On 11/8/2025 at 12:58 PM, Anton Rize said:

Thank you. The core of the issue is here: you stated, "In your case you have β=r⋅sinθ and κ=r⋅cosθ". With all due respect, this is a fundamental misinterpretation. I dont know where you get it from but not from me.

I'm sorry if I misunderstood your maths.

I do however, hope that you are not denying the standard mathematical trigonometric parametrisation of a circle in terms of the radius and an angle.

That was correct.

Since you mention ontology several times and ontological relationships more recently, I have been looking into them.

Wikipedia is particularly interesting and suprising in this respect, as I have learned, is that there are more schools of thought of ontology and ontological relationships than there are 'interpretations' of quantum mechanics.

So it seems that there really is a language barrier involved, but not between English and other languages, but between mathematics and philosophy.

I hold that whenever you wish to do mathematics you should employ the definitions of mathematics, not of philosophy or any other discipline.

I note that you and KJW are working towards the analysis that I offered several times, particularly to start at the beginning with set theory.

I now see that this is even more important because of the fundamental difference between the precise mathematical definition of a relation (or relationship) and the philosophical one.

On 11/8/2025 at 3:02 PM, Anton Rize said:

after @studiot explicitly showed me that the Senior Member of this forum can spend 3 weeks criticising the model developed inside his own head without looking in to presented equations and multiple derivations

I will also thank you to discontinue the inappropriate personal comments.

You have no way of knowing what goes on inside another's head.

5 hours ago, Anton Rize said:

if you want me to continue dialog with you first need to answer my questions

You are deflecting from the actual point of my post, which was the energy-momentum tensor.

You are committing a category error known as Paradigm Incommensurability.

You are judging my Relational Geometry (RG) model using the axioms of General Relativity (GR) model as dogma. This is a circular argument (petitio principii).

• It's like rejecting GR because it violates Newton's axioms (e.g., "Where is instantaneous action at a distance?").

• It's like rejecting Quantum Mechanics because it violates GR's axioms (e.g., "Where is the deterministic trajectory?").

You demand my model must first obey GR's axioms (e.g., "there must be a metric" or "you must use the 10 components of [math]T_{\mu\nu}[/math]") before you will even analyze its math.

The only logical way to evaluate a model is by two criteria:

1. Internal Consistency: Is its own algebra contradictory?

2. External Verification: Do its final numerical predictions match reality? ( @KJW confirmed they do).

I ask you to focus on these two points, not on whether my axioms match yours.

---

@studiot , You repeatedly ignored my definitions for [math]\beta[/math] and [math]\kappa[/math]. Instead, you invented your own flawed definition ([math]\beta = r \cdot \sin\theta[/math]) - because your paradigm requires [math]\beta[/math] to be a "length variable" - and then attacked your own misinterpretation. This is a textbook example of unproductive dialog.
You were so incensed that I was supposedly ignoring your questions that you yourself ignored my direct questions to you:

On 11/6/2025 at 12:48 PM, Anton Rize said:

I got few questions for you:
1. Do you find my derivation of statement "Complex mathematics is the consequence of bad philosophy." rigours?
2. Do you find my "Relational Displacement as Q Parameter" operationally useful?

In order to show that you engaging in dialog please answer this questions.

20 minutes ago, Anton Rize said:

You are judging my Relational Geometry (RG) model using the axioms of General Relativity (GR) model as dogma

In my post I responded to your comments on the energy-momentum tensor, your model wasn’t mentioned. The point was that it isn’t an ad-hoc invention, but arises from Noether’s theorem.

25 minutes ago, Anton Rize said:

You demand my model must first obey GR's axioms

No, that’s not my intention, I’m aware that what you’re trying to do is different. You do have to remember though that GR is a very thoroughly tested model of gravity, so whatever model you propose should match its predictions, at the very least those that have been verified. If you choose to ignore all source terms other than energy density, you’re off to a bad start.

32 minutes ago, Anton Rize said:

( @KJW confirmed they do).

Did he? I seem to remember that he pointed out to you that the particular example at hand worked out only because it implicitly exploited a particular symmetry of the Schwarzschild metric; and also that he was of the opinion that the model was too simplistic to work for more complex scenarios.

11 hours ago, Anton Rize said:

In order to show that you engaging in dialog please answer this questions.

1. No, since I as I have already discovered we are working from different definitions.

2. No. what would I use it for ?

11 hours ago, Anton Rize said:

You were so incensed that I was supposedly ignoring your questions that you yourself ignored my direct questions to you:

Just as you have ignored my repeated exhortations to start at the beginning not in the middle.

So I will make a start, in good faith, and see where it leads us.

Let us start with the simplest most general concept, a non empty set.

Call it M.

The only relation possessed by members of such a set is that they belong to the set.

It is worth noting that this relation is an equivalence relation since if m,n , p ∈ M then

m ~ m (reflexive)

m ~ n ⇢ n ~ m (symmetric)

m ~ n and n ~ p ⇢ m ~ p (transitive)

I asked you and you didn't answer, what you think a relation is.

Mathematically a relation is defined as a set of pairs of elements.

So another way of defining the euqivalence realtion is to specify (list ) ll possible pairs and put them into another set (the realtion set).

Although this does not get us very far it does display that there is often more than one way of approaching/defining/specifying.

It also demonstrates the idea of a collection of sets, each with its own special purpose.

Such a collection put together in another set leads to the idea of a space (including spacetime).

Our simple set of course is too simple to be a 'space'.

A space is a set containing a set of elements, a set of rules and posasibly a set of supporting objects.

Alternatively we are imposing some rules on our simple set which give it a required 'structure.

For example a vector space comprise a set of vectors, a set of rules for manipulating these and a set of coefficients (usually forming a Field).

Each of these sets will be more complicated than our basic simple set and bring with them some structure of their own, implicit in their definitions.

For example the Field set reminds us that a Field in mathematics is a very different object from a field in Physics.

So you want to study relativity from the point of view of relational geometry.

So I propose to do exactly that, starting at the beginning, for a much simpler example in Chemistry.

The example will demonstrate what is necessary to make a proper fist of the job and bring out the 'wrinkles' that need to be ironed out.

16 minutes ago, studiot said:

1. No, since I as I have already discovered we are working from different definitions.

2. No. what would I use it for ?

Your answers are the clearest possible demonstration of the Paradigm Incommensurability I have been describing. You cannot see a "use" for my parameters because you are evaluating them from within your own axioms, not mine.

You explicitly showing that you will not participate in a dialog and you have no intension to discuss the presented topic. Why are you here then?

19 minutes ago, studiot said:

Just as you have ignored my repeated exhortations to start at the beginning not in the middle.

I did start at the beginning. The actual foundational axioms of this theory:

On 11/2/2025 at 10:45 PM, Anton Rize said:

Foundational Approach

Guiding Principle

[math]\boxed{\textbf{Nothing is assumed. Everything is derived.}}[/math]

Epistemic Hygiene — Refusal to Import Unjustified Assumptions

This framework is constructed under a single epistemic constraint: to derive all of physics by removing one hidden assumption, rather than introducing new postulates.

This construction is deliberate and contains zero free parameters.

No assumptions are introduced, and no constructs are retained unless they are geometrically or energetically necessary.

Principle — Ontological Minimalism

Any fundamental theory must proceed from the minimum possible number of ontological assumptions.

The burden of proof lies with any assertion that introduces additional complexity or new entities.

This is not a statement about the nature of reality, but a rule of logical hygiene for constructing a theory.

No Ontological Commitments

This model makes no ontological claims about the "existence" of space, particles, or fields.

Instead, all phenomena are treated as observer-dependent relational projections.

Principle — Relational Origin

All physical quantities must be defined by their relations.

Any introduction of absolute properties risks reintroducing metaphysical artefacts and contradicts the foundational insight of relativity.

Mathematical Transparency

"Mathematics is a language, not a world. Its symbols must never outnumber the physical meanings they encode."

1) Each mathematical object must correspond to an explicitly identifiable relation between observers with transparent ontological origin.

2) Every symbol must be anchored to a unique physical idea.

3) Introducing symbols without explicit necessity constitutes semantic inflation: the proliferation of symbols without corresponding physical meaning.

4) Number of symbols = Number of independent physical ideas.

Mathematical Hygiene

[math]\boxed{\textbf{Mathematical hygiene is the geometry of reason}}[/math]

Ontological Blind Spot in Modern Physics

The standard formulation of General Relativity often relies on the concept of an asymptotically flat spacetime, introducing an implicit external reference frame beyond the physical systems under study.

While some modern approaches (e.g. shape dynamics) seek greater relationality, we proceed from strict epistemic minimalism — disallowing all background structures, even hidden or asymptotic ones.

Historical Pattern: breakthroughs delete, not add

- Copernicus eliminated the Earth/cosmos separation.

- Newton eliminated the terrestrial/celestial law separation.

- Einstein eliminated the space/time separation.

- Maxwell eliminated the electricity/magnetism separation.

Each step widened the relational circle and reduced the number of unexplained absolutes.

The spacetime–energy split is the only survivor of this pruning sequence.

The Contemporary Split: An Unpaid Ontological Bill

All present-day theories (SR, GR, QFT, ΛCDM, Standard Model) are built with a bi-variable syntax:

[math]\underbrace{\text{fixed manifold + metric}}_{\text{structure}} \;+\; \underbrace{\text{fields + constants}}_{\text{dynamics}}[/math]

No observation demands this duplication; it is retained only because the resulting Lagrangians are empirically adequate inside the split.

The split is not an empirical discovery but an unpaid ontological debt.

Empirical Bankruptcy of the Separation

- Local energy conservation is verified only after the metric is declared fixed; no experiment varies the volume of flat space and checks calorimetry.

- Universality of free fall tests [math]m_i = m_g[/math] numerically, not the claim that inertia resides in the object rather than in a geometric scaling relation.

- Gravitational-wave polarisations test spin content, not ontology; extra modes can still be called "matter on spacetime".

- Casimir/Lamb shift measure differences of vacuum energy between two geometries; the absolute bulk term is explicitly subtracted, leaving the split intact.

In short, every “test” is an internal consistency check of a formalism that already presupposes two substances.

None constitute positive evidence for the split.

