8 hours ago, Anton Rize said:

I don't know. My best guess is no. But minimalization of assumptions to infetecimal is what Im trying to do.

From zero assumptions to minimal assumptions eh ??

Please make up your mind.

Yet you make an assumption when you write

8 hours ago, Anton Rize said:

You ask if my parameter Q means I'm using a metric because it measures "how far" two centers are apart.

Because that is not what I asked.

19 hours ago, studiot said:

21 hours ago, studiot said:

Please also tell me how you can arive at

(anything)², without a metric ?

I would be very happy to discuss metrics if you were prepared to answer any of my questions concerning metrics.

The rest of your response leads me to think that you do not know what a metric is.

A metric is a noun and as such it is the rule part of a function or mapping from it's domain set to the co- domain ( which is the union of the positive real numbers and zero.)

It is unfortunate that different (scientific ) disciplines use different definitions and that we actually have a recent current thread discussing that very subject.

Because most sets will admit many different metrics we also do what English does well and use the word metric as an adjective to distinguish these.

In particular in GR we define the metric tensor, which means we must also have predefined certain other structure on our domain set.

This is why, I keep saying, you need to start at the beginning not in the middle.

Perhaps you would like to go back and read those offers properly - Then you would be able to see how we build up from simple set theory to a domain set such as you and KJW are discussing that has enough structure to actually describe our observations of the world around us.

To be quite clear I contend that you are displaying a metric by using graph paper and placing numbers on it, as well as some greek letters you are squaring (and Q if you must).

A final point.

Invariants are not the same as metrics, both in the sense of Physics and of Mathematics.

A really useful proceedure is to divide the extra structure you are imposing on the domain in the following way.

The equations of constitution and the equations of compatibility.

'The equations' is a term from applied maths. We could equally generalise to the realtions of since some are not in the form of an equation.

For instance the second law of thermodynamics and the uncertainty principle are both in the form of an inequality.

A simple example to make this clear would be the physics and maths of the flow of an incompressible fluid.

The equations of constitution are the Physics and describe relations between variables such as velocity, pressure, and so on.

The equation of compatibility is geometric in nature and relies on an invariant - the volumetric flow rate - usually denote Q.

At all points in the flow system Q must be the same that is constant , as the flow is incompressible.
Compatibility gives us an equation relating the cross section at any place with the velocity at that section - a geometric constraint.

9 hours ago, Anton Rize said:

You ask if my parameter Q means I'm using a metric because it measures "how far" two centers are apart.

No, there is no metric here.

In GR a metric [ds² = g_{μν}dx^μdx^ν] is a background rule that defines distance inside a pre-existing space.

In Relational Geometry, Q does not come from any background. Each observer places itself at (β,κ)=(0,0) and measures the other by its own relational projections. The closure [Q² = β² + κ²] is not a metric contraction but the relational difference between two observers.

It looks like a distance, but it is a relational invariant, not a metric length.

22 hours ago, Anton Rize said:

Each observer places itself at the relational origin (β,κ)=(0,0).

On 11/3/2025 at 11:40 AM, Anton Rize said:

  On 11/3/2025 at 8:40 AM, KJW said:

In order to establish how much spacetime is curved by energy-momentum density, one needs to consider Newtonian theory in the weak-field limit. This provides the proportionality constant between the Einstein tensor and the energy-momentum density tensor in the Einstein equation.

It’s also worth noting that the “proportionality constant” between geometry and energy, which GR must import from the Newtonian limit, arises automatically in RG from internal normalization.

In GR this constant is written as:

[math]\alpha = \frac{8\pi G}{c^2}.[/math]

In RG the same quantity follows algebraically from the geometric normalization of energy density:

[math]\rho_{\max} = \frac{c^2}{8\pi G r^2},[/math]

hence

[math]\alpha = \frac{1}{\rho_{\max}\,r^2}.[/math]

This shows that the coupling between curvature and energy is not something imposed externally to fit Newtonian gravity - it’s already embedded in the relational structure itself.

The gravitational constant G comes from Newtonian gravitation theory, so if this constant is anywhere in your theory, it came from Newtonian gravitation theory, even if indirectly. And if you avoid using this constant, then any mass or energy that is related to geometry can't be expressed in terms of kilograms or other familiar units of mass or energy.

On 11/3/2025 at 11:40 AM, Anton Rize said:

Philosophically, this means that the very bridge GR must borrow from an older, dualistic theory based on independent background is in RG replaced by an intrinsic self-consistency: the measure of geometry and the measure of energy are the same thing, merely expressed through different projections of a single underlying relation.

