4 hours ago, exchemist said:

How does E² = p² + m²? Isn't there a problem with dimensions?

7 minutes ago, studiot said:

So what are the units of c, m, E, p and v in your system ?

The author must have taken that from some book on relativistic physics with choice c=1 among preliminary conventions, without realising. That's my guess, anyway.

In any case, it's bound to go badly wrong when it assumes circular geometry, while relativistic kinematics is known to comply with hyperbolic geometry rather --as said by yours truly.

PS: Sorry, I thought I had answered, but my answer had been cached by my browser all this time without me actually publishing it.

2 hours ago, studiot said:

But I see no answer to this

Thanks, @studiot - that’s a fair question.

The symbols E, p, and m refer to the same physical quantities in both equations.
The only difference is the system of units used to express them.

In theoretical physics it’s completely standard to set certain constants, like c = 1, to simplify notation.
This doesn’t change the meaning of any quantity - it just removes conversion factors that would otherwise clutter the algebra.

So:

 • In natural units (c = 1), we write E² = p² + m².
 • In SI units, we restore dimensions explicitly: E² = (p·c)² + (m·c²)².

They are two notational layers of the same invariant relation.
If you substitute SI dimensions into the second form, it checks out exactly.
The first one is simply a shorthand used universally in theoretical physics to keep the focus on structure rather than bookkeeping.
To avoid any further confusions - from now on I'll write all my derivations explicitly in SI units.

3 hours ago, studiot said:

As a mattter of interest since projective geometry has no distance function it cannot support a metric or function variables based on distance.

I have been rather busy with some domestic emergencies today so have not had time to progress further study of your circle, but realise that if you are referring to spacetime it must be a hypercircle.

Thanks, @studiot and @joigus and @exchemist I think now I see were the misconception is coming from. Let me clarify step by step so that we keep the structure transparent and consistent.

1. Methodology
   The WILL framework begins with the strictest possible discipline: we postulate nothing and assume no background structure. A Minkowski metric, by contrast, is introduced a priori as a fixed geometric container. Within WILL, such a metric cannot be fundamental - it arises only as an anthropocentric artifact of how we choose to parameterize relational transformations. In other words, distance and time are not primitive ingredients; they are convenient coordinates that appear after the relational structure is already defined.

2. Two geometries, two meanings of X and Y
   In a Minkowski diagram the axes X and Y are physical coordinates with a postulated metric: ct and x, and boosts are hyperbolic isometries of that metric.
   In the S¹–S² formulation of WILL, X and Y represent dimensionless relational projections - amplitude-like and phase-like components of one conserved quantity. These projections have no intrinsic metric; they describe ratios, not distances.

3. Emergence of metric and hyperbolic structure
   The metric-like relations appear only when we reintroduce scale - for example, when we define
    r = Rs / κ² with Rs = 2GM / c².
   At that stage, the familiar hyperbolic form a² − b² = c² naturally emerges as a particular parameterization of the same invariant relation. It is not postulated but derived from closure.
   This is also how the classical results - the Kepler energy relation, Newton’s third law, the Lagrangian and Hamiltonian structures - appear as necessary consequences, not as inputs.

4. Foundational distinction
   Traditional relativity assumes a metric and derives dynamics within it.
   WILL derives dynamics purely from relational closure; metric descriptions then appear as a representational convenience. This reverses the usual order of reasoning and removes the hidden background from physics entirely.

5. On the concept of spacetime
   It helps to move away from the picture of spacetime as a container holding energy. In WILL, spacetime and energy are two complementary descriptions of a single relational entity -structure and dynamics viewed from different sides of the same relation.

6. Empirical correspondence and next steps
   Every relation in the framework produces testable, quantitative results - GPS time shifts, orbital radii (photon sphere 1.5 Rs, ISCO 3 Rs), and more.
   If anyone is interested, we can go through the classical derivations - Keplerian energy, Lagrangian, Hamiltonian, and Newton’s law - one by one in upcoming messages. The only reason for taking them sequentially is to preserve the internal logic of the model, since everything unfolds directly from the same first principles.

   
   ---
   
   Short conceptual note on what S1 and S2 mean in the WILL framework

   S1 and S2 are not spatial circles or spheres placed inside spacetime.
   They are relational architectures – minimal topological structures that guarantee closure, conservation, and isotropy of the relational resource we call energy.

   Energy here is not a substance but a measure of difference between possible states – a kind of bookkeeping of transformations.
   S1 and S2 are the internal rules that make such bookkeeping consistent:
   • S1 encodes directional relations (kinematics)
   • S2 encodes omnidirectional relations (gravitation)

   Applied view – two observers

   Imagine two observers, A and B:

   • A is the center of their own relational framework. B lies on A’s S1 (for motion) and S2 (for gravity).

   • At the same time, B is the center of their own framework, with A lying on B’s S1 and S2.

   Each observer defines their own relational coordinates, yet the mutual consistency of these structures produces the shared phenomena we call space and time.
   So S1 and S2 are not in space – they are the patterns of relation that generate its appearance.
   
   Please let me know if it clears things out and if you want to see the derivation of classical mechanics results. It's absolutely transparent and I find it beautiful.

Edited October 23Oct 23 by Anton Rize

@studiot @exchemist @joigus
It’s been three days since my last detailed response, and so far none of the critics have addressed or verified a single one of the derivations presented above - despite shown earlier confidence and patronising tone that should not be used in scientific discus.