Consequence

Until an experiment varies the amount of space while holding everything else fixed, the spacetime–energy separation remains an un-evidenced metaphysical postulate — the last geocentric epicycle in physics.

Ontological Minimalism

If no empirical or logical ground justifies the distinction between structure and dynamics, the distinction must be dissolved.

[math]\textbf{SPACETIME} \equiv \textbf{ENERGY}[/math]

This equivalence is not algebraic but ontological: spacetime and energy are two descriptive projections of a single invariant entity we call WILL.

Unifying Principle Removing the Hidden Assumption

False Separation (Lemma)

Any model that treats processes as unfolding within an independent background necessarily assigns to that background structural features (metric, orientation, or frame) not derivable from the relations among the processes themselves. Such a background constitutes an extraneous absolute.

Proof

Suppose an independent background exists. Then at least one of its structural attributes — metric relations, a preferred orientation, or a class of inertial frames — remains fixed regardless of interprocess data. This attribute is not relationally inferred but posited a priori. It violates relational closure by introducing a non-relational absolute external to the system. Hence the separation is illicit.

Corollary (Structure–Dynamics Coincidence)

To avoid the artifact of False Separation, the structural arena and the dynamical content must be identified: geometry is energy, and energy is geometry.

Working Principle: Removing the Hidden Assumption (Principle)

[math]\boxed{\textbf{SPACETIME} \;\equiv\; \textbf{ENERGY}}[/math]

This is not introduced as a new ontological entity but as a principle with negative ontological weight: it removes the hidden unjustified separation between “geometry” and “dynamics.” Space and time are not containers but emergent descriptors of relational energy.

Remark (Auditability)

The Working Principle is foundational but testable: it is subject to (i) geometric audit (internal logical consequences) and (ii) empirical audit (agreement with empirical data).

Summary

This Principle does not add, it subtracts: it removes the hidden assumption. Structure and dynamics are two aspects of a single entity that we call — WILL.

What is Energy in a Relational Framework?

Across all domains of physics, one empirical fact persists: in every closed system there exists a quantity that never disappears or arises spontaneously, but only transforms in form. This invariant is observed under many guises — kinetic, potential, thermal, quantum — yet all are interchangeable, pointing to a single underlying structure.

Crucially, this quantity is never observed directly, but only through differences between states: a change of velocity, a shift in configuration, a transition of phase. Its value is relational, not absolute: it depends on the chosen frame or comparison, never on an object in isolation.

Moreover, this quantity provides continuity of causality. If it changes in one part of the system, a complementary change must occur elsewhere, ensuring the unbroken chain of transitions. Thus it is the bookkeeping of causality itself.

Definition (Energy)

Energy is the relational measure of difference between possible states, conserved in any closed whole. It is not an intrinsic property of an object, but comparative structure between states (and observers), always manifesting as transformation.

Deriving the WILL Structure

Having established the Working Principle by removing the illicit separation of structure and dynamics, we now derive its necessary geometric and physical consequences. This single principle enforces closure, conservation, and isotropy of the relational structure, leading to a unique set of geometric carriers for energy.

Definition (WILL)

WILL ≡ SPACE–TIME–ENERGY is the unified relational structure determined by the Working Principle. All physically meaningful quantities are relational features of WILL; no external container is permitted.

Lemma (Closure)

Under the Working Principle, WILL is self-contained: there is no external reservoir into or from which the relational resource can flow.

Lemma (Conservation)

Within WILL, the total relational transformation resource (energy) is conserved.

Lemma (Isotropy from Background-Free Relationality)

If no external background is allowed, then no direction can be a priori privileged. Thus the admissible relational geometry of WILL must be maximally symmetric (isotropic and homogeneous) at the level at which it encodes the conserved resource.

Classification of Minimal Relational Transformations

Given Closure, Conservation, and Isotropy, the minimal carriers are:

(a) Directional (Kinematic) Relation:

The simplest non-trivial relation is between two distinct states (A and B). The minimal description of this directed relation requires a single degree of freedom (the axis connecting A and B). For self-containment, the 1D geometry must be closed, uniquely specifying the circle.

[math]S^{1}[/math]

(b) Omnidirectional (Gravitational) Relation:

The simplest isotropic relation is between a central state (A) and the locus of all states equidistant from it. The minimal closed, maximally symmetric 2D carrier is the 2-sphere.

[math]S^{2}[/math]

Theorem (Minimal Relational Carriers of the Conserved Resource)

The only closed, maximally symmetric manifolds that can serve as minimal carriers of the conserved relational resource are:

(a) [math]S^{1}[/math] for directional (one-degree-of-freedom) relational transformation.

(b) [math]S^{2}[/math] for omnidirectional (central, all-directions-equivalent) relational transformation.

Proof

• For one relational degree of freedom, the classification of connected closed 1-manifolds yields [math]S^{1}[/math] as the unique option (up to diffeomorphism); its isometry group acts transitively with isotropy at each point.

• For an omnidirectional relation from a distinguished center, the encoding manifold must be a closed, simply connected, constant positive curvature 2-manifold with full isotropy at every point. By the classification of constant-curvature surfaces, [math]S^{2}[/math] is the maximally symmetric representative. Nontrivial quotients spoil global isotropy and are excluded.

Corollary (Uniqueness)

Under the Working Principle with Closure, Conservation, and Isotropy, [math]S^{1}[/math] and [math]S^{2}[/math] are necessary relational carriers for, respectively, directional and omnidirectional modes of energy transformation.

Remark (Non-spatial Reading)

Throughout, [math]S^{1}[/math] and [math]S^{2}[/math] are not spacetime geometries. They are relational manifolds encoding the closure, conservation, and isotropy of the transformational resource. Ordinary spatial and temporal notions are emergent descriptors of patterns within WILL.

Summary

By removing the hidden assumption (False Separation), we arrive at the Working Principle [math]\text{SPACETIME}\equiv\text{ENERGY}[/math]. From this we deduce: (i) closure, (ii) conservation, (iii) isotropy, and hence (iv) the unique selection of [math]S^{1}[/math] and [math]S^{2}[/math] as minimal relational carriers for directional and omnidirectional transformation. These objects are non-spatial encodings of conservation and symmetry; they are enforced by the principle rather than assumed independently.

Let me know if this format is suitable for you guys and ill continue.

My axioms are not "a non-empty set M" . My axioms are "Ontological Minimalism" and "Epistemic Hygiene".

Your attempt to reset the entire 120-post discussion to your preferred axioms (Set Theory) is a diversion. It ignores the actual purpose of this thread. Why are you here?

28 minutes ago, studiot said:

So I will make a start, in good faith, and see where it leads us.

Let us start with the simplest most general concept, a non empty set.

Call it M.

The only relation possessed by members of such a set is that they belong to the set.

It is worth noting that this relation is an equivalence relation since if m,n , p ∈ M then

m ~ m (reflexive)

m ~ n ⇢ n ~ m (symmetric)

m ~ n and n ~ p ⇢ m ~ p (transitive)

This has nothing to do with the theme of this thread (unless you convinced that there's only one possible way to build a theory - yours, and everyone should obey). Please create your own thread and we can discuss your ideas there.

34 minutes ago, studiot said:

So you want to study relativity from the point of view of relational geometry.

So I propose to do exactly that, starting at the beginning, for a much simpler example in Chemistry.

We cannot have a dialogue if you continue to ignore my foundational principles and attempt to substitute your own. Why are you here?

16 hours ago, Anton Rize said:

Why are you here?

17 hours ago, studiot said:

So you want to study relativity from the point of view of relational geometry.

So I propose to do exactly that, starting at the beginning, for a much simpler example in Chemistry.

If I take as my base set the set of all hydrocarbon molecules but restict it to three subsets I can show all I need to show/deduce.

For this I do not need a coordinate system

Nor do I need a metric

Nevertheless using (relational) geometry without these I can deduce two different relations for these subsets,

rather as you say you can do for GR and your proposal.

So yes I agree there is more than one approach and that it is especially pleasing when they end up with the same result.

So considering my three subsets as the alkanes, the alkenes and tha alkynes I observe that

The alkanes have a formula C_nH_2n+2

The alkenes have a formula C_nH_2n

The alkynes have a formula C_nH_2n-2

This is one relation deducible from geometry

Another relation is deducible from the Chemistry.

The alkanes have a only single bonds

The alkenes have a double bond

The alkynes have a triple bond

Various relational diagrams may be drawn to show these facts.

Now I said I don't need a metric for this.

This is actually slightly disingenuous since I really should say there is a metric but I am not using it.

It is in fact quite difficult to create a structure without a metric, whether you use that metric or not.

In this case there exists (in the mathematical sense of that statement) a metric that may be described as a modified Hamming metric on the set of hydrocarbons in general and my subsets in particular.

What's more you can even draw circles and define radii with this metric.

It is a very simple metric the modulus of the difference of the number of carbon atoms between two molecules in a subset.

This simple formula | C_a - C_b | where C is the number of carbon atoms and a, b are molecules in the set, satisfy the three metric axioms.

This is exactly what I think you are trying to do when you post a diagram with a coordinate grid and circles.

By doing this you have acknowledged that your schema (I use that word since you have something against sets) contains an implied metric.

On 11/13/2025 at 12:28 AM, Anton Rize said:

The only logical way to evaluate a model is by two criteria:

1. Internal Consistency: Is its own algebra contradictory?

2. External Verification: Do its final numerical predictions match reality? ( @KJW confirmed they do).

I ask you to focus on these two points, not on whether my axioms match yours.

If the final numerical predictions match reality, over the entire span of the theory, it suggests that the math is ultimately the same; it might differ in appearance only because equations can be rearranged. Do you have equations that differ from GR in substance (not merely form)?

You ask people to focus on this, here, but much of your thesis focuses on philosophy as well, which has nothing to do with the internal consistency of math or agreement with observation.

You have tendency to just repeat yourself rather than address questions, and do so in a dismissive tone. That can’t continue.

On 11/11/2025 at 12:09 PM, Anton Rize said:

For you, as far as I understood, this is not a statement about the structure of reality but "opening your toolbox" of sort.

That's a reasonable assessment. However, I see it as a way to get around the unprovability of physics to utilise the provability of mathematics. It's not perfect, the weakness is the limitation of the mathematics to describe physical reality. You might say that is the point you are making to me, but this is as much a question about physical reality as it is about mathematics. For example, can quantum theory be described with the same mathematics as general relativity?