As I said, general relativity must reduce to Newtonian theory in the weak-field limit. So, comparing general relativity in the domain where the relationship between geometry and energy has already been empirically established allows one to extend this relationship to the domain where general relativity is on its own. It is not about accepting Newtonian theory as an ontologically correct theory but accepting Newtonian theory as an empirically accurate theory within its domain of applicability.

16 hours ago, Anton Rize said:

1. Let's assume, for the sake of argument, that I am tacitly using some metric.

2. Yet I can't find it in my mathematics.

That's because you don't understand mathematics. "Relational Geometry" - your term. The word geometry implies a metric (-metry ; metric). Specifically, the geometry of a concrete space comes entirely from its metric. Different metric, different geometry. No metric, no geometry.

Others may have more patience with this, but I'm out

@studiot @KJW @Xerxes @joigus @Markus Hanke

Thank you all for your persistence - I finally realized where my mistake was.

Already on the very first page I defined what kind of question this discussion actually belongs to:

On 10/21/2025 at 10:51 PM, Anton Rize said:

Let me take a brief pause from the line-by-line debate to clarify what kind of question this discussion actually belongs to.

What we are discussing here is not really whether one can write “spacetime ≡ energy” or which algebraic symbol is more proper.

It is the continuation of a philosophical debate that began more than three centuries ago - between Newton’s substantivalism and Leibniz’s relationalism, later echoed in Mach and Einstein.

Newton’s view: space and time exist as independent containers - an absolute background in which events unfold.

Leibniz’s view: space and time are nothing over and above the relations among entities and their changes.

General Relativity kept Newton’s mathematics but never fully escaped his ontology: spacetime in GR still exists even in the absence of matter (the vacuum solution).

That is substantivalism in disguise - geometry treated as an independent thing.

Einstein himself was never satisfied with this. In his later correspondence with Schilpp and Besso, he called GR “a provisional synthesis” and wrote explicitly that a truly Machian theory would eliminate spacetime as an independent entity altogether.

RG continues that unfinished line - it is not a denial of GR, but the completion of Einstein’s own relational intent.

Its principle (Structure ⇔ Dynamics) simply removes the last residue of substantivalism: geometry and energy are two projections of a single relational resource.

So when someone objects that “spacetime = energy makes no sense,” they are unknowingly taking Newton’s side of the same 300-year-old debate.

The question is not whether GR’s equations work - of course they do.

The question is whether they describe the world as it is or a mathematical prosthesis that still carries Newton’s metaphysics.

That is the level at which RG operates. Everything else - tensors, topology, units - follows from that one epistemic decision.

After that message, I mistakenly assumed that the conceptual question was settled and that we had moved on to discussing the model itself.

I was wrong. All this time we have actually been debating substantivalism (SB) versus relationalism (RL).

Re-reading our exchange, I can now see clearly why questions about units and metrics keep appearing - and why mutual understanding there is so hard.

Given that this dispute is more than 300 years old, I incorrectly assumed that the basic operational distinctions between these two views did not need to be spelled out. That was my mistake.

Let’s fix it together and make these distinctions absolutely clear:

Operational distinction between SB and RL:

In Substance-Based (SB):

• space-time is a pre-existing metric manifold.

• A single object can be assigned coordinates on this manifold, and physical quantities (such as position, velocity, and distance) are defined with respect to that dynamical but still background.

• The origin (0,0) is arbitrary - a conventional point on an independent grid.

In Relational (RL):

• space and time have no existence apart from the relations among physical entities.

• An isolated object has no meaningful coordinates at all; every observer defines its own (0,0) relational origin.

• Only relations between such self-defined frames are physically meaningful.

[math]\boxed{Q^2 = \beta^2 + \kappa^2}[/math]
is not a metric distance, but a relational invariant between two self-centered observers.

Before we go any further, let’s make sure we are truly on the same page about this.

---

To solidate the fact that we are not distancing our selves from Einstein's ideas, I will put some of his quotes with sources and context from original source:

Quote: "It is contrary to the spirit of relativity theory to treat space as an independent thing."^*1

Quote: "Space-time does not claim existence on its own, but only as a structural quality of the field."^*2

Quote: "Time and space and gravitation have no separate existence from matter."^*3

Quote: "With every new important advance the researcher here sees his expectations surpassed, in that those basic laws are more and more simplified under the press of experience. With astonishment he sees apparent chaos resolved into a sublime order... this is what Leibniz so happily characterized as 'pre-established harmony."^*4

