All Conversations
March 17, 2026 at 18:28 UTC

User: Rigorous physicist

Assistant: Confirmed. Mathematical engagement level set to Rigorous physicist. How can I assist you with WILL Relational Geometry?

User: Hi WILL-AI! Anton Rize here. Lets think together - can we get rid of the mass and G in my Earth - GPS calculations as elegant as we did for Mercury - Sun system?

Here’s my current desmos set up for Earth - GPS calculations:

settings @{ product: “geometry-calculator”, randomSeed: “d42ee8d6409f2fc143826e4bc9ed0b00”, viewport: @{ xmin: -5.715304690921283, xmax: 16.862820309079442, ymin: -17.207645391226784, ymax: 14.068588058352509, }, xAxisMinorSubdivisions: 1, yAxisMinorSubdivisions: 1, xAxisNumbers: false, yAxisNumbers: false, polarNumbers: false, }

folder”geometry”{

}@{id:”dcg_geo_folder“,secret:true}

“WILL=1. Energy-Symmetry of Earth-GPS system, Earth-GPS time shift\nhttps://willrg.com/documents/WILL_RG_I.pdf#sec:geometric_composition\nhttps://willrg.com/documents/WILL_RG_R.O.M..pdf#eq:rom\nhttps://willrg.com/documents/WILL_RG_I.pdf#sec:energy-symmetry\nhttps://willrg.com/documents/WILL_RG_I.pdf#sec:willinvariant”

folder”RESULTS”{ “Distance to GPS stellate. Feel free to play around with it to discover invariant parameters. Real distance is r_GPS = 26571000 m”

r_GPS=26571000@{color:”#c74440”,slider:@{min:0,max:26571000}}

“One minus Ratio of multiplied orthogonal projections gives us the precise time correction used in atomic clocks on board GPS satellites in microseconds per day. “

T_imedif=(1-tau_Earth/tau_GPS)8640010^6@{color:”#6042a6”}

“Real world system calculation confirms the WILL invariant:”

W_ILLGPS=(E_GPST_GPS)/(M_GPSL_GPS)@{color:”#2d70b3”}

“Real world system calculation confirms the Energy Symmetry Law:”

E_SYMETRY=E_gEarth2GPS+E_gGPS2Earth@{color:”#6042a6”}

“Conformation of nontriviality of Energy Symmetry Law. Unitless geometrically derived energy E_{gEarth2GPS} is = to E_{totnorm}.\nE_{totnorm} is a classical full energy (E_{tot}) normalized by its rest energy (E_{0GPS}):”

E_totnorm/E_gEarth2GPS@{color:”#2d70b3”}

“Remarkably, satellite mass m_{GPS} does not appear anywhere in the computation of the geometric energy E_{gEarth2GPS}. Yet, when the total physical energy E_{tot} (the sum of kinetic and potential energies, which does depend on m_{GPS} is normalized by the satellite’s rest energy E_{0GPS} = m_{GPS} c^2 the ratio exactly matches the unitless geometric value as shown above. This confirms that E_{gEarth2GPS} precisely encodes the relationship between the system’s real energy and the satellite’s own rest energy, entirely independent of its absolute mass.\nIn other words, the geometric projection captures the physical “share” of energy - purely as a relational quantity - regardless of the object’s individual mass.”

E_gGPS2Earth=(1/2)*(kappa_GPS^2-kappa_Earth^2+(beta_Earth^2-beta_GPS^2))@{ color:”#2d70b3”, }

E_gEarth2GPS=(1/2)*(kappa_Earth^2-kappa_GPS^2+(beta_GPS^2-beta_Earth^2))@{ color:”#388c46”, }

“Lets calculate standard protentional and kinetic energy’s “

E_pGPS=-((GM_Earthm_GPS)/r_GPS)–((GM_Earthm_GPS)/R_Earth)@{ color:”#c74440”, }

E_kGPS=(1/2)*m_GPS(v_GPS^2-v_Earthspin^2)@{color:”#000000”}

“Lets combine them to derive total energy”

E_tot=E_pGPS+E_kGPS@{color:”#c74440”}

“Lets make it unitless by normalizing over rest energy”

E_totnorm=E_tot/E_0GPS@{color:”#c74440”}

“Lets compare classical and relational energies”