Nevertheless, it is my answer to the question of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" that by producing a mathematical description of physical reality, it is only natural that the mathematical properties of descriptions translate to observed properties of physical reality.

On 11/11/2025 at 12:09 PM, Anton Rize said:

My methodology assumes that all mathematical choices are ontological statements.

The problem with this is that, as suggested above, you do not know the ontology of reality. You have suggested that observed reality is a projection of a hidden reality that your theory uncovers, and that this hidden reality is the true ontology. But I would say that physical reality itself does not have mathematical properties or logic, and that strictly speaking, these can only be applied to mathematical descriptions. Physical reality is what it is, nothing more, nothing less. And because what you are describing is hidden, that description doesn't correspond to observed reality. In other words, you are overstating what you know about physical reality. That's not to say that mathematical descriptions have to correspond directly to observed reality. The power of mathematics is its ability to transform descriptions into some other form where obscure details can be revealed. Even the notion of "mathematical properties" of descriptions may involve substantial mathematical processing of a description to reveal. Because of this, the mathematics associated with the descriptions of physical reality can't really be said to be the ontology of reality.

For example, suppose one has a waveform over some region of spacetime. A Fourier transform of that waveform creates a description in the domain of wavenumber and frequency. Does that mean the ontology of physical reality is the domain of wavenumber and frequency? Can we even say that the ontology of physical reality is the domain of spacetime? What about a fractional Fourier transform?

To treat the observed physical reality as if it were a projection of a higher hidden reality is not necessarily wrong if this reveals a mathematical property of the observed physical reality, and one is not regarding the higher hidden reality as the true ontology. For example, one might choose to embed four-dimensional curved spacetime into a higher-dimensional flat spacetime (I believe it requires ten dimensions to embed an arbitrarily curved four-dimensional spacetime, though I've not seen a proof of this). Why would one do this? Because dealing with flat spaces is less problematic than dealing with curved spaces. Maybe. The mathematics is about extracting knowledge about the observable reality, not about describing an ontology.

On 11/11/2025 at 12:09 PM, Anton Rize said:

Your methodology seems to imply a split - that some math is just 'tools' , and some is 'physics' . This raises a critical question for me: In your view, how do we draw an objective line? Where does the 'toolbox' end and 'physical reality' begin? Without a clear dividing line, that distinction itself seems arbitrary and potentially misleading.

That's straightforward: When you're doing mathematics, you're doing mathematics, and when you're performing experiments, making observations, taking measurements, you're doing physics.

On 11/11/2025 at 12:09 PM, Anton Rize said:

On 11/11/2025 at 12:09 PM, Anton Rize said:

because reality looks like a "differentiable manifold", this can form the basis of a mathematical description of reality.

But how do you know that its not just "when holding a hammer everything around looks like a nail"?

By how effective the mathematical descriptions are at describing reality. On the other hand, a differentiable manifold might not be effective at describing reality at extremely small scale. That is, the best mathematical tools are those that match the physical reality being described.

On 11/11/2025 at 12:09 PM, Anton Rize said:

This "one step before" is the very methodology we are discussing. It's not another mathematical structure. It is the philosophical act of applying "Ontological Minimalism". My methodology (which you call "esoteric") is what determines which tools we are even allowed to pick up. You assume [math]\mathbb{R}^n[/math] is a free, "obvious" choice because it "looks like reality". I derive my S^1 and S^2 carriers as the only necessary and sufficient structures allowed after applying my method of removing all unnecessary postulates (like the [math]\mathbb{R}^n[/math] grid itself)

This is not entirely unlike me choosing to consider mathematical descriptions instead of physical reality itself as the notion to be studied. And we encounter more-or-less the same problem when we choose the mathematics to start from. However, I feel that choosing a mathematical notion that looks like physical reality is a good place to start.

On 11/11/2025 at 12:09 PM, Anton Rize said:

This question is backward. RG does not get to a differentiable manifold; it replaces it.

If you had said that RG augments a differentiable manifold, I could maybe accept this. But I don't think there is enough detail in RG to actually replace a differentiable manifold in real-world problems.

On 11/11/2025 at 12:09 PM, Anton Rize said:

The manifold (your [math]\mathbb{R}^n[/math] "grid") is the very a priori assumption that RG rejects.

Do you deny that an experimental physicist can set up a coordinate grid in the laboratory?

On 11/11/2025 at 12:09 PM, Anton Rize said:

1. The Lensing formula ([math]\alpha=2\kappa^2[/math]) is a general law for light, which follows from its single-axis state [math]\beta=1[/math] ⇒ Y axis disappears [math]\beta_Y=0[/math] ⇒ no projection partition leads to factor of 2. Light has no rest frame.

2. The Photon Sphere is a specific configuration. In RG, it defined by the unique equilibrium condition where the two relational angles are equal:
[math]\theta_1 = \theta_2[/math].

This angular equality implies condition on their projections: [math]\beta^2 + \kappa^2 = \cos^2(\theta_1) + \sin^2(\theta_2) = 1[/math] (not Pythagorean constant but specific configuration).

When we solve this equilibrium condition ([math]\beta^2 + \kappa^2 = 1[/math]) simultaneously with the system's closure law ([math]\kappa^2 = 2\beta^2[/math]), we get the precise solution for the Photon Sphere:

[math]\beta^2 = 1/3[/math] and [math]\kappa^2 = 2/3[/math] (which correctly gives [math]r = R_s / \kappa^2 = 1.5 R_s[/math]).

Surely, gravitational lensing and the photon sphere are based on the same principle? So why would gravitational lensing involve [math]2\kappa^2[/math], and the photon sphere involve only [math]\kappa^2[/math]? Also, for gravitational lensing, you said [math]\beta^2 = 1[/math], whereas for the photon sphere, you said [math]\beta^2 = \dfrac{1}{3}[/math]?

On 11/11/2025 at 12:09 PM, Anton Rize said:

Why are you ignoring nine of the independent components?

This is again the "toolbox" vs. "ontology" error. You are confusing a descriptive model of matter ([math]T_{\mu\nu}[/math]) with the theory of gravity itself. Your 10-component
[math]T_{\mu\nu}[/math] is not a fundamental law of nature; it is a phenomenological input - a placeholder where physicists insert simplified models like "ideal fluids" or "shear stress" to describe the state of matter. My Relational (RL) model is generative. It does not need 10 independent components as an input. Instead, it uses one parameter ([math]\rho[/math]) and derives other necessary properties (like Pressure) as emergent consequences.

One could extricate density from the energy-momentum tensor so that:

[math]T_{\mu\nu} = \rho c^2 U_{\mu\nu}[/math]

where [math]\rho[/math] is the mass part of the energy-momentum tensor, and [math]U_{\mu\nu}[/math] is the kinetic part of the energy-momentum tensor. Whereas [math]\rho[/math] would determine the overall strength of the gravitation, [math]U_{\mu\nu}[/math] would be about the directional aspect of the gravitation, which seems to be less important to you.

On 11/11/2025 at 12:09 PM, Anton Rize said:

By applying Hubble horizon ([math]r_H[/math]) as [math]r=r_H[/math] we can calculate [math]\Lambda(r) = 1/r^2=1.1941779885\times10^{-52} m^{-2}[/math]
Its matches closely with Λ≈1.36×10⁻⁵² m⁻² is standard in cosmology literature.

According to the Wikipedia article, "De Sitter space":

[math]\Lambda = \dfrac{3}{\alpha^2}[/math]

where [math]\alpha[/math] is the cosmological horizon.

Edited November 15Nov 15 by KJW

I want to revisit the perihelion shift, particularly for circular orbits. Thus, this post will investigate circular trajectories centred around a Schwarzschild black hole. However, to avoid working with Christoffel symbols, I use a formula that I've already mentioned on the forum some time ago, the relationship between an accelerated frame of reference and time dilation.

[math]a_\mu = -\dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\mu}[/math]

[math]a^\mu = -g^{\mu\nu} \dfrac{c^2}{T}\dfrac{\partial T}{\partial x^\nu}[/math]

[math]a = \sqrt{|a_\mu a^\mu|} = \sqrt{|g_{\mu \nu} a^\mu a^\nu|} = \sqrt{|g^{\mu \nu} a_\mu a_\nu|}[/math]

[math]= \dfrac{c^2}{T} \sqrt{\Big|g^{\mu \nu}\dfrac{\partial T}{\partial x^\mu}\dfrac{\partial T}{\partial x^\nu}\Big|}[/math]

[math]T[/math] is the magnitude of a timelike Killing vector, so this formula assumes a stationary frame of reference. But, note that an object on a circular trajectory centred around a Schwarzschild black hole is a stationary object. However, to make the stationary nature of this object explicit, the Schwarzschild metric is coordinate-transformed to a rotating coordinate system in which the object is at rest. The angular velocity of the object in the original non-rotating coordinate system is [math]\dfrac{d\phi}{dt} = \omega[/math].

[math]\text{Schwarzschild metric:}\ \ \ (ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\phi)^2[/math]

[math]\text{Let:}\ \ \ \phi = \varphi + \omega t\ \ \ ;\ \ \ d\phi = d\varphi + \omega dt[/math]

[math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi + \omega dt)^2[/math]

[math](ds)^2 = \Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt - \omega^2 r^2 \sin^2\theta (dt)^2[/math]

[math](ds)^2 = \Big(\Big(1 - \dfrac{2GM}{c^2 r}\Big) c^2 - \omega^2 r^2 \sin^2\theta\Big)(dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math]

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math]

Specifying the location of the object at rest in the rotating coordinate system, the acceleration of the object is derived, noting that in this coordinate system, the magnitude of the Killing vector can be taken to be [math]\sqrt{g_{tt}}[/math], and that this magnitude depends only on the radial coordinate. Thus, the acceleration vector has only the radial non-zero component. Only the covariant form of the vector and the invariant magnitude of the acceleration are provided.