---

^{*1 -} The original letter, dated June 25, 1913, is documented in The Collected Papers of Albert Einstein, Volume 5: The Swiss Years: Correspondence, 1902–1914 (Princeton University Press, 1995), Document 448.
Context: Einstein was responding to Mach's ideas on inertia and relativity. He praised Mach's critique of absolute space (from Newton's mechanics) and emphasized that relativity theory views space not as an absolute, independent entity but as relational, influenced by matter. This aligns with Einstein's push against "opportunism" in physics that avoids deeper thinking about space's nature. The letter reflects Einstein's early enthusiasm for incorporating Mach's principles into general relativity, though he later distanced himself somewhat from "Mach's principle."
Academic sources:

Einstein Papers Project (Princeton University): Primary archive with full transcript and translation.

John D. Norton, "Einstein's Conflicting Heuristics: The Discovery of General Relativity" (2012 PDF): Discusses the letter in the context of Einstein's development of general relativity, quoting similar phrasing about space's independence.

"Mach's Principle and the Origin of Inertia" (2003 PDF, eds. Sachs and Roy): References the letter directly in discussions of relativity's philosophical foundations.

^{*2 -} Appendix V ("Relativity and the Problem of Space"), which was added in the 5th edition (1920) and expanded in later editions (e.g., 1952). The core book was first published in 1916 in German as Über die spezielle und die allgemeine Relativitätstheorie.
Context: In this appendix, Einstein argues against viewing space-time as a substantive "container" independent of physical events (a view from Newtonian physics). Instead, he describes it as emergent from the gravitational field, determined by matter. This reflects his Machian influences and the philosophical shift in general relativity, where geometry (space-time) is dynamical and tied to physical content. He contrasts it with ether theories, emphasizing that "there exists no space empty of field."
Academic sources:

Wikisource/Gutenberg editions of the book: Full text available, confirming the quote in Appendix V.

"Relativity: The Special and the General Theory - 100th Anniversary Edition" (Princeton University Press, 2015): Includes the appendix with historical notes on editions.

John Earman, "The Hole Argument and Some Physical and Philosophical Implications" (Living Reviews in Relativity, 2014): Analyzes the quote in the context of space-time substantivalism debates.

^{*3 -} Albert Einstein, summary of general relativity, as cited in Ideas and Opinions (Crown Publishers, 1954), p. 376.
Context: A concise encapsulation of relativity's relational core: physical properties emerge from interactions, not absolutes, echoing Mach's and Leibniz's critiques.
Academic sources:
Appears in multiple editions of Ideas and Opinions; often referenced in biographical works like Abraham Pais's Subtle is the Lord (1982).

^{*4 -} "Physics and Reality" (1936), reprinted in Journal of the Franklin Institute 221 (1936), p. 313; also in Ideas and Opinions (1954).
Context: Einstein invokes Leibniz's concept to describe the relational harmony in physics, where laws emerge from empirical relations rather than absolutes.
Academic sources:
Full essay in journal archives; CPAE Vol. 6.

@Xerxes

On 11/6/2025 at 1:53 PM, Anton Rize said:

Would you call that fair reasoning?

Or are you going to make another self-confident statement to prove my point again?

15 hours ago, Xerxes said:

That's because you don't understand mathematics. "Relational Geometry" - your term. The word geometry implies a metric (-metry ; metric). Specifically, the geometry of a concrete space comes entirely from its metric. Different metric, different geometry. No metric, no geometry.

Others may have more patience with this, but I'm out

Edited November 7Nov 7 by Anton Rize

6 hours ago, Anton Rize said:

@studiot @KJW @Xerxes @joigus @Markus Hanke

Thank you all for your persistence - I finally realized where my mistake was.

Already on the very first page I defined what kind of question this discussion actually belongs to:

After that message, I mistakenly assumed that the conceptual question was settled and that we had moved on to discussing the model itself.

I was wrong. All this time we have actually been debating substantivalism (SB) versus relationalism (RL).

Re-reading our exchange, I can now see clearly why questions about units and metrics keep appearing - and why mutual understanding there is so hard.

Given that this dispute is more than 300 years old, I incorrectly assumed that the basic operational distinctions between these two views did not need to be spelled out. That was my mistake.

Let’s fix it together and make these distinctions absolutely clear:

Operational distinction between SB and RL:

In Substance-Based (SB):

• space-time is a pre-existing metric manifold.

• A single object can be assigned coordinates on this manifold, and physical quantities (such as position, velocity, and distance) are defined with respect to that dynamical but still background.

• The origin (0,0) is arbitrary - a conventional point on an independent grid.

