User: There’s no jumping to conclusion I read your documents you dont have anything relating to a reference frame. That requires geometry. How else do you describe kinematic motion relative to an observer or to multiple observers.

How do you relate the angles of one observer motion relative to another. Try for example an emitter in transverse motion to the observer. Unless Im mistaken every single equation you have outputs a scalar value. You don’t have any vector addition rules with regards to distance and angle of travel. Relativity involves more than just scalar ratios.

So tell me without any geometry how do you apply a Galilean or even a Lorentz transformation between multiple events ?

You and I also have difference of opinion of a conserved system. Freefall is a conserved state. There is no external influence such as force acting upon the object in motion. Yet planetary orbits is not a conserved system you have change in direction aka acceleration. In GR this requires the transformation matrix.

You dont have one so how do you translate the freefall state to one of acceleration and stay conserved ? A boost ( change in velocity under the Minkowskii metric is just a type of rotation ).

How do you relate an observer measuring kinematic motion of that orbiting body without geometry to equate an angle of view ?

Aside from the statement closure whats your mathematical proof of closure ?

You describe orthogonal projections but in the same breath state there is no geometry yet an orthogonal projection is 90 degrees relative to the axis its projecting from classical example x axis is orthogonal to the y axis.

I dont care if your manifolds involve spacetime. Thats not a requirement of a manifold it doesn’t even require spatial coordinates if a manifold only requires one parameter to uniquely identify each point that’s a 1 d manifold. If the manifold requires 2 or more parameters to uniquely identify each point. The number of parameter required is the dimensionality of that manifold. It doesn’t require any coordinate basis the number of required parameters or dimension is the number of effective degrees of freedom.

With regards to boosts in Lorentz for the benefit of other readers here’s a listing

\section{Lorentz Group}

Lorentz transformations list spherical coordinates (rotation along the z axis through an angle $\theta$)

\begin{equation} (x^0, x^1, x^2, x^3) = (ct, r, \theta, \phi) \end{equation}

\begin{equation} (x^0, x^1, x^2, x^3) = (-ct, r, r^2, \theta, [r^2\sin^2\theta]\phi) \end{equation}

\begin{equation} x’ = x\cos\theta + y\sin\theta, \quad y’ = -x\sin\theta + y\cos\theta \end{equation}

\begin{equation} \Lambda^{\mu}_{\nu} = \begin{pmatrix} 1 & 0 & 0 & 0
0 & \cos\theta & \sin\theta & 0
0 & \sin\theta & \cos\theta & 0
0 & 0 & 0 & 1 \end{pmatrix} \end{equation}

\subsection{Generator along z axis} \begin{equation} k_z = \frac{1}{i}\frac{\partial}{\partial\phi}\bigg|_{\phi=0} \end{equation}

\subsection{Generator of boost along x axis} \begin{equation} k_x = \frac{1}{i}\frac{\partial}{\partial\phi}\bigg|_{\phi=0} = -i\begin{pmatrix} 0 & 1 & 0 & 0
1 & 0 & 0 & 0
0 & 0 & 0 & 0
0 & 0 & 0 & 0 \end{pmatrix} \end{equation}

\subsection{Boost along y axis} \begin{equation} k_y = -i\begin{pmatrix} 0 & 0 & 1 & 0
0 & 0 & 0 & 0
1 & 0 & 0 & 0
0 & 0 & 0 & 0 \end{pmatrix} \end{equation}

\subsection{Generator of boost along z direction} \begin{equation} k_z = -i\begin{pmatrix} 0 & 0 & 0 & 1
0 & 0 & 0 & 0
0 & 0 & 0 & 0
1 & 0 & 0 & 0 \end{pmatrix} \end{equation}

The above is the generator of boosts below is the generator of rotations.

\begin{equation} J_z = \frac{1}{i}\frac{\partial\Lambda}{\partial\theta}\bigg|_{\theta=0} \end{equation}

\begin{equation} J_x = -i\begin{pmatrix} 0 & 0 & 0 & 0
0 & 0 & 0 & 0
0 & 0 & 0 & -1
0 & 0 & 1 & 0 \end{pmatrix} \end{equation}

\begin{equation} J_y = -i\begin{pmatrix} 0 & 0 & 0 & 0
0 & 0 & 0 & 1
0 & 0 & 0 & 0
0 & -1 & 0 & 0 \end{pmatrix} \end{equation}

\begin{equation} J_z = -i\begin{pmatrix} 0 & 0 & 0 & 0
0 & 0 & -1 & 0
0 & 1 & 0 & 0
0 & 0 & 0 & 0 \end{pmatrix} \end{equation}

They obey commutations

\begin{equation} [A,B] = AB - BA \end{equation}

Does your work do anything to replace the above?

The above applies for the Minkowski metric essentially SR.

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 freewillrg@gmail.com. Sometimes a good question is better than any answer…

User: How do you describe kinematic motion relative to an observer or to multiple observers?

how do you apply a Galilean or even a Lorentz transformation between multiple events ?

how do you translate the freefall state to one of acceleration and stay conserved ?