[math]\text{For:}\ \ \ r = R\ \ \ ;\ \ \ dr = 0\ \ \ ;\ \ \ \theta = \dfrac{\pi}{2}\ \ \ ;\ \ \ d\theta = 0\ \ \ ;\ \ \ \varphi = 0\ \ \ ;\ \ \ d\varphi = 0[/math]

[math](ds)^2 = \Big(c^2 - \dfrac{2GM}{R} - \omega^2 R^2\Big) (dt)^2[/math]

[math]-a_r = \dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = \dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r}[/math]

[math]= \dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}[/math]

[math]a = \dfrac{|a_r|}{\sqrt{|g_{rr}|}} = \left|\dfrac{\dfrac{GM}{R^2} - \omega^2 R}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\right| \sqrt{\left|1 - \dfrac{2GM}{c^2 R}\right|}[/math]

While the above provides the acceleration of circular trajectories centred around a Schwarzschild black hole, the derivations below are specifically for circular orbits for which the acceleration of the object is zero. Kepler's third law is obtained, specifying the orbital period in terms of time measured in the non-rotating coordinate system, both at infinity and at the orbit radius. Also, the radial coordinate of the orbit is obtained for a lightlike orbit, as well as for timelike orbits specified by the speed of the object as a fraction [math]\beta[/math] of [math]c[/math]. And just as [math]R[/math] is obtained in terms of [math]\beta^2[/math], [math]\beta^2[/math] is obtained in terms of [math]R[/math].

[math]\text{Let:}\ \ \ \omega = \dfrac{2\pi}{T_\infty}\ \ \ \text{where}\ \ T_\infty\ \ \text{is the orbital period observed from}\ \ r = \infty[/math]

[math]-a_r = \dfrac{\dfrac{GM}{R^2} - \dfrac{4\pi^2 R}{T_\infty^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}}[/math]

[math]a_r = 0\ \ \text{for}\ \ \dfrac{GM}{R^2} = \dfrac{4\pi^2 R}{T_\infty^2}[/math]

[math]\dfrac{R^3}{T_\infty^2} = \dfrac{GM}{4\pi^2}[/math]

[math]T_R^2 = T_\infty^2 \Big(1 - \dfrac{2GM}{c^2 R}\Big)\ \ \ \text{where}\ \ T_R\ \ \text{is the orbital period observed from}\ \ r = R[/math]

[math]\dfrac{R^3 \Big(1 - \dfrac{2GM}{c^2 R}\Big)}{T_R^2} = \dfrac{GM}{4\pi^2}[/math]

[math]\text{For a lightlike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2}[/math]

[math]R^3 - \dfrac{2GM}{c^2} R^2 = \dfrac{GM}{4\pi^2} T_R^2[/math]

[math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math]

[math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2}[/math]

[math]= \dfrac{3GM}{c^2}[/math]

[math]\text{For a timelike trajectory:}\ \ \ T_R = \dfrac{2\pi R}{c \beta}\ \ \ ;\ \ \ \dfrac{T_R^2}{R^2} = \dfrac{4\pi^2}{c^2 \beta^2}[/math]

[math]R = \dfrac{2GM}{c^2} + \dfrac{GM}{4\pi^2} \dfrac{T_R^2}{R^2}[/math]

[math]= \dfrac{2GM}{c^2} + \dfrac{GM}{c^2 \beta^2}[/math]

[math]= \Big(2 + \dfrac{1}{\beta^2}\Big) \dfrac{GM}{c^2}[/math]

[math]\dfrac{1}{\beta^2} = \dfrac{c^2 R}{GM} - 2 = \Big(\dfrac{GM}{c^2 R}\Big)^{-1} \Big(1 - \dfrac{2GM}{c^2 R}\Big)[/math]

[math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math]

Finally, the time dilation of the orbit is obtained compared to time at infinity. This comes from the [math]tt[/math]-component of the rotating metric. And because the [math]t[/math]-coordinate was unchanged by the coordinate transformation to the rotating metric, the rotating metric can be directly compared to the non-rotating metric.

[math]\text{Also:}\ \ \ 1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2} = 1 - \dfrac{2GM}{c^2 R} - \dfrac{4\pi^2 R^2}{c^2 T_\infty^2}[/math]

[math]= 1 - \dfrac{2GM}{c^2 R} - \dfrac{GM}{c^2 R}[/math]

[math]= 1 - \dfrac{3GM}{c^2 R}[/math]

[math]\text{Therefore, the time dilation for an object in a circular orbit:}\ \ \ \dfrac{\Delta t_R}{\Delta t_\infty } = \sqrt{1 - \dfrac{3GM}{c^2 R}}[/math]

From this, and comparing to the [math]tt[/math]-component of the non-rotating metric, gravitation contributes [math]\dfrac{2GM}{c^2 R}[/math] to the time dilation, and speed contributes [math]\dfrac{GM}{c^2 R}[/math] to the time dilation. But, we also obtained [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math]. That is because speed [math]c \beta[/math] was specified in terms of time at [math]r = R[/math] rather than time at [math]r = \infty[/math], as required by the invariance of the local speed of light, and as observed from [math]r = \infty[/math], it is slower by the time dilation factor [math]\sqrt{1 - \dfrac{2GM}{c^2 R}}[/math]. Therefore, the contribution of [math]\beta^2[/math], which is not time dilated as it is a ratio, to the time dilation of the object is reduced by factor [math]1 - \dfrac{2GM}{c^2 R}[/math], resulting in a contribution of [math]\dfrac{GM}{c^2 R}[/math], in agreement with the time dilation obtained for the object in a circular orbit. Thus, in terms of [math]T_\infty[/math], defining [math]\beta_\infty[/math] accordingly:

[math]\text{For a timelike trajectory:}\ \ \ T_\infty = \dfrac{2\pi R}{c \beta_\infty}\ \ \ ;\ \ \ \dfrac{T_\infty^2}{R^2} = \dfrac{4\pi^2}{c^2 \beta_\infty^2}[/math]

[math]\dfrac{R^3}{T_\infty^2} = \dfrac{GM}{4\pi^2}[/math]

[math]R = \dfrac{GM}{4\pi^2} \dfrac{T_\infty^2}{R^2} = \dfrac{GM}{c^2 \beta_\infty^2}[/math]

[math]\beta_\infty^2 = \dfrac{GM}{c^2 R}[/math]

Edited November 17Nov 17 by KJW

On 11/15/2025 at 4:20 AM, swansont said:

If the final numerical predictions match reality, over the entire span of the theory, it suggests that the math is ultimately the same; it might differ in appearance only because equations can be rearranged. Do you have equations that differ from GR in substance (not merely form)?

Yes. Here's one of them:

On 11/6/2025 at 3:48 PM, Anton Rize said:

Galactic rotation curves [math]V_{\mathrm{WILL}}^2(r) = Q^2 c^2 = 3\beta^2 c^2 = 3 V_{\mathrm{bary}}^2(r)[/math] yielding the final law:
[math]\boxed{V_{\mathrm{WILL}}(r) = \sqrt{3}\, V_{\mathrm{bary}}(r)}[/math] (with assumption of equilibrium state) tested on SPARC database 175 galaxies no adjustable parameters and fixed Y*=0.66 light/mass ratio for all galaxies yields RMSE=20.23 km/s

*Happy to provide detailed derivations for any claim in this list.

On 11/15/2025 at 4:20 AM, swansont said:

You ask people to focus on this, here, but much of your thesis focuses on philosophy as well, which has nothing to do with the internal consistency of math or agreement with observation.

You have tendency to just repeat yourself rather than address questions, and do so in a dismissive tone. That can’t continue.

Agreed. I spent too much time addressing philosophical objections instead of focusing on the mathematical derivation. It won't happen again.

Thank you @KJW for your detailed answer. I love how productive our discussion is. Let me address your first comment:

You argued that RG relies on a "hidden reality" while physics should rely on "observed reality".

I argue the exact opposite.

It is the Standard Method that relies on a "hidden reality" (the unobservable coordinate manifold, the metric tensor field, the arbitrary coordinates).

RG relies strictly on Direct Observables.

Every parameter in RG is a direct, measurable quantity. We do not "postulate" [math]\kappa[/math]; we measure it.

Here is the formal proof of Operational Measurability:

1. The Observable:

The most direct observable of a gravitational field is the frequency shift (redshift). We measure the ratio of observed clock rates to proper clock rates. In RG, for a stationary observer
([math]\beta=0 \Rightarrow \beta_Y=1[/math]), this ratio is exactly the projection [math]\kappa_X[/math].

This is an empirically measurable, pure dimensionless number.

2. The Geometric Constraint:

The [math]S^2[/math] carrier enforces quadratic closure (a Pythagorean identity) on its projections:

[math]\kappa_X^2 + \kappa^2 = 1[/math]

3. The Algebraic Consequence:

By substituting the measured value into the closure equation, we find the direct relationship between the observable ([math]\kappa_X[/math]) and the parameter ([math]\kappa[/math]):

[math]\kappa = \sqrt{1 - \kappa_X^2}[/math]

[math]R_s=\kappa^2 \cdot r [/math] - no mass no G required.

Conclusion:

An observer can empirically measure the pure number [math]\kappa_X[/math] (redshift) and algebraically find [math]\kappa[/math], [math]R_s[/math].

This entire operation requires zero knowledge of [math]G[/math], [math]c[/math], [math]m_0[/math], manifolds, or metrics.

Compare this to the "Hidden Reality" of GR:

To describe this same simple measurement, General Relativity demands that we postulate:

1. An invisible 4-dimensional differentiable manifold.

2. A metric tensor field [math]g_{\mu\nu}[/math].

3. Connection coefficients (Christoffel symbols [math]\Gamma^{\lambda}_{\mu\nu}[/math]) - non-tensorial mathematical artifacts that vanish in free fall yet are required to define "derivatives."

These entities (manifolds, symbols) are never observed; they are the mathematical scaffolding the "hidden reality".

So, who is really working with "hidden reality"?

The model that builds everything from the direct measurement ([math]\kappa_X \to \kappa[/math])?

Or the model that postulates invisible manifolds and abstract symbols to describe that measurement?


---

On 11/15/2025 at 2:40 PM, KJW said:

For example, can quantum theory be described with the same mathematics as general relativity?