In Relational (RL):

• space and time have no existence apart from the relations among physical entities.

• An isolated object has no meaningful coordinates at all; every observer defines its own (0,0) relational origin.

• Only relations between such self-defined frames are physically meaningful.

Q2=β2+κ2
is not a metric distance, but a relational invariant between two self-centered observers.

Before we go any further, let’s make sure we are truly on the same page about this.

---

To solidate the fact that we are not distancing our selves from Einstein's ideas, I will put some of his quotes with sources and context from original source:

Quote: "It is contrary to the spirit of relativity theory to treat space as an independent thing."^*1

Quote: "Space-time does not claim existence on its own, but only as a structural quality of the field."^*2

Quote: "Time and space and gravitation have no separate existence from matter."^*3

Quote: "With every new important advance the researcher here sees his expectations surpassed, in that those basic laws are more and more simplified under the press of experience. With astonishment he sees apparent chaos resolved into a sublime order... this is what Leibniz so happily characterized as 'pre-established harmony."^*4

---

^{*1 -} The original letter, dated June 25, 1913, is documented in The Collected Papers of Albert Einstein, Volume 5: The Swiss Years: Correspondence, 1902–1914 (Princeton University Press, 1995), Document 448.
Context: Einstein was responding to Mach's ideas on inertia and relativity. He praised Mach's critique of absolute space (from Newton's mechanics) and emphasized that relativity theory views space not as an absolute, independent entity but as relational, influenced by matter. This aligns with Einstein's push against "opportunism" in physics that avoids deeper thinking about space's nature. The letter reflects Einstein's early enthusiasm for incorporating Mach's principles into general relativity, though he later distanced himself somewhat from "Mach's principle."
Academic sources:

Einstein Papers Project (Princeton University): Primary archive with full transcript and translation.

John D. Norton, "Einstein's Conflicting Heuristics: The Discovery of General Relativity" (2012 PDF): Discusses the letter in the context of Einstein's development of general relativity, quoting similar phrasing about space's independence.

"Mach's Principle and the Origin of Inertia" (2003 PDF, eds. Sachs and Roy): References the letter directly in discussions of relativity's philosophical foundations.

^{*2 -} Appendix V ("Relativity and the Problem of Space"), which was added in the 5th edition (1920) and expanded in later editions (e.g., 1952). The core book was first published in 1916 in German as Über die spezielle und die allgemeine Relativitätstheorie.
Context: In this appendix, Einstein argues against viewing space-time as a substantive "container" independent of physical events (a view from Newtonian physics). Instead, he describes it as emergent from the gravitational field, determined by matter. This reflects his Machian influences and the philosophical shift in general relativity, where geometry (space-time) is dynamical and tied to physical content. He contrasts it with ether theories, emphasizing that "there exists no space empty of field."
Academic sources:

Wikisource/Gutenberg editions of the book: Full text available, confirming the quote in Appendix V.

"Relativity: The Special and the General Theory - 100th Anniversary Edition" (Princeton University Press, 2015): Includes the appendix with historical notes on editions.

John Earman, "The Hole Argument and Some Physical and Philosophical Implications" (Living Reviews in Relativity, 2014): Analyzes the quote in the context of space-time substantivalism debates.

^{*3 -} Albert Einstein, summary of general relativity, as cited in Ideas and Opinions (Crown Publishers, 1954), p. 376.
Context: A concise encapsulation of relativity's relational core: physical properties emerge from interactions, not absolutes, echoing Mach's and Leibniz's critiques.
Academic sources:
Appears in multiple editions of Ideas and Opinions; often referenced in biographical works like Abraham Pais's Subtle is the Lord (1982).

^{*4 -} "Physics and Reality" (1936), reprinted in Journal of the Franklin Institute 221 (1936), p. 313; also in Ideas and Opinions (1954).
Context: Einstein invokes Leibniz's concept to describe the relational harmony in physics, where laws emerge from empirical relations rather than absolutes.
Academic sources:
Full essay in journal archives; CPAE Vol. 6.

@Xerxes

This season of Guy Fawkes is is good time for all to examine the underpinning of their conclusions.

Sadly you have refused to do this.

In particular you have refused to explain how you can square anything that is not a number (with or without units),

despite being asked several times.

Instead your alleged response to any question has been repetition of the original unfounded presentation and all the self contradictions it contains.

Unfortunately the broken record technique is neither valid in Science nor Philosophy.

So I am faced with two options.

I could formally report this thread for total lack of support for the claims made.

Or I could simply walk away, shaking my head in sorrow, like others who also hoped for a useful and perhaps enlightening discussion.