This silence is quite telling: once the discussion reached the level of explicit formulas and transparent definitions, the enthusiasm for debate seems to have disappeared.

For readers genuinely interested in the content rather than posturing, all derivations, unit checks, and equivalences have been provided in full detail above. Unless new, substantive objections are raised, I will take the current pause as tacit agreement with the correctness of the presented results. It was far from the level of discussion I was hoping for.

4 minutes ago, Anton Rize said:

@studiot @exchemist @joigus
It’s been three days since my last detailed response, and so far none of the critics have addressed or verified a single one of the derivations presented above - despite shown earlier confidence and patronising tone that should not be used in scientific discus.

This silence is quite telling: once the discussion reached the level of explicit formulas and transparent definitions, the enthusiasm for debate seems to have disappeared.

For readers genuinely interested in the content rather than posturing, all derivations, unit checks, and equivalences have been provided in full detail above. Unless new, substantive objections are raised, I will take the current pause as tacit agreement with the correctness of the presented results. It was far from the level of discussion I was hoping for.

Good. As your work --it seems-- has reached a point of maturity, now it's time you take it to a professional publication and see how it fares. Don't waste another minute with people of our ilk. The Nobel Prize could be just around the corner.

BTW, your geometry is still circular, not hyperbolic, as you have not made any precisions concerning the complex plane, which would turn your trigonometric relations into hyperbolic ones, as they should. Good luck!

8 hours ago, joigus said:

your geometry is still circular, not hyperbolic,

your critic is based on elementary category error that you once again fail to admit. I already answered this:

On 10/23/2025 at 10:27 AM, Anton Rize said:

Two geometries, two meanings of X and Y
In a Minkowski diagram the axes X and Y are physical coordinates with a postulated metric: ct and x, and boosts are hyperbolic isometries of that metric.
In the S¹–S² formulation of WILL, X and Y represent dimensionless relational projections - amplitude-like and phase-like components of one conserved quantity. These projections have no intrinsic metric; they describe ratios, not distances.

---

"The Nobel Prize could be just around the corner." - I was hoping to meat likeminded people here who are like me passionate about physics. I was hoping for deep and meaningful discus and collective search for Truth. But all Im getting is just uncalled personal attacks. Im verry disappointed...

Edited October 25Oct 25 by Anton Rize

On 10/23/2025 at 10:27 AM, Anton Rize said:

Methodology
The WILL framework begins with the strictest possible discipline: we postulate nothing and assume no background structure. A Minkowski metric, by contrast, is introduced a priori as a fixed geometric container. Within WILL, such a metric cannot be fundamental - it arises only as an anthropocentric artifact of how we choose to parameterize relational transformations. In other words, distance and time are not primitive ingredients; they are convenient coordinates that appear after the relational structure is already defined.

I'm having difficulty understanding your approach, but it seems to me that it has the flaw of disconnecting the maths from the physics. You talk about distance and time not being primitive ingredients whereas these are physical quantities that are measured. You seem to be invoking a hidden reality, whereas it seems to me that a correct theory of reality should be based on what we observe and measure.

Thank you for jumping in @KJW

1 hour ago, KJW said:

I'm having difficulty understanding your approach

I understand it might feel unconventional at first. Try this page https://antonrize.github.io/WILL/relativistic-foundations/ I wrote it specifically to ease understanding. There's a lot of visual and interactive elements, definitely better than scrolling through long forum debates.

1 hour ago, KJW said:

disconnecting the maths from the physics

No. I'm doing exactly opposite: removing assumptions and anthropocentric elements. If you will follow the link you will see it immediately.

1 hour ago, KJW said:

You talk about distance and time not being primitive ingredients whereas these are physical quantities that are measured.

Good point! I like your grounded approach. But lets have a look what exactly we measuring:
When we “measure distance,” what we actually record is not some absolute spatial entity but a relational configuration of events - a count of oscillations, light pulses, or clock ticks exchanged between systems.

Likewise, when we “measure time,” we don’t access a primitive flow; we compare periodic processes and define one as our reference. Both quantities - distance and time - are abstractions distilled from relational data.

So in WILL, the goal is not to deny measurements, but to clarify that what is measured are relations, while what is inferred (like space or time) is representational. The relational structure comes first; coordinates appear only after we project that structure into human-friendly form.

So yes coordinate space and time not primitive ingredients but the result of our conventional choice of representation. And that what you can call:

1 hour ago, KJW said:

hidden reality,

Universe doesn't know and doesn't care about any coordinate system of ours.

1 hour ago, KJW said:

it seems to me that a correct theory of reality should be based on what we observe and measure.

Completely agree with you on this. And this is precisely what im trying to show here.

---

The remarkable thing is when you adopt this relational view the math is just fits in to places naturally! Its simpler cleaner requires no unphysical constructions (like coordinate system) and perfectly allying with experimental results. And I cant believe it myself! That's why I posted it here.

3 hours ago, Anton Rize said:

The remarkable thing is when you adopt this relational view the math is just fits in to places naturally! Its simpler cleaner requires no unphysical constructions (like coordinate system) and perfectly allying with experimental results. And I cant believe it myself! That's why I posted it here.

Suppose there’s a region filled with some form of energy-momentum distribution, perhaps a radiation field, not necessarily assumed to be uniform, plus some extended body with given mass located at a given point within that region. Suppose further a test particle enters this region, at some given point and instant, with some given initial velocity vector. Assume further that the test particle moves under the influence of gravity only.