Its funny that you asked. My results from document "WILL Part III QM" strongly suggesting that yes. I'm finding hard to believe it myself. If you want you can find all derivations in here: https://raw.githubusercontent.com/AntonRize/WILL/main/documents/WILL_PART_III_QM.pdf

On 11/15/2025 at 2:40 PM, KJW said:

However, I feel that choosing a mathematical notion that looks like physical reality is a good place to start.

Yes, I see your point, and it seems like a reasonable starting place intuitively.
However, I am skeptical of our ability to determine what reality "actually looks like" without distorting it through our human-centric lens.
My goal is to minimize anthropocentric contamination of reality.
I try to avoid any ontological claims about the "existence" of background entities like space, time, or manifolds. Instead, all phenomena are treated strictly as observer-dependent relational projections.
This constraint forces me to deal with directly measurable values only, stripping away the invisible scaffolding that we humans tend to impose on nature. Its like letting the Universe to unfold on its own terms.

On 11/15/2025 at 2:40 PM, KJW said:

If you had said that RG augments a differentiable manifold, I could maybe accept this. But I don't think there is enough detail in RG to actually replace a differentiable manifold in real-world problems.

That is exactly the boundary I am trying to find.

So far, I haven't found a real-world problem in the domain of GR that RG couldn't solve.

It handles:

1. Weak field solar system tests (Mercury, Light Bending).
2. Strong field orbital dynamics (ISCO, Photon Sphere).
3. Rotational systems (Kerr limits, Frame Dragging).
4. Galactic scale dynamics (Rotation Curves without Dark Matter).
5. Orbital Decay: Hulse–Taylor binary Pulsar (PSR B1913+16) and (PSR J0737-3039) (but this one I least happy with. It gives the right values but luck that simple elegans persistent in the rest of my results)
6. Strong field precession of S2 star on Sgt A orbit.

You can find my google colab python scripts here: https://antonrize.github.io/WILL/predictions/

If RG is merely an "approximation" or "lacks detail," it should break down somewhere.

I am genuinely asking: Can you propose a specific "Stress Test" - a complex real-world scenario - that you believe requires the full differentiable manifold and where RG would fail?

Let's test it.

On 11/15/2025 at 2:40 PM, KJW said:

Do you deny that an experimental physicist can set up a coordinate grid in the laboratory?

No, I do not deny that. We can physically set up a grid of rods and clocks anywhere we want.

The grid is real as a tool, just as a map is real as a piece of paper.

The error arises when we mistake the map for the territory. The grid is a human imposition onto reality, not the source of reality. In RG, I am trying to describe the territory (relations) directly, without forcing it to conform to the rectangular logic of our map (the grid). Again Its like letting the Universe to unfold on its own terms.

On 11/15/2025 at 2:40 PM, KJW said:

Surely, gravitational lensing and the photon sphere are based on the same principle? So why would gravitational lensing involve 2κ2, and the photon sphere involve only κ2? Also, for gravitational lensing, you said β2=1, whereas for the photon sphere, you said β2=13?

Its two different questions therefor we have two different answers:

1. Photon sphere: At what radial distance from M light can obtain circular orbit? [math]r=\frac{R_s}{\kappa^2} [/math]

2. Lensing: How lights path will be effected at the given radial distance from M? total geometric effect [math] 2\kappa^2 [/math]

Here, we ask: "At what radius does the geometry allow a closed circular orbit?" This is a state of marginal stability defined by specific geometric symmetry, not just kinematics. Input: The "Magic Angle" equilibrium [math]\theta_1 = \theta_2[/math] (Kinematic angle = Potential angle). Consequence: This symmetry forces the projections to balance in a specific ratio. Solving [math]\kappa^2 = 2\beta^2[/math] under the constraint [math]\theta_1 = \theta_2[/math] algebraically forces [math]\kappa^2 = 2/3[/math] and [math]\beta^2 = 1/3[/math]. Result: Substituting [math]\kappa^2=2/3[/math] into the field equation gives the exact radius: [math]r = \frac{R_s}{\kappa^2} = 1.5 R_s[/math].


Its almost midnight. will have to leave the rest for tomorrow.

On 11/6/2025 at 12:48 AM, Anton Rize said:

6. And finally Galactic rotation curves V2WILL(r)=Q2c2=3β2c2=3V2bary(r) yielding the final law:
VWILL(r)=3–√Vbary(r) (with assumption of equilibrium state) tested on SPARC database 175 galaxies no adjustable parameters and fixed Y*=0.66 light/mass ratio for all galaxies yields RMSE=20.23 km/s

I’d like to see the derivation, and also the justification that RMSE is the appropriate standard to apply. Also, if this calculation is including dark matter.

If no, and you’re just shifting the curve up, then the agreement is worse at small r. You get better agreement at large r, but the issue is that the shape is wrong, so you’re not solving the problem.

There’s also an issue that the SMBH masses are inferred from theory, so if the theory is wrong, the masses are wrong and everything has to be recalculated.

I'm currently writing replies to other things you've written, but I want to address this now:

7 hours ago, Anton Rize said:

Its two different questions therefor we have two different answers:

1. Photon sphere: At what radial distance from M light can obtain circular orbit? [math]r=\frac{R_s}{\kappa^2} [/math]

2. Lensing: How lights path will be effected at the given radial distance from M? total geometric effect [math] 2\kappa^2 [/math]

I know the answer to the Photon sphere question is a radius while the answer to the Lensing question is a deflection angle, but what is the result of inserting the Photon sphere radius [math]1.5 r_s[/math] into the Lensing formula?

7 hours ago, Anton Rize said:

At what radius does the geometry allow a closed circular orbit?

Have you read the circular orbit derivations in my previous post? How would you derive the radius of the circular orbit for an object moving at [math]\beta = \dfrac{v}{c}[/math]?

Ok lets get back to it. Its getting busy and I falling behind on responses. Great!

On 11/15/2025 at 2:40 PM, KJW said:

One could extricate density from the energy-momentum tensor so that:

Tμν=ρc2Uμν

where ρ is the mass part of the energy-momentum tensor, and Uμν is the kinetic part of the energy-momentum tensor. Whereas ρ would determine the overall strength of the gravitation, Uμν would be about the directional aspect of the gravitation, which seems to be less important to you.

You suspect that by relying on the scalar [math]\rho[/math], I lose the information about directionality (rotation, kinetic flow) that GR stores in [math]U_{\mu\nu}[/math].

The answer is that in RG, directionality is not an input property of matter (like a tensor component), but a geometric state of the system.

Specifically, Rotation is handled by the Kinematic Projection [math]\beta[/math].

When a system rotates, it acquires a non-zero [math]\beta[/math] potential that interacts with the gravitational [math]\kappa[/math] potential.

Here is how RG reproduces the complex directional structure of the Kerr Metric (Rotating Black Hole) purely from these scalar projections, without a stress-energy tensor:

1. Definition of Rotation:

Instead of an angular momentum density component [math]T_{0i}[/math], we define the relational rotation parameter:

[math]\beta = \frac{a c^2}{G m_0}[/math]

(where [math]a = J/mc[/math] is the standard Kerr parameter).

2. The Interaction (Directional Modification):

The rotation ([math]\beta[/math]) distorts the gravitational closure.

For a static system, the horizon is at [math]\kappa^2=1[/math].

For a rotating system, the interplay of [math]\beta[/math] creates two horizons and an ergosphere, derived directly from the projection geometry:

* Event Horizons:

[math]r_{\pm} = \frac{R_s}{2} \left(1 \pm \sqrt{1-\beta^2}\right)[/math]

(Notice how the kinematic term [math]\beta[/math] modifies the radial location).

* Ergosphere (Directional Dragging):

[math]r_{\text{ergo}} = \frac{R_s}{2} \left(1 + \sqrt{1 - \beta^2 \cos^2 \theta}\right)[/math]

Here, the angle [math]\theta[/math] appears naturally from the projection geometry on [math]S^2[/math], reproducing the exact shape of the ergosphere.

3. The Limit (Extreme Kerr):

In GR, an extremal black hole occurs when [math]a = M[/math].

In RG, this is the natural saturation of the kinetic projection:

[math]\beta = 1[/math] (Maximal Rotation).

The closure condition [math]\kappa^2 = 2\beta^2[/math] then forces [math]\kappa^2 = 2[/math], which corresponds to the collapsed horizon radius [math]r = R_s/2[/math].

Conclusion:

I do not ignore the "kinetic part" [math]U_{\mu\nu}[/math]. I map it to the Kinematic Projection [math]\beta[/math].

In standard GR, you input rotation via the tensor to warp the manifold.

In RG, rotation is the [math]\beta[/math]-projection, which naturally reshapes the causal boundaries (horizons) and potential surfaces. The "directional aspect" is fully preserved.

Edited November 18Nov 18 by Anton Rize

On 11/15/2025 at 2:40 PM, KJW said:

According to the Wikipedia article, "De Sitter space":

Λ=3α2

where α is the cosmological horizon.

You are absolutely right about the factor of 3 difference.

I suspect this isn't an error, but the precise "fingerprint" of the ontological difference we are discussing.

In standard theory, we assume vacuum energy fills the Volume. Mathematically, getting the total energy requires integrating the surface area over the radius:

[math]\int r^2 dr = r^3/3[/math]

That’s where the factor of 3 (or 1/3) comes from, right?

In RG, energy is defined by the Surface projection on [math]S^2[/math]. Since I don't assume a "bulk container" that needs to be filled, I don't integrate over [math]r[/math], so the 3 never appears.

What do you think? Could the factor of 3 be just a mathematical artifact of the "volume-filling" assumption?

On 11/17/2025 at 3:42 PM, KJW said:

I want to revisit the perihelion shift, particularly for circular orbits. Thus, this post will investigate circular trajectories centred around a Schwarzschild black hole. However, to avoid working with Christoffel symbols, I use a formula that I've already mentioned on the forum some time ago, the relationship between an accelerated frame of reference and time dilation.

aμ=−c2T∂T∂xμ

aμ=−gμνc2T∂T∂xν

a=|aμaμ|−−−−−√=|gμνaμaν|−−−−−−−√=|gμνaμaν|−−−−−−−−√

=c2T∣∣gμν∂T∂xμ∂T∂xν∣∣−−−−−−−−−−−√

T is the magnitude of a timelike Killing vector, so this formula assumes a stationary frame of reference. But, note that an object on a circular trajectory centred around a Schwarzschild black hole is a stationary object. However, to make the stationary nature of this object explicit, the Schwarzschild metric is coordinate-transformed to a rotating coordinate system in which the object is at rest. The angular velocity of the object in the original non-rotating coordinate system is dϕdt=ω.