3 hours ago, studiot said:

In particular you have refused to explain how you can square anything that is not a number (with or without units),

despite being asked several times.

Your "question" is nonsensical. It based on assumptions that you made by yourself and did not vocalize them. I genuinely do not understand what do you mean by "not a number". Im dealing only with numbers. What line of assumptions brought you to conclusion that im squaring "not numbers"? Please clarify your reasoning.

40 minutes ago, Anton Rize said:

Your "question" is nonsensical. It based on assumptions that you made by yourself and did not vocalize them. I genuinely do not understand what do you mean by "not a number". Im dealing only with numbers. What line of assumptions brought you to conclusion that im squaring "not numbers"? Please clarify your reasoning.

I have to agree with Xerxes. It is arrogant to say that because you do not understand it a statement or question is nonsense.

(Note I do not agree with his comment about geometry. There are areas of geometry that do not require a metric and the 'metry' refers to the process of measurement, not to a metric. In fact the full translation is the measurement of the Earth (Geo).

Of course the anything in (anything)² is a number.

There is nothing I said that suggests otherwise.

I was rather expecting you to tell me (correctly) that if one wants to take the square 'anything' must be a number.

That is exactly my point.

Of course, also, the anything may or may not have units.

If it does have units then squaring will also square the units.

It is also true that beta and kappa are length variables which in turn means that you must have a metric to assign numerical values to them.

In ordinary old fashioned geometry a metric allows the congruence relation, a lack of a metric is restricted to a similarity relation (which I have already mentioned without response).

In my last but one post I offered you a similar (comparable) situation in terms of the 'constitutive equations or relations' of fluid mechanics where a constraint is defined by an invariant constant.

Unfortunately the silence of your response to this was deafening.

41 minutes ago, studiot said:

It is arrogant to say that because you do not understand it a statement or question is nonsense.

agree. I apologise. What I meant is "It seems nonsensical to me".

43 minutes ago, studiot said:

It is also true that beta and kappa are length variables which in turn means that you must have a metric to assign numerical values to them.

Thank you for clarifying this. To make sure that there's no misunderstanding happening Could you also clarify hoe exactly you came to conclusion "beta and kappa are length variables" and what exactly do you mean by "length" please?

On 11/3/2025 at 8:40 AM, KJW said:

@Anton Rize, it seems to me that you do not have a proper understanding of general relativity. For example, you talk about spacetime being an independent background that you reject, whereas in general relativity, spacetime is not an independent background. For the Einstein equation, given the energy-momentum density distribution, one obtains the metric tensor field as a solution. The metric tensor field contains the information that determines both the energy-momentum density field and the gravitational field.

Now I see where this comment came from. You misunderstood me, and it helps perfectly isolate the root of our disagreement. You are absolutely correct: in GR, SPACETIME is a dynamic solution, determined by the energy-momentum tensor ([math]T_{\mu\nu}[/math]). I completely agree with this. My argument is not about the metric but about

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

the postulate of a background

It is about the underlying 4D manifold - the "arena" or "stage" on which the metric is defined in the first place. My thesis is that this manifold itself is the "background" that GR postulates (assumes a priori), but does not derive. Therefore, my question to you as a GR expert is:
Do you agree that the 4D manifold itself is a non-dynamic, a priori presupposition in the standard formulation of GR, separate from the metric that it supports?

On 11/6/2025 at 9:11 PM, KJW said:

What you are essentially saying is that some observations are invalid because they don't conform to some theoretical framework. However, it's important to note that any geocentric coordinate system needs to be constructed in accordance with general relativity, and in such a case will never disagree with observation in any way that a heliocentric coordinate system agrees with observation. So, it isn't a case of heliocentrism verses geocentrism because general relativity isn't choosing one over the other.

Yes I agree geocentric coordinate system can be constructed within GR and will remain coherent. Similar way how Geocentric model was agreeing with observations thruogh epicycles adding new levels of mathematical complexity. This example clearly shows the value of ontological transparency and mathematical simplicity in physics. It leads directly to the "Operational differences" I pointed above..

On 11/6/2025 at 11:50 PM, KJW said:

The gravitational constant G comes from Newtonian gravitation theory, so if this constant is anywhere in your theory, it came from Newtonian gravitation theory, even if indirectly. And if you avoid using this constant, then any mass or energy that is related to geometry can't be expressed in terms of kilograms or other familiar units of mass or energy.