Can you show us exactly how you calculate the trajectory of this test particle? Does it hit the body, or not? Remember you cannot use any concept of coordinates, you don’t have a field equation, you don’t have a metric, you don’t have a concept of geodesics, nor do you have access to tensors or any other type of covariant object.

I’m interested to see how you encode the gravitational sources, and how you find the trajectory of the particle, as test whether or not it hits the solid body.

9 hours ago, Markus Hanke said:

Suppose there’s a region filled with some form of energy-momentum distribution, perhaps a radiation field, not necessarily assumed to be uniform, plus some extended body with given mass located at a given point within that region. Suppose further a test particle enters this region, at some given point and instant, with some given initial velocity vector. Assume further that the test particle moves under the influence of gravity only.

Can you show us exactly how you calculate the trajectory of this test particle? Does it hit the body, or not? Remember you cannot use any concept of coordinates, you don’t have a field equation, you don’t have a metric, you don’t have a concept of geodesics, nor do you have access to tensors or any other type of covariant object.

I’m interested to see how you encode the gravitational sources, and how you find the trajectory of the particle, as test whether or not it hits the solid body.

This is the "describe a simple instance" that I like to use from time to time.

16 hours ago, Anton Rize said:

"The Nobel Prize could be just around the corner." - I was hoping to meat likeminded people here who are like me passionate about physics. I was hoping for deep and meaningful discus and collective search for Truth. But all Im getting is just uncalled personal attacks. Im verry disappointed...

You started taking it personal. Here:

On 10/25/2025 at 2:55 PM, Anton Rize said:

For readers genuinely interested in the content rather than posturing, all derivations, unit checks, and equivalences have been provided in full detail above. Unless new, substantive objections are raised, I will take the current pause as tacit agreement with the correctness of the presented results. It was far from the level of discussion I was hoping for.

You implied that all the answers were mere "posturing" (and not "genuine") and so that legitimised your theory.

If the bundle of nonsense you are proposing were correct, energies and momenta would be periodic quantities of their arguments (circular topology), instead of being hyperbolically fashioned, so to speak.

I'm sorry if I wasn't clear.

You still haven't answered this. And there is little doubt in my mind you never will. Unless you correct your blunder at some point.

22 hours ago, Markus Hanke said:

Suppose there’s a region filled with some form of energy-momentum distribution, perhaps a radiation field, not necessarily assumed to be uniform, plus some extended body with given mass located at a given point within that region. Suppose further a test particle enters this region, at some given point and instant, with some given initial velocity vector. Assume further that the test particle moves under the influence of gravity only.

Can you show us exactly how you calculate the trajectory of this test particle? Does it hit the body, or not? Remember you cannot use any concept of coordinates, you don’t have a field equation, you don’t have a metric, you don’t have a concept of geodesics, nor do you have access to tensors or any other type of covariant object.

I’m interested to see how you encode the gravitational sources, and how you find the trajectory of the particle, as test whether or not it hits the solid body.

Thanks for jumping in @Markus Hanke Great question!

Your premise is a bit flowed. Within WILL RG we not dealing with non physical imaginary regions. Region is not a physical object - you cant measure it. You'll have to postulate one and we don't want to postulate non physical made up stuff. Instead we can work with relations within given system.

---

Hit-or-miss in WILL (short, coordinate-free)

• We do not use metrics or geodesics. Dynamics = a single invariant budget:

ΔE = 0.5*(β² − κ²).

• “Radial” is defined without coordinates: it is the direction of ∇κ (meridian on S²). “Tangential” is orthogonal to it on the relational sphere.

• Case A (stationary spherical source): the field is one scalar,

κ²(r) = Rₛ / r, equivalently r = Rₛ / κ².

Given initial state at r₀ with total speed β₀ (where β₀² = βᵣ,₀² + βₜ,₀²) and tangential part βₜ,₀, two constants follow:

ΔE = 0.5*(β₀² − κ₀²), and h = r₀ * βₜ,₀ (specific angular momentum; conserved by spherical symmetry).

• Periapsis is where βᵣ = 0, hence βₚ = βₜ,ₚ = h / rₚ and κₚ² = Rₛ / rₚ.

Invariance gives a single algebraic equation for rₚ:

0.5*(h² / rₚ² − Rₛ / rₚ) = ΔE,

i.e. (2ΔE)·rₚ² + Rₛ·rₚ − h² = 0, solve for rₚ > 0. Hit if rₚ ≤ R_body.

• Case B (non-uniform/radiation field): replace the scalar by a directional map κ(r, Ω). ΔE remains invariant; “radial” stays ∇κ. There is no global conservation of h without symmetry, so propagate the tangential budget step-by-step relative to ∇κ. The same hit test applies once rₚ is obtained.

Takeaway: WILL computes trajectories by conserving algebraic budgets (ΔE, and h only when symmetry allows). A metric can be added later as a descriptive overlay, not as the engine.

---

PARAMETER GLOSSARY

• c - speed of light (units: length/time).
• G - gravitational constant.
• M - central mass (Case A).
• R_s - Schwarzschild radius, R_s = 2 G M / c^2 (units: length).

• r - areal radius (units: length), defined by the observable area A via A = 4π r^2.
In stationary spherical exterior: r = R_s / κ^2.

• r_0 - initial areal radius (units: length).
• r_p - periapsis (closest-approach) radius (units: length).
• R_body - physical radius of the extended body (units: length).