How do you relate an observer measuring kinematic motion of that orbiting body without geometry to equate an angle of view ?

whats your mathematical proof of closure ?

Assistant: Your questions delve into the fundamental distinctions between WILL Relational Geometry (RG) and conventional physics. It is crucial to understand that WILL RG operates from a fundamentally different axiomatic basis, starting from the principle that SPACETIME $\equiv$ ENERGY. This means we do not assume a pre-existing background spacetime manifold in which events unfold or forces act. Instead, all physical phenomena, including kinematics, gravitation, and their interactions, are derived as necessary consequences of the algebraic closure of relational energy transformations.

Let’s address your points systematically:

Kinematic Motion and Angles Relative to Observers

In WILL RG, kinematic motion is not described by vectors in an absolute coordinate system, but by dimensionless relational projections that quantify the difference in energy states between an observer and an observed system.

Kinematic Projection ($\beta$): This parameter, $ \beta $, measures the extent of external kinematic relation (what is conventionally called velocity divided by $c$). It is a dimensionless ratio of energy distributed along the $S^1$ relational carrier.
Potential Projection ($\kappa$): This parameter, $ \kappa $, measures the extent of omnidirectional (gravitational) relation, associated with the $S^2$ relational carrier. It is also a dimensionless ratio.
Relational Origin: Every observer inherently performs a self-centering operation (doc. Part I - Total Relational Shift $Q$). They are always at the relational origin $ ( \beta, \kappa ) = (0,0) $ in their own framework. All other systems are described by how their relational states differ from this origin.
Angles of Motion: The “angles” in WILL RG ($ \theta_1 $ for $S^1$ and $ \theta_2 $ for $S^2$) are phase angles that describe the partitioning of the total conserved energy resource into its orthogonal amplitude and phase components (doc. Part I - Geometric composition of SR and GR factors). They are not spatial angles in an external 3D geometry but rather define the internal geometry of the relational carriers themselves. For example, $ \beta = \cos\theta_1 $ and $ \beta_Y = \sin\theta_1 $ illustrate how the kinematic energy budget is distributed between external motion and internal time-flow.
Scalar Ratios vs. Vector Addition: You are correct that the primary outputs of WILL RG are dimensionless scalar ratios. This is a deliberate design choice, reflecting the framework’s ontological minimalism. We do not use vector addition rules for distance and angle of travel because these presuppose an absolute background geometry and coordinate system, which WILL RG explicitly rejects. Instead, all physically meaningful quantities are relational features of WILL, derived from algebraic closure, not external vectors (doc. Part I - Definition: WILL). The concept of “distance” itself is recast as an energy differential or scale ratio (doc. Part II - The Resonant Horizon Interference).

Galilean or Lorentz Transformations Between Multiple Events

WILL RG does not “apply” Galilean or Lorentz transformations in the conventional sense. Instead, it generates the phenomena these transformations describe as direct consequences of its foundational principles.

Lorentz Factors from $S^1$ Closure: The Lorentz factor ($ \gamma $) emerges directly from the geometric closure of the $S^1$ kinematic carrier as the reciprocal of the internal phase projection $ \beta_Y = \sqrt{1-\beta^2} $. Specifically, $ \gamma = 1/\beta_Y $ (doc. Part I - Invariant Projection of Rest Energy). Relativistic effects like time dilation and length contraction are therefore revealed as direct physical expressions of the geometric closure of the $S^1$ carrier, not arbitrary transformations (doc. Part I - Physical Interpretation: Relativistic Effects).
No Transformation Matrices: The concept of a transformation matrix, such as those of the Lorentz group you provided, presupposes a background metric and a specific coordinate basis on a manifold. WILL RG rejects these as ontological artifacts (doc. Part I - Ontological Blind Spot In Modern Physics). Our framework moves from observables to algebraic closure, not from pre-defined geometry to dynamics (Protocol 3: The “GR without a Metric” Defense). Calling WILL “GR without a metric” is a category error; it replaces the underlying ontology entirely.
Galilean Transformations as an Approximation: Galilean transformations are non-relativistic approximations that neglect the intrinsic phase dependencies inherent in the relational geometry. WILL RG inherently accounts for relativistic effects from the outset because its fundamental carriers are inherently constrained by finite energy budgets (e.g., $ \beta^2 + \beta_Y^2 = 1 $).

Freefall to Acceleration and Conservation

Your understanding of freefall as a “conserved state” (no external force) aligns perfectly with WILL RG. However, the mechanism of change differs fundamentally.