You are correct that [math]G[/math] comes from Newton, but you are mistaken about its role in RG. My model is fundamentally dimensionless, built on the ratios [math]\beta[/math] and [math]\kappa[/math]. [math]G[/math] is not an axiom. It is a translation constant. I use it only to connect my dimensionless model to our historical, "cultural" unit: the kilogram ([math]m_0[/math]). I can prove [math]\kappa[/math] is fundamental and [math]G[/math] is not. We can measure [math]\kappa[/math] without using [math]G[/math]. Gravitational time dilation ([math]\tau[/math]) is a pure, measurable number (e.g., from redshift). In my model (for a stationary observer), [math]\tau = \kappa_X = \sqrt{1 - \kappa^2}[/math]. An observer can measure [math]\tau[/math] and algebraically find [math]\kappa[/math] without ever knowing [math]G[/math]. This proves [math]\kappa[/math] is the real, measurable physical quantity, while [math]G[/math] is just a historical "converter". I am happy to share all the detailed derivations if you are interested.

On 11/6/2025 at 9:11 PM, KJW said:

Thus, I put it back to you: How can the observation of a space in which we exist be invalid?

I didn't say "invalid". As the name of this thread states: "Simplifying SR and GR with Relational Geometry - Algebraic Derivations Without Tensors. Testing and discussion."
My argument lays in the realm of ontological transparency and mathematical simplicity maintaining same calculational results with potential for some unique predictions as I showed with Galactic Rotation Curves results above.

On 11/6/2025 at 11:50 PM, KJW said:

As I said, general relativity must reduce to Newtonian theory in the weak-field limit. So, comparing general relativity in the domain where the relationship between geometry and energy has already been empirically established allows one to extend this relationship to the domain where general relativity is on its own. It is not about accepting Newtonian theory as an ontologically correct theory but accepting Newtonian theory as an empirically accurate theory within its domain of applicability.

This argument only proves my point. I already showed how Newtonian mechanics derived from RG as ontologically collapsed approximations without need for any calibrations:

On 10/30/2025 at 11:32 PM, Anton Rize said:

Derivation of statement:
"Complex mathematics is the consequence of bad philosophy"

Classical Keplerian Energy as a WILL-Minkowski Projection

For a test body of mass m on a circular orbit of radius a about a central mass M⊕, classical Newtonian mechanics gives:

ΔU=−GM⊕ma+GM⊕mR⊕

K=12mGM⊕a

Adding these and dividing by the rest-energy E0=mc2 yields the dimensionless total:

EtotE0=GM⊕R⊕c2−12GM⊕ac2

---

Projection Parameters and Minkowski-like Form

Define the WILL projection parameters for the surface and the orbit:

κ2⊕≡2GM⊕R⊕c2

β2orbit≡GM⊕ac2

Substituting into the classical equation gives the exact identity:

EtotE0=12(κ2⊕−β2orbit)

This is already in the form of a hyperbolic difference of squares. If we set x≡κ⊕ and y≡βorbit, then:

EtotE0=12(x2−y2)

This is structurally identical to a Minkowski interval in (1+1) dimensions.

---

Physical Interpretation

In classical derivations, this is just the sum ΔU+K with a particular choice of potential zero. In WILL Relational Geometry, this form emerges directly from the Energy-Symmetry Law:

ΔEA→B=12((κ2A−κ2B)+β2B)

(with (A,B)=(surface, orbit))

This shows that the Keplerian total energy is not an isolated Newtonian artifact but a special case of a deeper geometric structure. The emergence of this Minkowski-like structure from purely energetic principles is a powerful indicator of the deep identity between the geometry of spacetime and the geometry of energy transformation.

---

Lagrangian and Hamiltonian as Ontologically Corrupted Approximations

The familiar Lagrangian and Hamiltonian formalisms are not fundamental principles but approximations that arise from collapsing the two-point relational structure of the Energy-Symmetry Law into a single-point description.

The "Ontological Collapse"

If one commits the ontological violation by identifying the two distinct points, rA=rB=r, the relational structure degenerates into a local, single-point function:

L(r,r˙,ϕ˙)=12m(r˙2+r2ϕ˙2)+GMmr

This is precisely the standard Newtonian Lagrangian.

Defining the Hamiltonian via the Legendre transformation H=prr˙+pϕϕ˙−L evaluates to the total energy of the collapsed system:

H=T+U=12m(r˙2+r2ϕ˙2)−GMmr

Interpretation

In terms of the collapsed WILL projections β2=v2/c2 and κ2=2GM/(rc2), the match becomes explicit:

L=12mv2+GMmr⟷12mc2(β2+κ2)

H=12mv2−GMmr⟷12mc2(β2−κ2)

Here the + or - signs do not come from κ2 itself (which is always positive), but from the ontological collapse of the two-point relational energy law into a single-point formalism.