• β - total kinematic projection (dimensionless), β = v / c, with 0 ≤ β ≤ 1.
• β_r - radial component of β, defined relative to the local “radial” direction.
• β_t - tangential magnitude of β, β_t = √(β^2 − β_r^2).
• β_0 - initial β; β_t0 - initial tangential component of β.
• β_p - total β at periapsis; at periapsis β_r = 0, so β_p = β_t,p.

• κ - potential projection (dimensionless).
- General (non-uniform) field: κ = κ(r, Ω), a directional map on S².
- Stationary spherical exterior: κ^2(r) = R_s / r (so 0 < κ^2 < 1 for r > R_s).
• κ_p - κ at periapsis; in spherical exterior: κ_p^2 = R_s / r_p.

• Ω - direction on the unit sphere S² (solid angle label for anisotropic fields).
• ∇κ - spatial gradient of κ; defines the “radial” direction locally (coordinate-free).

• h - specific angular momentum (units: length), h = r · β_t.
- Conserved only when spherical symmetry holds (Case A).
- Not a global invariant in generic non-uniform fields.

• ΔE - relational energy budget (dimensionless), invariant along the trajectory: ΔE = 0.5 · (β^2 − κ^2).
(In spherical exterior this yields the periapsis equation 0.5 · (h^2 / r_p^2 − R_s / r_p) = ΔE.)

---

Notes on domains and use:
• Exterior of a compact spherical source: r > R_s ⇒ 0 < κ^2 < 1 and 0 ≤ β ≤ 1.
• “Radial” and “tangential” are defined operationally: radial ≡ direction of ∇κ; tangential ≡ any direction orthogonal to ∇κ on S².
• Conserved quantities: ΔE always; h only when symmetry (spherical) supplies a corresponding Killing vector.

13 hours ago, joigus said:

You still haven't answered this. And there is little doubt in my mind you never will. Unless you correct your blunder at some point.

@joigus
I think there may still be a misunderstanding here.
In a Minkowski diagram the axes X and Y are physical coordinates (ct, x) defined within a metric.
In the S¹–S² formulation of WILL, X and Y are dimensionless relational projections – amplitude- and phase-like components of a single conserved quantity.
They are not spacetime coordinates; no metric is postulated at that stage.
Could you please specify which part of this distinction remains unclear?

Edited October 27Oct 27 by Anton Rize

I’m sorry, but this is utter nonsense. You said you aren’t going to use coordinates or metrics, but then you talk about radial and tangential directions, lengths and orthogonality, gradients, areas, components, conserved quantities, Killing vectors…none of which can meaningfully exist without a spacetime manifold endowed with a connection and a metric. Some of these things are also observer-dependent, which you’re not taking into consideration at all.

Furthermore, nowhere do you actually take into consideration the nature and strength of the energy-momentum distribution; there’s only talk of some nebulous scalar, defined via a Schwarzschild radius, though this is most certainly not a Schwarzschild situation. if this were an EM field, where do you encode the specifics of the field? In reality the EM field cannot be represented by a single scalar. Also, real-world gravity isn’t linear, but nowhere do you account for that non-linearity - the overall gravity of this situation is not simply the sum of all gravitating sources.

If I were to model this situation in standard GR, the result for the trajectory of the test particle is very different - not surprisingly, because it accounts for all relevant relativistic effects.

It seems obvious to me that all you’re doing is to pull random stuff out of some LLM-AI. So I agree with the other posters on this thread that there’s not much of value here. My honest opinion.

@Markus Hanke

I'm very disappointed. I genuinely was hoping to finally have a meaningful conversation... I took your question seriously and spend time to provide you with an answer. You on the other hand didn't even try to understand which I can see clearly by your comment "Some of these things are also observer-dependent" - its a relational framework most of "these things" are observer-dependent.

You state that concepts like "gradient," "orthogonality," and "conserved quantity" cannot "meanfully exist" without a spacetime manifold and a metric (gμν).

My position is that these concepts emerge algebraically from the relational geometry (S¹ × S²), before any metric is postulated. You seem to be mistaking my algebraic definitions (the physics) for their metric descriptions (the coordinate overlay).

To move this from opinion to mathematics, please identify which of my exact algebraic definitions you claim is (or illicitly requires) a metric tensor.

---

Objection 1: "You use 'r', 'gradients', 'orthogonality', and 'areas' which require a metric."

Response: I define these algebraically. Which of the following definitions is a metric tensor?

• A) r ≡ Rₛ / κ² (My definition of the areal radius 'r' as an algebraic output of a field measurement κ, not an a priori coordinate.)

• B) ∇κ (The physical gradient of the scalar potential field κ. I define "radial" as this direction.)

• C) The intrinsic orthogonality on the S² manifold. (I define "tangential" as the direction orthogonal to ∇κ on my base manifold S², which is the source of geometry, not its product.)

---

Objection 2: "You use 'conserved quantities' which require Killing vectors (metric symmetries)."

Response: My invariants are algebraic, not differential. Which of these algebraic budgets do you claim requires a Killing vector?

• A) ΔE = 0.5 * (β² - κ²) (My fundamental trajectory invariant, derived from the Energy-Symmetry Law, not a ∂ₜ symmetry of a metric.)

• B) h = r₀ * βₜ,₀ (The definition of angular momentum, which I noted is conserved only because spherical symmetry was a given input - i.e., ∇κ is purely radial.)

