15 hours ago, Anton Rize said:

You claim Rs explicitly depends on M and G.

That's incorrect.

No it is not. Rs of the sun is not the same as Rs of earth, for example. It depends on the mass of the body in question, as well as the relative strength of gravity.

15 hours ago, Anton Rize said:

You are mistaken. z_sun is a raw observable we take from spectroscopy. Neither G or M needed for its measurement.

Again, the gravitational potential depends on both the mass of the body as well as the relative strength of gravity. The potential function of the sun isn’t the same as that of earth. I don’t know what “measurement” has to do with the basic fact that not all bodies share the same potential.

15 hours ago, Anton Rize said:

The system scale is determined by the orbit itself, not a "Mass" label.

The orbit itself depends on M and G.

15 hours ago, Anton Rize said:

Both are valid boundary conditions. But mine do not require the assumption of a "Mass" entity

So why do you observe those quantities, orbits etc to be different for different bodies? What is it about those bodies that makes them different?

You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

@Mordred Great! Now I can see that you actually red something.
Let me answer your questions:

2 hours ago, Mordred said:

You have parameters that do vary by definition you have effective degrees of freedom. They may or may be independent degrees of freedom.

The system is algebraically locked. So the change in 1 parameter inevitably triggering the cascade of changes in all codependent parameters in order to conserve invariants.

2 hours ago, Mordred said:

You also have trigonometric relations between your effective degrees of freedom within your article. So direction is inherent in your S^2 manifold regardless of what parameters you use to determine each unique point on said 2d manifold.

Not quiet. This is a very common misreading. That's exactly why I'm emphasising in the document: https://willrg.com/documents/WILL_RG_I.pdf#rem:nonspatial
"Throughout this paper, S^1 and S^2 are not to be interpreted as spacetime geometries. They are relational carriers that encode the closure, conservation, and isotropy of the transformational resource. Ordinary spatial and temporal notions are emergent descriptors of patterns within WILL.".

Here's the ontology clarification:

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Ontological Status of the Relational Carriers [math]S^1[/math] and [math]S^2[/math]

A natural question arises regarding the ontological status of the circle [math]S^1[/math] and the sphere [math]S^2[/math]: What are they, and where do they "exist"?

The answer requires a shift in perspective. In WILL Relational Geometry, [math]S^1[/math] and [math]S^2[/math] are not spatial entities existing within a pre-defined container. They are the necessary relational architectures that implement the core identity [math]\text{SPACETIME} \equiv \text{ENERGY}[/math].

Energy as Relational Transformation Capacity

Recall that energy is defined as the relational measure of difference between possible states. It is not an intrinsic property but a relational potential for change. It is never observed directly, only through transformations.

The Carriers as Protocols of Interaction

The Carriers [math]S^1[/math] and [math]S^2[/math] are the minimal, unique mathematical structures capable of hosting this relational "bookkeeping" for directional and omnidirectional transformations, respectively. They enforce closure, conservation, and symmetry by their very topology.

Imagine two observers, [math]A[/math] and [math]B[/math]:

* Observer [math]A[/math] is the center of their own relational framework. Observer [math]B[/math] is a point on [math]A[/math]'s [math]S^1[/math] (for kinematic relations) and [math]S^2[/math] (for gravitational relations).

* Simultaneously, observer [math]B[/math] is the center of their own framework. Observer [math]A[/math] is a point on [math]B[/math]'s [math]S^1[/math] and [math]S^2[/math].

There is no privileged "master" carrier. Each observable interaction is structured by these mutually-centered relational protocols. The parameters [math]\beta[/math] and [math]\kappa[/math] are the coordinates within these relational dimensions, and the conservation laws (e.g., [math]\beta^2 + \beta_Y^2 = 1; \quad \kappa_X^2 + \kappa^2 = 1[/math]) are the innate accounting rules of these protocols.


So its like Im looking at you and asking how is your state is different from mine? Because my velocity and potential in my rest frame are always [math](\beta, \kappa)= (0, 0) [/math] - Im always the origin of my relational frame. Therefor I can map your state reletive to mine as a point on [math](\beta, \kappa) [/math] plane. And because the same rules applies to you, you can map me as a point on [math](\beta, \kappa) [/math] plane of your relational frame. So the total relational shift [math]Q=\sqrt{\beta^2+\kappa^2}[/math] stays invariant between frames. And that's what Principal of Relational Reciprocity is all about. Its way simpler then boosts, spacetime intervals, geodesics, Christoffel symbols, tensors etc...

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3 hours ago, Mordred said:

Now why am I pointing this out. Well fundamentally your going to eventually want to take these relations and eventually apply them to some coordinate system otherwise how do you determine the force the sun exerts on mercury when you have a force with a 1/r^2 relation.

