Space” of reality.

How to use:

1. Drag the Triangle ($\triangle Q_p$): This represents the state of the orbit at its closest point (Periapsis).
2. Watch the Orbit: See how moving one single point changes the shape ($e$), stability ($\delta$), and precession ($\Delta\varphi$) of the entire system.
3. Find the Zones:
   • Green Dashed Line: The “Golden Ratio” $\kappa = \beta\sqrt{2}$. Perfect circles live here.
   • Red Dashed Line: The “Escape Horizon” $\kappa = \beta$. Below this, the bond breaks.
   • Inversion: Try dragging the point above the green line. See how the orbit flips!