User: How did you derive this R_s = \frac{\beta_1^2 - \beta_2^2}{\frac{r_2-r_1}{r_1 r_2}} = \frac{r_1 r_2}{r_2 - r_1} (without invoking G or M)?

Assistant: The derivation of the Schwarzschild radius $R_s$ without invoking the gravitational constant $G$ or mass $M$ in WILL Relational Geometry (RG) comes directly from the principle of the Conservation of the Relational Invariant ($W$). This principle states that for any closed system, the specific energy difference between the potential ($\kappa^2$) and kinematic ($\beta^2$) projections remains constant throughout the orbit (doc. Part I - Section Energy-Symmetry Law).

Here’s the step-by-step derivation, as presented in WILL Part I (Section 15.2):

1. The Relational Invariant: The core of this derivation is the conserved energy invariant $W$. For a bound system, the specific energy invariant (per unit rest energy) is given by:

   \[\boxed{W = \frac{1}{2}(\kappa^2 - \beta^2)}\]

   This invariant is conserved at every point along an orbit.

2. Applying the Invariant to Two Points: Consider any two arbitrary points, 1 and 2, along the trajectory of an orbiting object. Since $W$ is conserved, the invariant at point 1 must equal the invariant at point 2:

   \[\frac{1}{2}(\kappa_1^2 - \beta_1^2) = \frac{1}{2}(\kappa_2^2 - \beta_2^2)\]

   We can simplify this by canceling the factor of $1/2$:

   \[\kappa_1^2 - \beta_1^2 = \kappa_2^2 - \beta_2^2\]

   Rearranging the terms to group potential and kinematic projections:

   \[\kappa_1^2 - \kappa_2^2 = \beta_1^2 - \beta_2^2\]

3. The Potential Projection ($\kappa^2$) Definition: In WILL RG, the potential projection $\kappa^2$ is defined as the dimensionless ratio of the system’s Schwarzschild radius ($R_s$) to the current radial distance ($r$) (doc. Part I - Section Potential Energy Projection on $S^2):$

   \[\kappa^2 = \frac{R_s}{r}\]

   It’s crucial to understand that $R_s$ here is treated as a geometric scale factor, not as a parameter directly dependent on $G$ or $M$. Its value is inferred from relational observations.

4. Substitution and Algebraic Solution: Now, substitute the definition of $\kappa^2$ into the invariant equation:

   \[\frac{R_s}{r_1} - \frac{R_s}{r_2} = \beta_1^2 - \beta_2^2\]

   Factor out $R_s$ from the left-hand side:

   \[R_s \left( \frac{1}{r_1} - \frac{1}{r_2} \right) = \beta_1^2 - \beta_2^2\]

   Combine the terms within the parentheses:

   \[R_s \left( \frac{r_2 - r_1}{r_1 r_2} \right) = \beta_1^2 - \beta_2^2\]

   Finally, solve for $R_s$:

   \[\boxed{R_s = \frac{\beta_1^2 - \beta_2^2}{\frac{r_2-r_1}{r_1 r_2}} = \frac{r_1 r_2}{r_2 - r_1}(\beta_1^2 - \beta_2^2)}\]

This derivation (doc. Part I - Section 15.2, Theorem 15.2) shows that $R_s$ can be determined purely from two measurements of radial position ($r_1, r_2$) and their corresponding kinematic projections ($\beta_1, \beta_2$). These kinematic projections ($\beta = v/c$) are directly observable through Doppler shifts, and radii ($r$) are observable through astrometry. No information about the central mass $M$ or the gravitational constant $G$ is required as input for this calculation. $G$ and $M$ become output calibration tools if one wishes to translate $R_s$ into conventional units, rather than fundamental inputs (doc. Part I - Section The Role of G as Translation Constant).

You can explore this method and test it yourself using the interactive Desmos calculators available on the Relativistic Foundations page of the WILL RG website.