> Key Message: The Lagrangian and Hamiltonian are not fundamental principles. They are degenerate shadows of a deeper relational Energy-Symmetry Law. Classical mechanics, Special Relativity, and General Relativity all operate within this corrupted approximation. WILL restores the underlying two-point relational clarity.

---

Third Ontological Collapse: Derivation of Newton's Third Law

Newton's Third Law is another "degenerate shadow" that arises as a necessary mathematical consequence of this same ontological collapse.

Theorem: Newton's Third Law as a Degenerate Consequence

The Energy-Symmetry Law (ΔEA→B+ΔEB→A=0) mathematically necessitates Newton's Third Law (F⃗ AB=−F⃗ BA) in the classical limit where the two-point relational energy budget is collapsed into a single-point potential function U(r⃗ ).

Proof:

Begin with the foundational Energy-Symmetry Law:

ΔEA→B+ΔEB→A=0

In the classical limit, this is "corrupted" into a single-point potential function U that depends only on the relative positions:

U=U(r⃗ )wherer⃗ =r⃗ B−r⃗ A

The force F⃗  is defined as the negative gradient of this potential.

(1) Force on B by A:

F⃗ AB=−∇BU(r⃗ B−r⃗ A)=−(dUdr⃗ )⋅(∂r⃗ ∂r⃗ B)=−∇U(r⃗ )

(2) Force on A by B:

F⃗ BA=−∇AU(r⃗ B−r⃗ A)=−(dUdr⃗ )⋅(∂r⃗ ∂r⃗ A)=+∇U(r⃗ )

(3) Conclusion:

By direct comparison:

F⃗ AB=−∇U(r⃗ )andF⃗ BA=+∇U(r⃗ )

Therefore, it is a mathematical tautology of the collapsed formalism that:

F⃗ AB=−F⃗ BA

This completes the proof. The law of "equal and opposite forces" is revealed to be a degenerate approximation of the more fundamental, generative law of Relational Geometry.

---

General Consequence

Bad philosophy (the ontological separation of Structure and Dynamics) has measurable effects:

1. Inflated Formalism: Equations multiply to compensate for the error.

2. Loss of Transparency: Physical meaning is hidden behind coordinate dependencies.

By contrast, good philosophy (**epistemic hygiene**) enforces relational closure and yields simplicity through necessity.

> Daring Remark: The historical escalation of mathematical complexity in physics did not reveal deeper reality—it compensated for a philosophical mistake. Once the ontological symmetry is restored, Nature’s laws reduce to algebraic self-consistency.

>

> Bad Philosophy⇒Ontological Duplication⇒Mathematical Inflation

> Complex mathematics is the consequence of bad philosophy.



Yes I know its to daring and to radical. But this is exactly why I love it so much. It makes you think it makes you feel and maybe it will help someone to see the bigger picture...

12 hours ago, Anton Rize said:

agree. I apologise. What I meant is "It seems nonsensical to me".

Thank you for clarifying this. To make sure that there's no misunderstanding happening Could you also clarify hoe exactly you came to conclusion "beta and kappa are length variables" and what exactly do you mean by "length" please?

Given that you (hopefully) have some respect for the intellect of the other person the rational consequence of "It seems nonsensical to me" is to ask what they mean.

This is exactly what I am trying to do for you, and why I say your second paragraph is so much better +1.
Asking about Length is an excellent question.

In the context of your proposal and our discussion, Length is a specific case of distance, Distance being the more general.

Length is the distance between two points on the same object, usually 'start' and 'end' points.

More generally distance includes the separation of two different objects, or the same object after movement.

The thing about both Length and distance is that they can be considered very generally, without numbers, and some relations concerning them can be studied.

For instance in elementary plane geometry we learn about similar triangles.

Without numbers we can deduce that they are the same shape and that the (Lengths) of the sides are all in the same proportion.

(Which is what I think you are trying to do with your diagram).

The instant we introduce numbers, some very interesting things happen.

We can give the concept (length)² a meaning and see that it is different from the concept of (Length) in a very special way.

If we put successive numbers into (Length) we obtain an arithmetic series.

If we put successive numbers into (Length)² we obtain a geometric series.

In your case you have β = r*sinθ and κ = r*cosθ or the other way round I can't remember which so please correct me if I got it wrong.

so (r*sinθ)² + ( r*cosθ)² = r²(sin²θ + cos²θ)

By introducing the object (radius) r you have introduced Length and and thereby introduced a metric.