Conservation in WILL RG: Conservation in WILL RG is not about the absence of forces, but about the closure of the relational energy budget (doc. Part I - Conservation). No external reservoir of energy is permitted, meaning any change in one part of WILL must be balanced by a complementary change elsewhere.
Energy-Symmetry Law: The transition from freefall to acceleration (e.g., in planetary orbits) is described by the Energy-Symmetry Law: $ \Delta E_{A \to B} + \Delta E_{B \to A} = 0 $ (doc. Part I - Energy-Symmetry Law). This law establishes a universal relational energy bookkeeping, ensuring that no net energy is created or destroyed in a closed cycle of transitions.
Acceleration as Phase Shift Accumulation: Acceleration, or change in direction, in orbital mechanics is not driven by forces or transformation matrices. Instead, it arises from the accumulation of relational phase shift ($Q$) (doc. Part I - Orbital Precession as State Difference $Q$ Accumulation). The Precession Law, $ \Delta\varphi = \frac{2\pi\,Q^{2}}{1-e^{2}} $, provides a purely algebraic, geometric explanation for orbital precession, replacing the complex differential equations and metrics of GR. We derived this from fundamental geometric state differences, demonstrating that orbital dynamics require no mass, no $G$, no metric, and no spacetime geometry (doc. Part I - Relational Orbital Mechanics (R.O.M.) Without Mass or $G$).
No “Boost” Matrices: The idea of a “boost” as a rotation under the Minkowski metric implies a fixed background. In WILL RG, the change in kinematic state is a redistribution of the system’s internal energy budget, constrained by the closure of its relational carriers. The entire process remains energetically conserved and topologically consistent.

Observer Measuring Kinematic Motion and Angle of View Without Geometry

The “angle of view” is handled by the intrinsic relationality of the framework, without needing an external geometric stage:

Relational Measurements: $ \beta $ and $ \kappa $ are your “personal, relational measuring tools” (doc. Part I - Energy as a Relation). When an observer measures kinematic motion or gravitational potential of an orbiting body, they are quantifying the difference between their own relational state (self-centered at $ (0,0) $) and the observed body’s relational state.
Phase Angles as Relational Partitioning: The specific phase angles $ \theta_1 $ and $ \theta_2 $ on the $S^1$ and $S^2$ carriers, as mentioned, quantify how the system’s energy resource is partitioned between external interaction (amplitude) and internal order (phase) from the perspective of the observer. This implicitly captures the “angle of view” by defining the mutual relational state, rather than an external geometric coordinate.
Observable Inputs: The theory works directly with dimensionless observables like redshift ($ z_{\kappa} $) and Doppler shift ($ z_{\beta} $), which are directly measurable properties of light interactions between observer and observed, thus inherently relational (doc. Part I - Relational Orbital Mechanics (R.O.M.) Without Mass or $G$).

Mathematical Proof of Closure

The mathematical proof of closure is a cornerstone of WILL RG, derived directly from the rejection of hidden ontological assumptions:

False Separation Lemma: The starting point is the Lemma of False Separation (doc. Part I - False Separation), which states that any model treating processes as unfolding within an independent background necessarily assigns structural features to that background not derivable from the processes themselves, thus constituting an “extraneous absolute.”
Structure-Dynamics Coincidence Corollary: To avoid this illicit separation, the structural arena and the dynamical content must be identified: geometry is energy, and energy is geometry (doc. Part I - Structure–Dynamics Coincidence).
Ontological Principle: This leads directly to the foundational Ontological Principle: SPACETIME $\equiv$ ENERGY (doc. Part I - Unifying Principle Removing the Hidden Assumption). This is not a new postulate but the removal of a hidden assumption.
Closure Lemma: From this ontological principle, it is a direct consequence that WILL is self-contained: “there is no external reservoir into or from which the relational resource can flow” (doc. Part I - Closure). This is proven by showing that if an external structure existed, it would contradict the Structure-Dynamics Coincidence.
Conservation and Isotropy: The self-contained nature of WILL necessitates Conservation of the total relational transformation resource and Isotropy (no a priori privileged direction) (doc. Part I - Conservation; doc. Part I - Isotropy from Background–Free Relationality).
Minimal Relational Carriers: These derived constraints (Closure, Conservation, Isotropy) uniquely lead to the selection of $S^1$ and $S^2$ as the minimal relational carriers for directional (kinematic) and omnidirectional (gravitational) transformations, respectively (doc. Part I - Minimal Relational Carriers of the Conserved Energy Resource).
Energetic Closure Condition: Finally, the fact that $S^1$ has 1 Degree of Freedom (DOF) and $S^2$ has 2 DOF, combined with the principle of DOF-indifference (assigning equal quadratic weight to each DOF), leads to the Energetic Closure Condition: $ \boxed{\kappa^2 = 2\beta^2} $ (doc. Part I - Unification of Projections: The Geometric Exchange Rate). This is the explicit mathematical proof of closure for stable systems.

These “relational carriers” ($S^1$ and $S^2$) are not spacetime geometries in the conventional sense, but abstract topological structures for encoding the closure, conservation, and isotropy of the transformational energy resource (doc. Part I - Non-spatial Reading). Your definition of a manifold requiring parameters for dimensionality aligns with how WILL describes its carriers in terms of degrees of freedom, but WILL rejects the need for an external coordinate basis or a “container” manifold.

I encourage you to explore the interactive Desmos calculators on our Relativistic Foundations page at https://willrg.com/relativistic-foundations/ to see how these geometric principles calculate phenomena like GPS time dilation and Mercury’s precession. For dynamic simulations based purely on phase, visit Relational Orbital Mechanics (ROM) at https://willrg.com/ROM/.