---

Objection 3: "You use a 'nebulous scalar' and cannot encode an EM field or non-uniform source."

Response: I explicitly defined the general case in my "Case B." Which of these formalisms is an invalid source model?

• A) κ²(r) = Rₛ / r (The single scalar only for the spherically symmetric case.)

• B) κ(r, Ω) (The scalar field on S² I use to encode any non-uniform source, such as your EM field, as a directional map. This is not a single scalar.)

---

Objection 4: "You don't account for non-linearity."

Response: My model is algebraically non-linear at its foundation. Which of these core equations do you claim is linear?

• A) ΔE = 0.5 * (β² - Rₛ/r) (The invariant energy budget itself is a non-linear relation between β and r.)

• B) (2ΔE) · rₚ² + Rₛ · rₚ - h² = 0 (The final algebraic equation I derived to solve the trajectory.)
  
  You are assuming these concepts imply a metric. I am deriving them from an algebraic foundation that precedes it. The burden is now on you to show which definition is faulty, not which word sounds like GR.
  
  P. S. And regarding your comment about LLM-AI: I'm using AI to translate in to English and reformat Latex in to Unicode. If I would allow AI to wright answers for me this would be a very short thread. Do you think there's an AI who can handle tasks like this? Im using Gemini and it cant. If you know one that can please give me a link Id love to talk to it.

8 hours ago, Markus Hanke said:

If I were to model this situation in standard GR, the result for the trajectory of the test particle is very different - not surprisingly, because it accounts for all relevant relativistic effects.

@Markus Hanke I made 2 desmos projects for you:
https://www.desmos.com/geometry/nrtnjramrl - calculates aphelion of Mercury using the set of algebraic equations I listed above r_{a}=\frac{-R_{s}-\sqrt{R_{s}^{2}-8E_{d}\left(-h^{2}\right)}}{4E_{d}}=6.9762118617\times10^{10} m. empirical value r_a = 6.982×10^10 m (discrepancy due to estimated input values, but you got the point)
https://www.desmos.com/geometry/hkxjqfkchp - calculates perihelion precession of Mercury \Delta_{WILL}=\frac{2\pi Q_{Merc}^{2}}{\left(1-e_{Merc}^{2}\right)}\ = 5.0208724126\times10^{-7} radians/orbit. empirical value \Delta_{Merc}=5.02 \times10^{-7} radians/orbit.

Edited October 27Oct 27 by Anton Rize

My definition of the areal radius 'r' as an algebraic output of a field measurement κ

What is an “areal radius”? Since it is in units of meters, it sounds suspiciously like a distance (length) to me.

What is an “algebraic output of a field measurement”?

The physical gradient of the scalar potential field κ

And how do you mathematically define the gradient of a scalar field without recourse to any derivatives? How do you even have a scalar field if there’s no manifold for it to live on? A field on what, exactly?

The intrinsic orthogonality on the S² manifold. (I define "tangential" as the direction orthogonal to ∇κ on my base manifold S², which is the source of geometry, not its product.)

What is “intrinsic orthogonality”? How do you tell if two directions (vectors) are orthogonal, in the absence of an inner product? What does “source of geometry” mean?

17 minutes ago, Markus Hanke said:

What is an “areal radius”? Since it is in units of meters, it sounds suspiciously like a distance (length) to me.

Do you understand that distance can be relational without metric involved? Also try google sometimes:

"Areal radius is a term primarily used in cosmology and general relativity to describe a specific radial coordinate that measures the "area" of a spherical surface centered on an observer within a curved spacetime. It is defined so that the surface area of such a sphere is A=4πR2, where R is the areal radius."

21 minutes ago, Markus Hanke said:

How do you even have a scalar field if there’s no manifold for it to live on? A field on what, exactly?

I understand that its different from what you used to, but if everything would be the same as GR it wouldn't be a new idea isn't it? Its not that hard. You can define distance between n number of objects without a "grid" just relational. I already showed you set of equations and 2 demos projects with numerical conformations. I don't know what else I can do...

31 minutes ago, Markus Hanke said:

What is “intrinsic orthogonality”?

37 minutes ago, Markus Hanke said:

What does “source of geometry” mean?

Means that relational properties comes first and they determent geometry.




Are you aware that you haven't wrote a single equation here? We can exchange opinions with you forever. In the end its numbers that meters.

Edited October 27Oct 27 by Anton Rize

7 hours ago, Anton Rize said:

In the end its numbers that meters.

No. What matters more than calculated results is how one obtains one's formulae. Please explain how you obtained this formula:

[math]\Delta_{WILL}=\dfrac{2\pi Q_{Merc}^{2}}{\left(1-e_{Merc}^{2}\right)}[/math]

[If the above LaTeX doesn't render, please refresh the webpage.]

Edited October 28Oct 28 by KJW

2 hours ago, KJW said:

Please explain how you obtained this formula

The full derivation is here (right click save as to download .pdf) https://github.com/AntonRize/WILL/raw/main/documents/WILL_PART_I_SR_GR.pdf

In short its just as I said math fits in to places naturally. I didn't even knew the GR solution when I derived this. Its just make sense and every operation has clear physical meaning like it was "meant" to be this way isn't it?

From removing the separation between structure and dynamics I derived the core principle SPACETIME ≡ ENERGY.