Im not really using the concept of force but we can translate in to force if you want:
[math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math]

Where:

[math]\frac{F}{E_{0}} = \frac{R_{s}}{2r_{o}(o)^{2}} \equiv \frac{\kappa_{o}(o)^{2}}{2r_{o}(o)}[/math]

Where:

[math]E_{0}[/math] = rest energy

[math]o[/math] = orbital phase in radians

[math]\kappa_{o} = \sqrt{\frac{R_{s}}{r}} = 1-(1+z_{ko}(o))^{-2}[/math] (local potential projection at phase [math]o[/math])

[math]r = r_o(o)= a\frac{1-e^{2}}{1+e\cos o} = \frac{R_s}{\kappa_o^2}[/math] (radial distance at phase [math]o[/math])

[math]R_{s} = \kappa^{2}a = \frac{2Gm_{0}}{c^{2}} = \frac{2}{3}Q_{o}(O_{o})^{2}a = \frac{r_{1} r_{2}}{r_{2} - r_{1}}(\beta_{1}^{2} - \beta_{2}^{2}) =[/math]
[math] \frac{a}{2}(3-\sqrt{1+8\tau_{Wo}(O_{o})^{2}}) = \frac{r_{o}(o)}{2(2a-r_{o}(o))}(4a-r_{o}(o)-\sqrt{(4a-r_{o}(o))^{2}-8a(2a-r_{o}(o))(1-\tau_{Wo}(o)^{2})})[/math] (Schwarzschild radius - system scale)
[math]z_{ko}(o) = \frac{1}{\kappa_{Xo}(o)}-1[/math] (gravitational redshift at phase o)

4 hours ago, Mordred said:

The title of your thread specifically states "Simplifying SR and GR" yet I don't anything relating to observer effects and what different observers will see or measure. Relative motion from one frame of reference to another etc. I find that curious as well evidently its not in the scope of the work

I guess I already answered this one above with relational reciprocity. In your case you jumping to conclusions about the scope without any proper reasoning. Look I understand its a lot to take in and you dont have to if you dont want. The only thing I demand is scientific honesty. So jumping in to conclusion after misreading few pages is scientifically dishonest.

54 minutes ago, Markus Hanke said:

You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

Thats ok we tried second time and result is the same. Im just fascinated how can't you see that all your arguments are based on just an interpretation. Regardless...
this is mathematical and empirical fact [math]\Delta_{precession} = \frac{3\pi}{2} (\frac{R_{ratio}}{\beta_{p}} (1-(1+z_{sun})^{-2}))^2=5.0175347157\times10^{-7}[/math] if you instead of facing it and openly discuss it prefer to hide behind dogmatic thinking - its your choice.

4 hours ago, Mordred said:

I don't anything relating to observer effects and what different observers will see or measure.

I dont get it. What's your reasoning here? Dont you think that if you dont see it doesnt mean that it doesnt exist? Dont you think that look again or ask me would be more reasenoble then jump in to conclusion based on you glassing through the paper? If thats how you do science Id better doublecheck the accuracy of your cosmic calculator...

Edited 16 hours ago16 hr by Anton Rize

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

Lorentz group

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

\[(x^0,x^1,x^2,x^3)=(ct,r,\theta\phi)\]

\[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\]

 

\[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\]

\[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\]

generator along z axis

\[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\]

generator of boost along x axis::

\[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\]

boost along y axis\

\[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\]

generator of boost along z direction

\[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\]

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

\[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\]

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

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

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

they obey commutations

\[[A,B]=AB-BA\]

Does your work do anything to replace the above ?

The above applies for the Minkowskii metric essentially SR.

@Markus Hanke, I believe what @Anton Rize is saying is that Schwarzschild radius can act as a substitute for mass. In other words, mass and Schwarzschild radius are equivalent, and this equivalence can be applied even if the mass is not of a spherical object.

[math]r_s = \dfrac{2GM}{c^2}\ \ \ \ \ ;\ \ \ \ \ M = \dfrac{c^2 r_s}{2G}\ \ \ \text{or}\ \ \ GM = \dfrac{c^2}{2}\ r_s[/math]

Thus, any formula with [math]M[/math] or [math]GM[/math] can be replaced with the corresponding formula with [math]r_s[/math].

It is interesting to note that the radius of a Newtonian "black hole" (where the escape velocity at the surface is [math]c[/math]) is the Schwarzschild radius of its mass. I think this and other formulae for which the Newtonian version is the same as general relativity is due to the special nature of the [math]r[/math] coordinate of the Schwarzschild metric (the surface area at [math]r[/math] is equal to the Euclidean surface area). In this discussion, this appears to create an uncertainty as to whether @Anton Rize is in the realm of general relativity.