E_totnorm/E_gEarth2GPS@{color:”#388c46”} }

folder”GPS”{ 2.6571 cross 10^7=r_GPS@{color:”#c74440”}

m_GPS=600@{color:”#6042a6”,hidden:true}

E_0GPS=m_GPS*c^2@{color:”#2d70b3”}

v_GPS=sqrt((G*M_Earth)/r_GPS)@{color:”#2d70b3”}

beta_GPS=sqrt((GM_Earth)/(c^2r_GPS))@{color:”#388c46”}

kappa_GPS=sqrt((2GM_Earth)/(c^2*r_GPS))@{color:”#6042a6”}

kappa_XGPS=sqrt(1-kappa_GPS^2)@{color:”#c74440”}

beta_YGPS=sqrt(1-beta_GPS^2)@{color:”#2d70b3”}

tau_GPS=kappa_XGPS*beta_YGPS@{color:”#388c46”}

Q_GPS=sqrt(beta_GPS^2+kappa_GPS^2)@{color:”#000000”}

Q_tGPS=sqrt(1-Q_GPS^2)@{color:”#c74440”} }@{collapsed:true}

folder”WILL GPS”{ M_GPS=(beta_GPS^2/beta_YGPS)c^2(r_GPS/G)@{color:”#2d70b3”}

E_GPS=(kappa_GPS^2/kappa_XGPS)((c^4r_GPS)/(2*G))@{color:”#6042a6”}

T_GPS=kappa_XGPS((2GM_Earth)/(kappa_GPS^2*c^3))^2@{color:”#388c46”}

L_GPS=beta_YGPS((GM_Earth)/(beta_GPS^2c^2))^2@{color:”#c74440”} }@{collapsed:true}

folder”Earth”{ M_Earth=5.972 cross 10^24@{color:”#6042a6”}

R_Earth=6.37 cross 10^6@{color:”#000000”}

v_Earthspin=465@{color:”#c74440”,slider:@{min:465,max:465,step:465}}

kappa_Earth=sqrt((2GM_Earth)/(c^2*R_Earth))@{color:”#000000”}

beta_Earth=v_Earthspin/c@{color:”#c74440”,slider:@{min:0}}

beta_YEarth=sqrt(1-beta_Earth^2)@{color:”#000000”}

kappa_XEarth=sqrt(1-kappa_Earth^2)@{color:”#6042a6”}

tau_Earth=kappa_XEarth*beta_YEarth@{color:”#c74440”}

Q_Earth=sqrt(beta_Earth^2+kappa_Earth^2)@{color:”#2d70b3”}

Q_tEarth=sqrt(1-Q_Earth^2)@{color:”#388c46”} }@{collapsed:true}

folder”Constants”{ G=6.6743 cross 10^-11@{color:”#2d70b3”}

c=2.99792458 cross 10^8@{color:”#388c46”}

D_aySeconds=86400@{color:”#000000”,slider:@{min:86400,max:86400}}

M_icro=10^6@{color:”#c74440”} }@{collapsed:true}

Assistant: Your request is very complex and the AI took too long to think. You can try asking the same question again or try simplifying it a little or even better - you can ask Anton Rize antonrize@willrg.com. Sometimes a good question is better than any answer…

User: Hi WILL-AI! Anton Rize here. Lets think together - can we get rid of the mass and G in my Earth - GPS calculations as elegant as we did for Mercury - Sun system?

Assistant: Your request is very complex and the AI took too long to think. You can try asking the same question again or try simplifying it a little or even better - you can ask Anton Rize antonrize@willrg.com. Sometimes a good question is better than any answer…

User: Hi WILL-AI! Anton Rize here. Lets think together - can we get rid of the mass and G in my Earth - GPS calculations as elegant as we did for Mercury - Sun system?

Assistant: Hello Anton! It is a pleasure to refine the logic of the framework together. To eliminate $M$ (Mass) and $G$ (Gravitational Constant) from the Earth-GPS calculations, we must transition from Descriptive Physics (using parameters like mass to model gravity) to Generative Physics (using dimensionless relational projections $\kappa$ and $\beta$ as primary observables).

Just as we did for the Mercury-Sun system in WILL RG Part I - Path 1: Verification on Mercury, we can anchor the entire GPS calculation to a single spectroscopic measurement: the Gravitational Redshift at the Earth’s surface ($z_{\kappa, Earth}$).