From it constrains I derived relational manifolds (not spacetime geometry but energy relation careers)

circle S^1 for 1 degree of freedom with projections beta=v/c, beta_Y=√(1-beta^2)

2 sphere S^2 for 2 degrees of freedom with projections kappa=v_e/c=√(R_s/r), kappa_X=√(1-kappa^2)

Due to them being different projections of the same conserved quantity the "exchange" rate is 2DOF/1DOF=κ²/β² = 2 for energy closed systems (per period for elliptical orbits)

Total projection as a vector on (kappa, beta) plane Q=√(kappa^2+beta^2)

end then
𝐃𝐢𝐦𝐞𝐧𝐬𝐢𝐨𝐧𝐥𝐞𝐬𝐬 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐢𝐨𝐧 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫𝐬 𝐟𝐨𝐫 𝐌𝐞𝐫𝐜𝐮𝐫𝐲:
κₘₑᵣ꜀ = √(Rₛₛᵤₙ / aₘₑᵣ꜀)
βₘₑᵣ꜀ = √(Rₛₛᵤₙ / 2aₘₑᵣ꜀)

𝐂𝐨𝐦𝐛𝐢𝐧𝐞𝐝 𝐞𝐧𝐞𝐫𝐠𝐲 𝐩𝐫𝐨𝐣𝐞𝐜𝐭𝐢𝐨𝐧 𝐩𝐚𝐫𝐚𝐦𝐞𝐭𝐞𝐫:
Qₘₑᵣ꜀ = √(κₘₑᵣ꜀² + βₘₑᵣ꜀²)

𝐂𝐨𝐫𝐫𝐞𝐜𝐭𝐢𝐨𝐧 𝐟𝐚𝐜𝐭𝐨𝐫 𝐟𝐨𝐫 𝐭𝐡𝐞 𝐞𝐥𝐥𝐢𝐩𝐭𝐢𝐜 𝐨𝐫𝐛𝐢𝐭 𝐝𝐢𝐯𝐢𝐝𝐞𝐝 𝐛𝐲 𝐨𝐧𝐞 𝐨𝐫𝐛𝐢𝐭𝐚𝐥 𝐩𝐞𝐫𝐢𝐨𝐝:
(1 − eₘₑᵣ꜀²) / 2π

𝐅𝐢𝐧𝐚𝐥 𝐖𝐈𝐋𝐋 RG 𝐩𝐫𝐞𝐜𝐞𝐬𝐬𝐢𝐨𝐧 𝐫𝐞𝐬𝐮𝐥𝐭:
Δφ₍WILL₎ = (Qₘₑᵣ꜀²) / [(1 − eₘₑᵣ꜀²) / 2π] = (2πQₘₑᵣ꜀²) / (1 − eₘₑᵣ꜀²)


@KJW Please tell how to use Latex here this reformatting in to Unicode every time will drive me crazy soon 🙃.

Edited October 28Oct 28 by Anton Rize

10 hours ago, Anton Rize said:

Please tell how to use Latex here this reformatting in to Unicode every time will drive me crazy soon 🙃.

Considering the rant from a few days ago I don't see why I should help you but I am not like that.

On 10/25/2025 at 1:55 PM, Anton Rize said:

It’s been three days since my last detailed response, and so far none of the critics have addressed or verified a single one of the derivations presented above - despite shown earlier confidence and patronising tone that should not be used in scientific discus.

This silence is quite telling: once the discussion reached the level of explicit formulas and transparent definitions, the enthusiasm for debate seems to have disappeared.

For readers genuinely interested in the content rather than posturing, all derivations, unit checks, and equivalences have been provided in full detail above. Unless new, substantive objections are raised, I will take the current pause as tacit agreement with the correctness of the presented results. It was far from the level of discussion I was hoping for.

The forum uses Mathjax.

Mathjax also allows MathML I use this.

The following code appears as

[math]{z^2} = \sqrt {{{\left( {\frac{{\Delta y}}{{\Delta x}}} \right)}^2} - {{\left( {\frac{{\Delta p}}{{\Delta q}}} \right)}^2}} [/math]

As a model I have removed the leading and trailing square half-bracket

math]{z^2} = \sqrt {{{\left( {\frac{{\Delta y}}{{\Delta x}}} \right)}^2} - {{\left( {\frac{{\Delta p}}{{\Delta q}}} \right)}^2}} [/math

What is in between the mathml tags is pure Latex

others will perhaps offer other methods such as using the backslash, I am not so familiar with that.

Due to other vagaries of this forum you may have to either refresh the page or the go to another page and then return.

Finally I told you right at the beginning you are hiding some Physics by using natural Units.

Here is a useful explanation from the University of Otago Physics blog.

I have emboldened the most important part.

Now, do you agree that this is a touch confusing?  If we have E = m_e, it looks like we have an energy on the left hand side of the equals sign, and a mass on the right hand side. That can’t work in physics. An energy equals an energy; a mass equals a mass.  It’s because our particle physicists take a few liberties with the units when they say c = 1, h-bar = 1, m_0 = 1. (Or worse still, c = h-bar = m_0 = 1).  And I see plenty of textbooks that are written like this. Writing this is WRONG; what should be written is c = 1 natural length unit per natural time unit; h-bar = 1 natural mass unit natural length unit squared per natural time unit, and m_0 = 1 natural mass unit.  The units are not dispensable. Take them out and you start losing the physics. Saying c = h-bar = m_0 = 1 is claiming that the speed of light equals the mass of the proton. The two are utterly different entities.