@Mordred
Thank you for this detailed and rigorous comment. It is refreshing to see someone who clearly understands the Group Theory and the mathematical foundations of SR/GR. I appreciate the time you took to write out the Lorentz generators.

I have a detailed derivation ready that addresses your questions regarding kinematic motion, vectors, and boosts solely through the algebraic relations of WILL RG.

However, I suspect we are operating on different ontological premises. You seem to view the mathematical artifact (the transformation matrix/geometry) as the physics itself, whereas I view it as a descriptive tool for the physics.

Before I provide my answer, I want to establish a clear criterion for its acceptance to avoid moving the goalposts later. I need to know if an answer that satisfies you is even possible within your current view.

The Question: If I can demonstrate - using strictly algebraic relations of the S1/S2 projections - that I can derive the exact same empirical results as the Lorentz transformation (e.g., the exact numerical values for time dilation, aberration of light, and Doppler shift observed by a moving observer), but without invoking a 4D manifold or a transformation matrix...

...will you accept this as a valid physical solution?

Or is your position that unless the specific mathematical formalism of the Lorentz Group (matrices/tensors) is used, the result is invalid regardless of its agreement with experiment?

Put simply: Are we judging the theory by its ability to predict observable data (Empiricism), or by its adherence to a specific geometric tradition (Formalism)?

Once you clarify your standard of evidence, I will post the derivation.

All mathematics is nothing more than a useful tool. To me the mathematical methodology isn't an issue. I don't have strict adherence to any formalism. My comments above relate to versatility of a methodology or ontology.

The question is " does your methodology have the same predictive and descriptive ability as that of the entirety of SR or GR"

If you can demonstrate that it does and still follow mathematic rules then great. If its lacking then that's something to improve upon.

19 minutes ago, Mordred said:

The question is " does your methodology have the same predictive and descriptive ability as that of the entirety of SR or GR"

Agreed. please provide the list of empirically measured phenomena post diction of witch will be a sufficient evidence of predictive power, for you. Don't be shy. Any SR, GR but no grav waves please I don't want to sit here till morning.
For now here's a quick derivation of Constant of Aberration:

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In standard SR, this requires calculating the angle transformation between reference frames using vectors. In WILL RG, this is treated as a projection of the observer's kinetic intensity onto the transverse signal carrier.

Here is the derivation using strictly the algebraic relations of the [math]S^1[/math] carrier:

1. The Physical Setup

Consider a star (e.g., Gamma Draconis) positioned at the true geometric zenith relative to Earth's orbital plane.

Input Data (Empirical): Earth's mean orbital velocity [math]v \approx 29.78[/math] km/s.

Relativistic Kinetic Projection ([math]\beta[/math]):

[math]\beta = \frac{v}{c} = \frac{29.7827}{299792.458} \approx 0.0000993444[/math]

2. The Generative Logic (WILL RG)

In my framework, "angles" are not primary entities. They are derived ratios of projections.

When an observer possesses a kinetic projection [math]\beta[/math] (along the line of motion), it "contaminates" the purely transverse signal from the star. The observer perceives their own kinetic intensity as a longitudinal component in the incoming light.

On the unitary circle [math]S^1[/math] (where the signal radius is always 1), the relationship between the Transverse Projection (Line of Sight) and the Kinetic Projection (Motion) is fixed by the Pythagorean theorem.

The "Aberration Angle" [math]\alpha[/math] is simply the angle whose sine equals the kinetic projection:

[math]\sin(\alpha) = \beta[/math]

Note: There are no vectors here. This is a direct conversion of Energy ([math]\beta[/math]) into Geometry ([math]\alpha[/math]).

3. The Calculation

[math]\alpha_{rad} = \arcsin(0.0000993444)[/math]

[math]\alpha_{rad} \approx 0.0000993445 \text{ radians}[/math]

Converting to arcseconds ([math]1 \text{ rad} \approx 206264.8"[/math]):

[math]\alpha \approx 20.4913"[/math]

4. Comparison

WILL RG Prediction: [math]20.491"[/math]

Standard IAU Constant of Aberration: [math]20.495"[/math]

The result is effectively identical (variance is due to averaging Earth's orbital eccentricity).

Conclusion

I have derived the exact observational consequence of Special Relativity (Stellar Aberration) using zero vectors, zero tensors, and zero coordinate transformations.

I simply treated the observer's velocity as a spectral intensity ([math]\beta[/math]) and calculated its projection on the [math]S^1[/math] carrier.

This answers your question: Yes, the methodology has the same predictive ability, but it achieves it with significantly fewer mathematical entities (Occam's Razor).

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Does this derivation satisfy your requirement for 'predictive ability'? If not - provide me the list of phenomena and I'll provide the derivations.