Schwarzschild metric:   (ds)2=(1−2GMc2r)c2(dt)2−(1−2GMc2r)−1(dr)2−r2(dθ)2−r2sin2θ(dϕ)2

Let:   ϕ=φ+ωt   ;   dϕ=dφ+ωdt

(ds)2=(1−2GMc2r)c2(dt)2−(1−2GMc2r)−1(dr)2−r2(dθ)2−r2sin2θ(dφ+ωdt)2

(ds)2=(1−2GMc2r)c2(dt)2−(1−2GMc2r)−1(dr)2−r2(dθ)2−r2sin2θ(dφ)2−2ωr2sin2θ dφdt−ω2r2sin2θ(dt)2

(ds)2=((1−2GMc2r)c2−ω2r2sin2θ)(dt)2−(1−2GMc2r)−1(dr)2−r2(dθ)2−r2sin2θ(dφ)2−2ωr2sin2θ dφdt

(ds)2=(c2−2GMr−ω2r2sin2θ)(dt)2−(1−2GMc2r)−1(dr)2−r2(dθ)2−r2sin2θ(dφ)2−2ωr2sin2θ dφdt

Specifying the location of the object at rest in the rotating coordinate system, the acceleration of the object is derived, noting that in this coordinate system, the magnitude of the Killing vector can be taken to be gtt−−√, and that this magnitude depends only on the radial coordinate. Thus, the acceleration vector has only the radial non-zero component. Only the covariant form of the vector and the invariant magnitude of the acceleration are provided.

For:   r=R   ;   dr=0   ;   θ=π2   ;   dθ=0   ;   φ=0   ;   dφ=0

(ds)2=(c2−2GMR−ω2R2)(dt)2

−ar=c2gtt−−√∂gtt−−√∂r=12c2gtt∂gtt∂r

=GMR2−ω2R1−2GMc2R−ω2R2c2

a=|ar||grr|−−−−√=∣∣∣∣∣∣GMR2−ω2R1−2GMc2R−ω2R2c2∣∣∣∣∣∣∣∣∣1−2GMc2R∣∣∣−−−−−−−−−√

While the above provides the acceleration of circular trajectories centred around a Schwarzschild black hole, the derivations below are specifically for circular orbits for which the acceleration of the object is zero. Kepler's third law is obtained, specifying the orbital period in terms of time measured in the non-rotating coordinate system, both at infinity and at the orbit radius. Also, the radial coordinate of the orbit is obtained for a lightlike orbit, as well as for timelike orbits specified by the speed of the object as a fraction β of c. And just as R is obtained in terms of β2, β2 is obtained in terms of R.

Let:   ω=2πT∞   where  T∞  is the orbital period observed from  r=∞

−ar=GMR2−4π2RT2∞1−2GMc2R−4π2R2c2T2∞

ar=0  for  GMR2=4π2RT2∞

R3T2∞=GM4π2

T2R=T2∞(1−2GMc2R)   where  TR  is the orbital period observed from  r=R

R3(1−2GMc2R)T2R=GM4π2

For a lightlike trajectory:   TR=2πRc   ;   T2RR2=4π2c2

R3−2GMc2R2=GM4π2T2R

R=2GMc2+GM4π2T2RR2

=2GMc2+GMc2

=3GMc2

For a timelike trajectory:   TR=2πRcβ   ;   T2RR2=4π2c2β2

R=2GMc2+GM4π2T2RR2

=2GMc2+GMc2β2

=(2+1β2)GMc2

1β2=c2RGM−2=(GMc2R)−1(1−2GMc2R)

β2=GMc2R(1−2GMc2R)−1

Finally, the time dilation of the orbit is obtained compared to time at infinity. This comes from the tt-component of the rotating metric. And because the t-coordinate was unchanged by the coordinate transformation to the rotating metric, the rotating metric can be directly compared to the non-rotating metric.

Also:   1−2GMc2R−ω2R2c2=1−2GMc2R−4π2R2c2T2∞

=1−2GMc2R−GMc2R

=1−3GMc2R

Therefore, the time dilation for an object in a circular orbit:   ΔtRΔt∞=1−3GMc2R−−−−−−−−√

From this, and comparing to the tt-component of the non-rotating metric, gravitation contributes 2GMc2R to the time dilation, and speed contributes GMc2R to the time dilation. But, we also obtained β2=GMc2R(1−2GMc2R)−1. That is because speed cβ was specified in terms of time at r=R rather than time at r=∞, as required by the invariance of the local speed of light, and as observed from r=∞, it is slower by the time dilation factor 1−2GMc2R−−−−−−−−√. Therefore, the contribution of β2, which is not time dilated as it is a ratio, to the time dilation of the object is reduced by factor 1−2GMc2R, resulting in a contribution of GMc2R, in agreement with the time dilation obtained for the object in a circular orbit. Thus, in terms of T∞, defining β∞ accordingly:

For a timelike trajectory:   T∞=2πRcβ∞   ;   T2∞R2=4π2c2β2∞

R3T2∞=GM4π2

R=GM4π2T2∞R2=GMc2β2∞

β2∞=GMc2R

This derivation is impressive. Thank you, @KJW, for taking the time to write this out explicitly.

This allows for a precise, line-by-line comparison between the two frameworks.

Let's analyze your final result for the velocity parameter [math]\beta^2[/math] for a timelike circular orbit:

[math]\beta_{GR}^2 = \frac{GM}{c^2 R} \left(1 - \frac{2GM}{c^2 R}\right)^{-1}[/math]

This result is fascinating because it allows us to translate directly into WILL RG terms.

1. The term [math]\frac{GM}{c^2 R}[/math] is exactly the RG local invariant [math]\beta_{local}^2[/math], derived immediately from the closure condition [math]\kappa^2=2\beta^2[/math].

2. The term [math]\left(1 - \frac{2GM}{c^2 R}\right)[/math] is exactly the RG potential projection [math]\kappa_X^2=1-\kappa^2=cos^2(\theta_2)[/math] (which is the gravitational time dilation factor).

So, your complex GR formula collapses into a remarkably simple ratio of projections:

[math]\beta_{GR}^2 = \frac{\beta_{local}^2}{\kappa_X^2}[/math]

Interpretation:

Your derivation mathematically proves that the "coordinate velocity" measured from infinity ([math]\beta_{GR}[/math]) is simply the local invariant velocity ([math]\beta_{local}[/math]) scaled by the gravitational dilation ([math]\kappa_X[/math]).

This confirms that RG generates the core dynamic invariant ([math]\beta_{local}[/math]) directly in two lines of algebra. GR requires a full metric derivation to obtain the same value, wrapped in the necessary coordinate transformations to relate it to a distant observer.

This effectively demonstrates "Ontological Minimalism": RG yields the naked invariant, while GR dresses it in coordinate effects.

On 11/17/2025 at 3:42 PM, KJW said:

I want to revisit the perihelion shift, particularly for circular orbits.

Oh I think you'll find it interesting. I derived it just today after thinking "Is mass fundamental or maybe its just our human-centric artifact?..." I want to share this result with you that I suspect might be significant:

Massless Orbital Reconstruction (S2 Star Test)

Using the relational framework, it is possible to reconstruct the orbital dynamics (specifically precession) using only dimensionless observables, without ever knowing the Mass ([math]M[/math]), Gravitational Constant ([math]G[/math]), or even the physical size of the orbit ([math]a[/math]).

Here is the step-by-step logic:

1. The Operational Inputs:

We rely on two dimensionless ratios obtained from kinematics and astrometry:

* Eccentricity ([math]e[/math]**):** The shape of the orbit.

* Periapsis Velocity Projection ([math]\beta_p = v_p/c[/math]**):** The maximum redshift/Doppler shift at the closest approach.

(Note: The absolute radius [math]a[/math] is not needed to find the precession angle, only to find the physical value of [math]R_s[/math] in meters).

2. The Derivation:

In standard dynamics, velocity is governed by the Vis-Viva equation. In RG, we express this purely through relational projections.

The kinetic projection [math]\beta^2[/math] at periapsis relates to the semi-major geometric potential [math]\kappa^2(a)[/math] as:

[math]\beta_p^2 = \frac{\kappa^2(a)}{2} \left( \frac{2a}{r_p} - 1 \right)[/math]

Since [math]r_p = a(1-e)[/math], the scale [math]a[/math] cancels out of the bracket, leaving only the shape [math]e[/math]:

[math]\beta_p^2 = \frac{\kappa^2(a)}{2} \left( \frac{2}{1-e} - 1 \right)[/math]

Solving for [math]\kappa^2(a)[/math] gives the defining relation purely in terms of [math]\beta_p[/math] and [math]e[/math]:

[math]\boxed{\kappa^2(a) = \frac{\beta_p^2}{\frac{1}{1-e} - \frac{1}{2}}}[/math]

**3. The Test (Star S2 around Sgr A*):**

Let's plug in the observational data for S2:

* [math]e \approx 0.88466[/math]

* [math]v_p \approx 7.7 \times 10^3 \text{ km/s} \implies \beta_p \approx 0.02568[/math]

First, we compute the geometric scale [math]\kappa^2(a)[/math] (which represents the ratio [math]R_s/a[/math]):

[math]\kappa^2(a) = \frac{(0.02568)^2}{\frac{1}{0.11534} - 0.5} \approx \frac{6.59 \times 10^{-4}}{8.167} \approx 8.07 \times 10^{-5}[/math]

Then, we compute the orbit-level displacement norm [math]Q_{orbit}^2 = \frac{3}{2}\kappa^2(a)[/math]:

[math]Q_{orbit}^2 \approx 1.21 \times 10^{-4}[/math]

Finally, the precession follows from the geometric closure:

[math]\Delta\varphi = \frac{2\pi Q_{orbit}^2}{1-e^2} \approx \frac{2\pi (1.21 \times 10^{-4})}{1 - 0.88466^2}[/math]

Result:

[math]\Delta\varphi \approx 12.0 \text{ arcmin/orbit}[/math]

Conclusion:

This matches the GR prediction exactly. However, at no point did I use the Mass of the Black Hole ($4.3 \times 10^6 M_\odot$) or [math]G[/math].