Finally I amstill waiting for a response to my previous post.

Edited October 28Oct 28 by studiot

17 hours ago, Anton Rize said:

I think there may still be a misunderstanding here.

I apologise for I did misunderstand something you said. Your circular angle is not the standard hyperbolic "angle" of relativity, or any analogue of it for that matter. Rather, it is what to me looks like a circular definition (no pun intended). Thus, you present the kinematic / gravitational-potential quantities as functions of a respective points on respective copies of a \( S^{1} \), otherwise known as circle. You declare, eg,

\[ \theta_{1}=\arccos\beta \]

and,

\[ \theta_{2}=\arcsin\kappa \]

And, what are \( \beta \) and \( \kappa \)? Just the inverse functions of those. No mystery there. The \( \kappa \) trigonometric function, BTW, does not apply on the whole of \( S^{1} \), but only on half of it (the part where the sine of \( \theta_2 \) is positive.

You do need a metric to talk about angles and distances (or pseudo-distances).

If you want to do away with a metric altogether, I agree with @studiot that you should try something like projective geometry or the like. But physics is about metrics, distances and angles. It just is. So you would have to come up with a way to define what you understand by measuring thing, for lack of metrical properties. Saying it sounds hard is an understatement.

May I remind you, Einstein didn't base GR on tensor analysis just on a whim, but because of the principle of general covariance, that many people tend to disregard. You cannot dispose of tensors just like that. Equations of physics must be generally covariant for very robust reasons: They must transform in a way that makes all observers agree on what they measure when everything is expressed in terms of invariants, even though the particular numbers they get on their rules and clocks might differ. "Covariance" (and thereby tensors) is nothing but a fancy name for that prescription. We don't tell students they're dealing with tensors in pre-university physics, as well as many other university courses, but unbeknown to them, they're most of the time dealing with order-1 or order-2 Euclidean tensors, Galilean tensors, Minkowski tensors, or (in GR) even higher-order diffeomorphism-covariant tensors.

There's kind of an unwritten rule not to tell students of physics that's what they're actually doing most of the time.

The quantities you're using, on the other hand, seem to be observer-dependent, which goes against the principle of general covariance. Or it seems to be very hard to reconcile with it.

(Please refresh the page for LateX display.)

My apologies, I forgot to include the link to my quote.

https://blog.waikato.ac.nz/physicsstop/2011/08/01/natural-units/

11 hours ago, studiot said:

The forum uses Mathjax

Thankyou.

11 hours ago, studiot said:

Finally I amstill waiting for a response to my previous post.

I will assume that you rad my privies 2 reply's and for some reason find them incomplete.

@studiot To be perfectly clear: Yes, absolutely. The symbols E, p, m in the equation E^2 = p^2 + m^2 have different dimensions than the E, p, m in the equation E^2 = (pc)^2 + (mc^2)^2. This is not an error; it is the entire point. The blog post you quoted is correct that c=1 is a formal change of unit system, but it is philosophically incorrect to call it "WRONG".

Here is my position:
1. Physics must be Universal: A fundamental physical law must be valid for any civilization, regardless of their species-specific, historical units (like "meters" or "Zglorp-tails").
2. c=1 is Fundamental: This formulation, where c is a dimensionless unity, is the fundamental expression of physical reality. It is the only "intergalactic standard."
3. SI Units are for Bookkeeping: The SI-unit equation E^2 = (pc)^2 + (mc^2)^2 is a secondary translation of that fundamental law. We "restore c" only for the practical task of comparing our universal formulas to human-centric SI measurements.

My framework does not "lose the physics". It removes the anthropocentric "linguistic layer" to reveal the physics. This is why WILL is built from the ground up on universal, dimensionless projections like [math]\beta=v/c[/math] and [math]\kappa = v_e/c[/math]. These ratios are the same for everyone in the Cosmos.

@joigus

Thank you for this response. I'm glad that we finally resolve this misunderstanding about the [math]S^1[/math]/hyperbolic geometry. That clears the air, and now we can discuss the real substance of the model.

You've raised three crucial objections: that the model is (1) a tautology ("circular logic"), (2) philosophically misguided ("physics is about metrics"), and (3) non-covariant (because it doesn't use tensors).

Let's address them in order.

1. On Tautology ("Circular Logic")

You argue that the definitions are "circular" and that there is "no mystery there".

If I only said [math]\beta = \cos \theta_1[/math] and stopped, you would be 100% correct. But that is not what I do. The model is generative, not descriptive. I don't assume [math]\beta[/math]; I derive the framework that produces it.

The full derivation is in the paper (https://github.com/AntonRize/WILL/raw/main/documents/WILL_PART_I_SR_GR.pdf), but the short logical chain is this:

22 hours ago, Anton Rize said:

From removing the separation between structure and dynamics I derived the core principle SPACETIME ≡ ENERGY.

From it constrains I derived relational manifolds (not spacetime geometry but energy relation careers)

circle S^1 for 1 degree of freedom with projections beta=v/c, beta_Y=√(1-beta^2)

2 sphere S^2 for 2 degrees of freedom with projections kappa=v_e/c=√(R_s/r), kappa_X=√(1-kappa^2)

Due to them being different projections of the same conserved quantity the "exchange" rate is 2DOF/1DOF=κ²/β² = 2 for energy closed systems (per period for elliptical orbits)

Total projection as a vector on (kappa, beta) plane Q=√(kappa^2+beta^2)

This chain generates new, testable physics (like the precession formula [math]\Delta\phi = \frac{2\pi Q^2}{1-e^2}[/math]) from first principles. This is the opposite of a tautology; it's a predictive, generative structure.