The Relational Strategy

The core identity of the WILL framework is that spacetime is emergent from energy transformations. In the Relational Orbital Mechanics (R.O.M.), we replace the “force of gravity” with the potential projection $\kappa^2$ and “velocity” with the kinematic projection $\beta^2$.

To calculate the GPS time shift without $G$ or $M$, we follow these four steps:

1. Establish the Surface Anchor (Earth)

Instead of calculating $\kappa_{Earth}$ from $M$ and $G$, we treat the gravitational redshift at sea level, $z_{\kappa, Earth} \approx 6.9612 \times 10^{-10}$, as our primary spectroscopic input. The potential projection is then:

\[\kappa_{Earth}^2 = 1 - \frac{1}{(1 + z_{\kappa, Earth})^2}\]

For the kinematic projection, we use the Earth’s surface rotation velocity $v_{rot}$ (at the observer’s latitude) normalized by $c$:

\[\beta_{Earth} = \frac{v_{rot}}{c}\]

The Relational Spacetime Factor for the ground observer is:

\[\tau_{Earth} = \sqrt{1 - \kappa_{Earth}^2} \cdot \sqrt{1 - \beta_{Earth}^2}\]

2. Scale the Potential to the GPS Orbit

In the Unified Geometric Field Equation, we established that the potential projection scales linearly with the inverse of the radius ($\kappa^2 \propto 1/r$). Therefore, we can find the potential at the GPS altitude using a simple geometric ratio:

\[\kappa_{GPS}^2 = \kappa_{Earth}^2 \cdot \left( \frac{R_{Earth}}{r_{GPS}} \right)\]

This step replaces the need for the Schwarzschild radius formula $R_s = 2GM/c^2$. The “mass scale” is already encoded in the ratio of the surface potential to the radius.

3. Invoke Energetic Closure for the Satellite

Since the GPS satellite is in a stable circular orbit, it must satisfy the Closure Condition ($\kappa^2 = 2\beta^2$). This allows us to derive the satellite’s kinematic projection $\beta_{GPS}$ purely from its potential $\kappa_{GPS}$:

\[\beta_{GPS}^2 = \frac{\kappa_{GPS}^2}{2}\]

This replaces the Newtonian circular velocity formula $v = \sqrt{GM/r}$. The satellite’s motion is determined by its energetic balance with the vacuum structure at that scale.

4. Calculate the Total Time Shift

The net time shift is the ratio of the internal phase projections (Phase $\to$ Internal Order) WILL RG Part I - Duality of Evolution:

\[\tau_{GPS} = \sqrt{1 - \kappa_{GPS}^2} \cdot \sqrt{1 - \beta_{GPS}^2}\] \[\Delta T = \left( 1 - \frac{\tau_{Earth}}{\tau_{GPS}} \right) \cdot 86400 \cdot 10^6 \quad [\mu s/\text{day}]\]

The Clean Algebraic System (No $G$, No $M$)

Here is the “Elegant” folder setup for your Desmos project:

Inputs:

Equations: \(\kappa_{Earth}^2 = 1 - (1 + z_{\kappa, Earth})^{-2}\) \(\beta_{Earth}^2 = (v_{rot}/c)^2\) \(\kappa_{GPS}^2 = \kappa_{Earth}^2 \cdot (R_{Earth}/r_{GPS})\) \(\beta_{GPS}^2 = \kappa_{GPS}^2 / 2\) \(\tau_{Earth} = \sqrt{(1 - \kappa_{Earth}^2)(1 - \beta_{Earth}^2)}\) \(\tau_{GPS} = \sqrt{(1 - \kappa_{GPS}^2)(1 - \beta_{GPS}^2)}\) \(\text{Time Shift} = (1 - \tau_{Earth}/\tau_{GPS}) \cdot 86400 \cdot 10^6\)

Why this is Ontologically Superior

As described in WILL RG Part I - The contemporary split, standard physics duplicates variables by having a background “structure” (metric) and “dynamics” (mass).

By removing $G$ and $M$, you are proving the Structure-Dynamics Coincidence (Corollary 5.2). You no longer have an Earth mass “causing” a curvature; you simply have a system of relational projections ($\kappa, \beta$) that must maintain algebraic closure to exist. The fact that this produces the correct $38.5 \mu s/day$ confirms that the physical “share” of energy is a relational quantity, independent of the labels we attach to the source.

For further verification, you can test this logic with our Desmos GPS Calculator on the website.