The precession emerges directly from the dimensionless relationship between velocity ([math]\beta[/math]) and shape ([math]e[/math]). Mass is not a primary cause here; it is a secondary description of this geometric closure.

Edited November 18Nov 18 by Anton Rize

So can we still interpret gravity as bending of spacetime due to mass or not? What do you think?

Edited November 18Nov 18 by Anton Rize

1 hour ago, Anton Rize said:

This matches the GR prediction exactly. However, at no point did I use the Mass of the Black Hole ($4.3 \times 10^6 M_\odot$) or G.

This is just some sleight of hand, though. You used a measured value that depends on G and M.

1 hour ago, Anton Rize said:

You are absolutely right about the factor of 3 difference.

I suspect this isn't an error, but the precise "fingerprint" of the ontological difference we are discussing.

In standard theory, we assume vacuum energy fills the Volume. Mathematically, getting the total energy requires integrating the surface area over the radius:

[math]\int r^2 dr = r^3/3[/math]

That’s where the factor of 3 (or 1/3) comes from, right?

In RG, energy is defined by the Surface projection on [math]S^2[/math]. Since I don't assume a "bulk container" that needs to be filled, I don't integrate over [math]r[/math], so the 3 never appears.

What do you think? Could the factor of 3 be just a mathematical artifact of the "volume-filling" assumption?

No. Four-dimensional de Sitter space is defined as a four-dimensional "surface" in five-dimensional Minkowskian spacetime that is a constant distance from the origin, with the Minkowskian metric of the five-dimensional spacetime inducing a metric on the four-dimensional de Sitter space. From this metric, one directly obtains the cosmological horizon. But also from the metric, one obtains the various curvature fields. A de Sitter space is a space of constant curvature and is therefore a space with a cosmological constant. Thus, we have the relationship between the cosmological constant and the cosmological horizon. The factor of "3" does depend on the dimensionality of the de Sitter space (for n-dimensional de Sitter space, [math]\Lambda = \dfrac{(n-1)(n-2)}{2 \alpha^2}[/math]). No energy is involved. The cosmological constant is on the geometric side of the Einstein field equations.

3 hours ago, Anton Rize said:

Let's analyze your final result for the velocity parameter [math]\beta^2[/math] for a timelike circular orbit:

[math]\beta_{GR}^2 = \frac{GM}{c^2 R} \left(1 - \frac{2GM}{c^2 R}\right)^{-1}[/math]

This result is fascinating because it allows us to translate directly into WILL RG terms.

1. The term [math]\frac{GM}{c^2 R}[/math] is exactly the RG local invariant [math]\beta_{local}^2[/math], derived immediately from the closure condition [math]\kappa^2=2\beta^2[/math].

2. The term [math]\left(1 - \frac{2GM}{c^2 R}\right)[/math] is exactly the RG potential projection [math]\kappa_X^2=1-\kappa^2=cos^2(\theta_2)[/math] (which is the gravitational time dilation factor).

So, your complex GR formula collapses into a remarkably simple ratio of projections:

[math]\beta_{GR}^2 = \frac{\beta_{local}^2}{\kappa_X^2}[/math]

Interpretation:

Your derivation mathematically proves that the "coordinate velocity" measured from infinity ([math]\beta_{GR}[/math]) is simply the local invariant velocity ([math]\beta_{local}[/math]) scaled by the gravitational dilation ([math]\kappa_X[/math]).

This confirms that RG generates the core dynamic invariant ([math]\beta_{local}[/math]) directly in two lines of algebra. GR requires a full metric derivation to obtain the same value, wrapped in the necessary coordinate transformations to relate it to a distant observer.

This effectively demonstrates "Ontological Minimalism": RG yields the naked invariant, while GR dresses it in coordinate effects.

You have it backwards. [math]\beta^2 = \dfrac{GM}{c^2 R} \Big(1 - \dfrac{2GM}{c^2 R}\Big)^{-1}[/math] is the local value of [math]\beta^2[/math].

[math]\beta_\infty^2 = \dfrac{GM}{c^2 R}[/math] is the value observed from infinity.

Edited November 18Nov 18 by KJW

10 hours ago, KJW said:

Four-dimensional de Sitter space is defined as a four-dimensional "surface" in five-dimensional Minkowskian spacetime that is a constant distance from the origin, with the Minkowskian metric of the five-dimensional spacetime inducing a metric on the four-dimensional de Sitter space.

To put this into a more familiar context, a sphere is a two-dimensional surface in three-dimensional Euclidean space that is a constant distance from the origin, with the metric of the three-dimensional Euclidean space inducing a metric on the two-dimensional surface of the sphere:

[math]x^2 + y^2 + z^2 = r^2[/math]

where [math]r[/math] is a constant.

Defining latitude [math]\phi[/math] and longitude [math]\lambda[/math] coordinates on the sphere:

[math]x = r \cos\phi \cos\lambda[/math]

[math]y = r \cos\phi \sin\lambda[/math]

[math]z = r \sin\phi[/math]

Differentials:

[math]dx = -r\ (\cos\phi \sin\lambda\ d\lambda + \sin\phi \cos\lambda\ d\phi)[/math]

[math]dy = r\ (\cos\phi \cos\lambda\ d\lambda - \sin\phi \sin\lambda\ d\phi)[/math]

[math]dz = r \cos\phi\ d\phi[/math]

Square of differentials:

[math](dx)^2 = r^2\ (\cos^2\phi \sin^2\lambda\ (d\lambda)^2 + 2 \cos\phi \sin\lambda \sin\phi \cos\lambda\ d\lambda\ d\phi + \sin^2\phi \cos^2\lambda\ (d\phi)^2)[/math]

[math](dy)^2 = r^2\ (\cos^2\phi \cos^2\lambda\ (d\lambda)^2 - 2 \cos\phi \cos\lambda \sin\phi \sin\lambda\ d\lambda\ d\phi + \sin^2\phi \sin^2\lambda\ (d\phi)^2)[/math]

[math](dz)^2 = r^2\ \cos^2\phi\ (d\phi)^2[/math]

The Euclidean metric of the three-dimensional space induces the (non-Euclidean) metric on the two-dimensional sphere:

[math](ds)^2 = (dx)^2 + (dy)^2 + (dz)^2[/math]

[math]= r^2\ ((\cos^2\phi \sin^2\lambda + \cos^2\phi \cos^2\lambda)\ (d\lambda)^2 + (\sin^2\phi \cos^2\lambda + \sin^2\phi \sin^2\lambda + \cos^2\phi)\ (d\phi)^2)[/math]

[math]= r^2\ (\cos^2\phi\ (d\lambda)^2 + (d\phi)^2)[/math]

A sphere is a two-dimensional space of constant curvature. One thing to note is that the notion of a curved differential manifold with coordinate system and metric is actually familiar as we are living on such a thing. And we draw maps of such a thing too. A Mercator projection distorts scale but preserves angles, making it a conformal mapping to a flat two-dimensional surface. Thus, a sphere (like all two-dimensional spaces) is a conformally flat space. Conformally flat spacetimes also exist. A de Sitter space is an example. A Friedmann–Lemaître–Robertson–Walker (FLRW) metric with flat three-dimensional space is another example. Conformally flat spacetimes have the property that the Weyl conformal tensor field, the curvature tensor field that describes gravitation away from energy-momentum, is zero. By contrast, the curvature tensor field of the Schwarzschild black hole is entirely of the Weyl type. The point I'm making is that the mathematics of general relativity is quite extensive and also touches upon familiar notions such as for example the Mercator projection of the world.

20 hours ago, Anton Rize said:

Here is how RG reproduces the complex directional structure of the Kerr Metric (Rotating Black Hole) purely from these scalar projections, without a stress-energy tensor:

In Kerr spacetime you have

\[T^{\mu \nu}=0\]

everywhere, so nothing there arises from the stress-energy tensor. Equivalently, one can say that the dynamics of the Kerr spacetime arise entirely from the requirement that all curvature be purely of the Weyl type:

\[R^{\mu \nu}=0\]

as well as appropriate boundary conditions.

Edited November 19Nov 19 by Markus Hanke
Typo

@swansont

Thank you for the structured critique. You raise specific, testable objections regarding the statistical standard and the physical shape of the rotation curves.

On 11/18/2025 at 7:21 AM, swansont said:

justification that RMSE is the appropriate standard to apply.

I have completed the full statistical analysis to address your concerns about the metric and the fit quality.

You rightly pointed out that RMSE can mask shape mismatches, so I ran the test using Reduced Chi-Squared ([math]\chi_\nu^2[/math]).

The results reveal exactly what is happening physically. I compared my strict Zero-Parameter geometric model against a minimal One-Parameter variation (standard astrophysical practice).

--- COMPARISON OF PREDICTIVE POWER (SPARC, 175 Galaxies) ---

1. FIXED QWILL (0 Free Parameters):

Constraint: Fixed Global [math]\Upsilon_* = 0.66[/math]

Law: [math]V = \sqrt{3} V_{bary}[/math]

Median [math]\chi_\nu^2[/math]: 34.47

Median RMSE: 20.23 km/s

2. TUNED QWILL (1 Free Parameter per galaxy):

Constraint: [math]\Upsilon_*[/math] allowed to vary (representing stellar population differences)

Law: [math]V = \sqrt{3} V_{bary}[/math]

Median [math]\chi_\nu^2[/math]: 6.52 <-- THE SIGNAL

Median RMSE: 11.62 km/s

The Smoking Gun:

The massive drop in [math]\chi_\nu^2[/math] (from ~34 down to ~6.5) when allowing just one degree of freedom (mass-to-light ratio) proves that the "shape problem" you suspected is not intrinsic to the geometric law [math]\sqrt{3}[/math].

If the geometric law were wrong (e.g., wrong shape at small [math]r[/math]), adjusting the amplitude [math]\Upsilon_*[/math] would NOT fix the [math]\chi^2[/math] so dramatically. The fact that it [i]does[/i] drop to near-acceptable levels implies that the geometric profile is correct, and the residuals in the Fixed Model are dominated purely by astrophysical scatter (old vs. young stellar populations).