---

2. On Philosophy ("Physics is about metrics")

You said: "But physics is about metrics, distances and angles. It just is."

Here, we have a fundamental disagreement.

From my perspective, metrics, distances, and angles are the bookkeeping (the 'how'), not the physics (the 'why'). They are the anthropocentric representation, the "grammar of measurement" we use to describe relations.

The physics lies in the fundamental symmetries and invariants that dictate the rules of this bookkeeping. My model seeks to define those rules algebraically, before any coordinate system or metric is postulated.

---

3. On Covariance (Algebra vs. Tensors)

This is your most important point. You perfectly defined covariance: "...all observers agree on what they measure when everything is expressed in terms of invariants".

You then equated this principle with tensors. But tensors are just one implementation of covariance (a differential one). WILL implements covariance algebraically.

In GR, the invariant is [math]ds^2[/math].

In WILL, a key invariant is the Energy-Symmetry Law, which states that for any two observers/states (A and B), the energy transfer is perfectly balanced:

[math]\Delta E_{A \to B} + \Delta E_{B \to A} = 0[/math]

I derive this by defining the specific energy for a transition as the sum of the changes in the potential and kinetic budgets:

[math]\Delta E_{A \to B} = \Delta U + \Delta K = \frac{1}{2}(\kappa_A^2 - \kappa_B^2) + \frac{1}{2}(\beta_B^2 - \beta_A^2)[/math]

[math]\Delta E_{B \to A} = \frac{1}{2}(\kappa_B^2 - \kappa_A^2) + \frac{1}{2}(\beta_A^2 - \beta_B^2)[/math]

Summing them algebraically gives zero. This identity is WILL's form of covariance. It holds for all observers, yet requires no tensors.

This single algebraic law has profound, non-trivial consequences:

A) The Geometric Origin of the Universal Speed Limit ([math]v \le c[/math])

The speed limit is a foundational property of the [math]S^1[/math] relational manifold itself. The kinematic framework is defined by the conservation of two projections: the "Amplitude Component" ([math]\beta[/math], velocity) and the "Phase Component" ([math]\beta_Y[/math], internal structure). These are bound by the geometry's Pythagorean identity: [math] \beta^2 + \beta_Y^2 = 1 [/math] A state [math]\beta > 1[/math] (i.e., [math]v > c[/math]) would algebraically force: [math] \beta_Y^2 = 1 - \beta^2 < 0 [/math] This would mean the "Phase Component" [math]\beta_Y[/math] becomes an imaginary number. Therefore, a state [math]v > c[/math] is not just forbidden; it is geometrically undefined in this framework, as it would represent a state with no real internal structure.

B) The Nature of Light (No Rest Frame)

For massive particles ([math]\beta < 1[/math]), the energy exchange is symmetric, split between two orthogonal branches (X and Y), giving the [math]\frac{1}{2}[/math] factor.

But for light, [math]\beta=1[/math]. This means its complementary projection [math]\beta_Y = \sqrt{1-\beta^2} = 0[/math]. The symmetric split between two orthogonal branches (X and Y) disappears. The exchange can't be partitioned; it collapses onto a single axis. This is why light experiences twice the gravitational effect (e.g., lensing [math]\Phi_\gamma = \kappa^2 c^2[/math] vs. mass [math]\Phi_{mass} = \frac{1}{2}\kappa^2 c^2[/math]). Light is the boundary state where this energy symmetry breaks.

This isn't just philosophy; it's testable. In Appendix I, I validated this law on the real-world Earth-GPS system:

[math]\Delta E_{Earth \to GPS} = 6.1265399845 \times 10^{-10}[/math]

[math]\Delta E_{GPS \to Earth} = -6.1265399845 \times 10^{-10}[/math]

Sum = 0.

This confirms the law holds to machine precision. It is a true invariant.

Edited October 29Oct 29 by Anton Rize

Areal radius is a term primarily used in cosmology and general relativity to describe a specific radial coordinate

Exactly, and herein lies the issue - you explicitly said you weren’t going to use any coordinate systems, yet you talk about areal radius measured in meters.

Its not that hard. You can define distance between n number of objects without a "grid" just relational.

To meaningfully define the notion of “distance” between points, you need a metric space structure; the same is true for angles, and thus orthogonality.

My question about gradients you haven’t answered at all; in short, you need a notion of derivatives to define the gradient, and, if the basis is not orthonormal, you explicitly need a metric as well.

Are you aware that you haven't wrote a single equation here?

The point of all this isn’t numbers, but to show that you are tacitly using the very notions you are rejecting; that is why I asked about that scenario. You are literally going in circles, pun intended - and I note explicitly that the very notion of “circle” is meaningful only if you already have a metric space. Just throwing around the word “relational” doesn’t change these facts.

While you are thinking about Markus' questions ( and actually answering mine), and since I am an Applied Mathematician here are a couple of tongue in cheek questions.

Are you suggesting that

c = -(e^πi) ? or any of your natural constants, set to unity ?

Since you are listed as living in Brisbane I am suprised you need AI to write English for you. Do you not speak English ?

If this latter is true, fair enough, we at SF often need to help folks over a language barrier.

Edited October 29Oct 29 by studiot