Context on Complexity:

Standard Dark Matter halo models typically employ 3 free parameters per galaxy (halo scale, density, plus [math]\Upsilon_*[/math]) to achieve [math]\chi_\nu^2 \approx 1[/math].

WILL RG achieves [math]\chi_\nu^2 \approx 6.5[/math] and RMSE [math]\approx 11[/math] km/s with only 1 parameter.

Conclusion:

The fact that a parameter-free geometric law performs comparably to tuned Dark Matter models suggests that the $\sqrt{3}$ factor captures the fundamental driver of galactic dynamics, while astrophysical variations account for the residuals. The [math]\sqrt{3}[/math] factor potentially might replace the multi-parameter Dark Matter halo. The remaining deviation is just standard astrophysics.

Open-source:
You welcome to test it yourself. All my google colab notebooks you can find here: https://antonrize.github.io/WILL/predictions/

On 11/18/2025 at 7:21 AM, swansont said:

Also, if this calculation is including dark matter.

I didn't had to speculate any "dark" entities.

On 11/18/2025 at 7:21 AM, swansont said:

If no, and you’re just shifting the curve up, then the agreement is worse at small r. You get better agreement at large r, but the issue is that the shape is wrong, so you’re not solving the problem.

This is a physically sound intuition based on the "Maximum Disk" hypothesis, assuming galaxy centers are always baryon-dominated ([math]V_{obs} \approx V_{bary}[/math]). If this were universally true, a uniform scaling of [math]\sqrt{3} \approx 1.73[/math] would systematically overshoot the centers.

However, the data shows something unexpected.

I invite you to look at the actual SPARC profiles using the open visualizer I built for this verification: https://antonrize.github.io/WILL/calculator/

Case 1: The Counter-Example (Low Surface Brightness)

Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.

My parameter-free prediction [math]V = \sqrt{3} V_{bary}[/math] tracks the observed data perfectly from [math]r \to 0[/math] outwards.

If your intuition were universally correct, I should see a massive overshoot here. I do not.

Case 2: The Mixed Bag (High Surface Brightness)

Yes, for some dense galaxies (like NGC0801), there is indeed an overshoot at the bulge. This suggests that the transition from "Newtonian" (center) to "Relational/Dark" (outskirts) dynamics might depend on the local potential depth (the "internal observer" effect I mentioned in the paper that Ill link below).

The Verdict:

If my formula were systematically wrong at small [math]r[/math], the global Median RMSE would be inflated by these "center errors" across the board. The fact that the Global Median RMSE is only 20.23 km/s proves that for a significant portion of the dataset, the geometric relation [math]V = \sqrt{3} V_{bary}[/math] holds surprisingly well even at small radii.

I am not "guessing" the shape. I am reporting that the geometric factor [math]\sqrt{3}[/math] fits the data of diverse galactic morphologies better than the standard assumption that "baryons must dominate the center" or that magic invisible "dark matter" is a real thing.

On 11/18/2025 at 7:21 AM, swansont said:

There’s also an issue that the SMBH masses are inferred from theory, so if the theory is wrong, the masses are wrong and everything has to be recalculated.

This premise relies on a misunderstanding of my claim.

I do not argue that General Relativity is "wrong" in its predictions; I argue that it is ontologically redundant.

Therefore, a mass [math]M[/math] inferred via standard Keplerian/GR dynamics is algebraically consistent with the mass inferred via WILL.

Since the predictive equations converge, the "inferred values" do not need to be recalculated - they are valid inputs for both frameworks.

Furthermore, even if there were a higher-order divergence between the theories in the strong-field regime, it would be irrelevant for this specific test:

* SMBHs dominate kinematics only within a few parsecs (sphere of influence).

* SPARC rotation curves measure dynamics at kiloparsecs ([math]R > 1[/math] kpc).

* At these radii, the potential is dominated by the Stellar Disk and Gas ($10^{9}-10^{11} M_\odot$). The contribution of the central SMBH is vanishingly small.

The masses are safe. The geometric factor [math]\sqrt{3}[/math] is tested on scales where the specific model of the central black hole acts merely as a point-source correction, negligible compared to the galaxy's bulk mass.

On 11/18/2025 at 7:21 AM, swansont said:

I’d like to see the derivation

Since dropping a link to a PDF is often where discussion dies, I will write the explicit short algebraic derivation right here.

It requires no metric tensors, only the conservation of the relational energy budget.

The Geometric Derivation of [math]V_Q = \sqrt{3} V_{bary}[/math]

1. Inputs:

In WILL, the state of any system is defined by projections:

[list]

[*] Kinetic projection (Motion on [math]S^1[/math]): [math]\beta = v/c[/math]

[*] Potential projection (Gravity on [math]S^2[/math]): [math]\kappa = \sqrt{R_s/r}[/math]

[/list]

2. The Closure Condition (Geometric "Virial-like"):

For a self-contained, gravitationally bound system in equilibrium, the energy capacity of the potential field ([math]S^2[/math], 2 degrees of freedom) must balance the kinetic capacity ([math]S^1[/math], 1 degree of freedom).

This enforces the exchange rate of 2:1.

[math]\boxed{\kappa^2 = 2\beta^2}[/math]

3. The Observable (Total Projection):

An external observer (inter-galactic) measures the total energy budget [math]Q^2[/math] required to maintain this structure against the vacuum.

[math]Q^2 = \kappa^2 + \beta^2[/math]

4. Substitution:

Substitute the closure condition (2) into the total budget (3):

[math]Q^2 = (2\beta^2) + \beta^2 = 3\beta^2[/math]

5. Velocity Translation:

Convert back to velocities ([math]V = c \cdot \text{projection}[/math]):

[list]

[*] Baryonic velocity (visible matter): [math]V_{bary} = c \cdot \beta[/math]

[*] Total Observed velocity (dynamic mass): [math]V_{Q} = c \cdot Q[/math]

[/list]

[math]V_{Q}^2 = c^2 Q^2 = c^2 (3\beta^2) = 3 (c\beta)^2 = 3 V_{bary}^2[/math]

Final Result:

[math]\boxed{V_{Q} = \sqrt{3} \cdot V_{bary}}[/math]

---

Why this matters:

This factor [math]\sqrt{3} \approx 1.73[/math] is not a fitted parameter. It is a geometric constant arising from the topology of a closed system ([math]S^1 + S^2[/math]).

Standard Dark Matter models must add an invisible halo with 2-3 free parameters to bridge the gap between [math]V_{bary}[/math] and [math]V_{obs}[/math].

WILL derives the gap as a necessary geometric consequence of the system's unity.


You can download .pdf with all the details here: https://antonrize.github.io/WILL/results/


P.S. Forgot to highlight: Its the same Q parameter that predicts orbital perihelium. Isn't it fascinating!?

Edited November 19Nov 19 by Anton Rize

4 hours ago, Anton Rize said:

Look at galaxy IC2574 (and many LSB galaxies like it). Here, the baryonic contribution is low even at small radii. The "missing mass" problem appears immediately near the center.

My parameter-free prediction V=3–√Vbary tracks the observed data perfectly from r→0 outwards.

That tends to happen when you cherry-pick data.

But for galaxies who have little DM at small r, the orbits are just Newtonian, which suggests that your theory doesn’t predict such orbits properly. What happens when you calculate the orbit of the earth about the sun using the same method?

This premise relies on a misunderstanding of my claim.

I do not argue that General Relativity is "wrong" in its predictions; I argue that it is ontologically redundant.

And just a few days ago you said “I spent too much time addressing philosophical objections instead of focusing on the mathematical derivation. It won't happen again.” and here you are arguing philosophy.

No matter. If the numbers disagree, and one argues that one is correct, the other must be wrong. You can’t claim otherwise.

7 hours ago, swansont said:

But for galaxies who have little DM at small r, the orbits are just Newtonian, which suggests that your theory doesn’t predict such orbits properly. What happens when you calculate the orbit of the earth about the sun using the same method?

Hypothesis: Internal vs. External Observation (The "Carousel" Effect)

• Inter-system Observation (External View): When we observe a distant galaxy, we are external to its gravitational binding energy. We are not part of its "system." Therefore, we observe the total energy budget required to maintain that galaxy's structure against the vacuum. We see both the kinetic motion ([math]\beta^2[/math]) and the structural tension ([math]\kappa^2[/math]) required for closure. [math]Q^2_{\mathrm{ext}} = \beta^2 + \kappa^2 = 3\beta^2 \quad \Longrightarrow \quad V = \sqrt{3} V_{\mathrm{bary}}[/math]

• Intra-system Observation (Internal View): When we observe the Solar System, we are embedded within the same gravitational potential well ([math]\kappa_{\mathrm{local}}[/math]) as the planets. We are, effectively, "riding the same carousel." The background potential [math]\kappa^2[/math] is a shared baseline for both the observer (Earth) and the target (Jupiter).

Potential Screening Principle: For an observer embedded within the system, the binding potential [math]\kappa^2[/math] acts as a common background frame, not as an observable kinematic difference. The relative measurement cancels out the structural tension, leaving only the kinetic differential:
[math]Q^2_{\mathrm{int}} \approx \beta^2 \quad \Longrightarrow \quad V \approx V_{\mathrm{bary}}[/math]

Thus, the factor [math]\sqrt{3}[/math] is the signature of a holistic observation of a closed system from the outside (Galactic Scale), while Newtonian dynamics represents the differential observation from the inside (Local Scale).

---

7 hours ago, swansont said:

That tends to happen when you cherry-pick data.

This is a formal accusation of scientific dishonesty. It is a serious charge. I will not take such false accusation lightly. I demand that you substantiate it or retract it.

1. The Data: I presented a statistical analysis of 175 galaxies . This is the entirety of the SPARC database.

2. The Method: I used the Global Median RMSE. By definition, a median over the full dataset cannot be "cherry-picked."

3. The Code: The analysis scripts are open-source and linked .

   "Cherry-picking" means selecting only data that fits. Using every single data point available is the exact opposite. Making baseless accusations of data manipulation against a transparent, full-dataset analysis reflects poorly on the accuser, not the accused. If you cannot point to a specific galaxy I excluded or a specific line of code that filters the data, then your accusation is factually false and scientifically unethical.