Note that a metric may be formed by many different distance functions, which may produce different results.

The metric must contain only one distance function.

In your case the Manhatten metric coincides with the euclidian metric for Length, but not for distance.

This is because Length is always measured along a line.

I will be going out soon for the rest of Saturday.

Edited November 8Nov 8 by studiot

16 minutes ago, studiot said:

n your case you have β = r*sinθ and κ = r*cosθ

so (r*sinθ)² + ( r*cosθ)² = r²(sin²θ + cos²θ)

Thank you. The core of the issue is here: you stated, "In your case you have [math]\beta = r \cdot \sin\theta[/math] and [math]\kappa = r \cdot \cos\theta[/math]". With all due respect, this is a fundamental misinterpretation. I dont know where you get it from but not from me.

[math]\beta[/math] and [math]\kappa[/math] are the primary, ratios (pure, dimensionless numbers) geometric projections
My equation [math]Q^2 = \beta^2 + \kappa^2[/math] is a purely algebraic relationship between these two ratios. The diagrams I've used are just one possible visualization of this algebraic budget. This is the very essence of the Relationalism (RL) that I am trying to explain, and which - despite all my attempts - you seem to refuse to acknowledge:

18 hours ago, studiot said:

Instead your alleged response to any question has been repetition of the original unfounded presentation and all the self contradictions it contains.

Unfortunately the broken record technique is neither valid in Science nor Philosophy.

The ratios are primary; and geometry is just a conventional visual presentation and does not imply or require a specific, pre-existing metric.

Out of respect for your intellect, I must assume you can understand this fundamental distinction, but for reasons I cannot grasp, you are choosing not to. Im starting to suspect that you just trolling me. Its been almost 3 weeks and you still have no clue what Im talking about. That is quite telling... Your next response might be the last one that I will not ignore.

Edited November 8Nov 8 by Anton Rize

9 hours ago, Anton Rize said:

Do you agree that the 4D manifold itself is a non-dynamic, a priori presupposition in the standard formulation of GR, separate from the metric that it supports?

The whole thing starts with two fundamental observations about the world we live in:

1. Events in the real world are separated - they can be spatially separated (things don’t all happen at the same place), temporally separated (not everything happens simultaneously), or both.

2. Not all events are capable of influencing all other events - only events separated in particular ways are causally connected. In other words, the world is endowed with a causal structure (at least classically).

The job of physics is to make models of aspects of the world we live in. Einstein, using pre-existing work by Minkowski, Riemann and others, realised that, if one uses a semi-Riemannian manifold endowed with a metric and a connection, and allows the metric to vary while holding the connection fixed (the Levi-Civita connection), one obtains a mathematical model wherein the points on the manifold are related in the same way as events in the real world are observed to be causally related. In small enough local patches this gives you SR with its light cones, whereas globally it gives you GR and hence gravity. In either case it boils down to causal structure.

So the assumption that GR makes is that one can use a semi-Riemannian manifold endowed with the Levi-Civita connection and a varying metric as a good model for the causal structure of the world we live in. I highlighted “can” because no claim is made that this is the only possible way to model these dynamics. Many other models of gravity are known nowadays, not all of which are metric models either; but GR seems to be the simplest, and the one that works best AFAWCT.

BTW, the Einstein equations don’t a priori assume any specific dimensionality. One advantage of tensor equations other than general covariance is that their form remains the same in any number of dimensions. The real world looks 4D, but GR works just fine in other situations too.

1 hour ago, Anton Rize said:

Its been almost 3 weeks and you still have no clue what Im talking about

Seems to me that it’s rather the other way around - despite multiple posters having patiently attempted to explain to you why, in your proposal, you are implicitly using some of the concepts you initially rejected, you’re still not getting it. I strongly suspect that’s because you don’t want to get it.

To be honest, I think we’re done here.

19 minutes ago, Markus Hanke said:

Seems to me that it’s rather the other way around - despite multiple posters having patiently attempted to explain to you why, in your proposal, you are implicitly using some of the concepts you initially rejected, you’re still not getting it. I strongly suspect that’s because you don’t want to get it.

To be honest, I think we’re done here.

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, and you @Markus Hanke for 2 weeks yapping about "using some of the concepts you initially rejected" but failing to show me any mathematical form that could support your claim or at least make it less vague - it becomes clear that none of you accept @KJW actually looked at my math. So it rise the question:

Why are you here? What are you trying to achieve? Show the math or leave. I'm sick of this bullshit! For three weeks I was answering nicely every question and now I see that you haven't even looked at my answers and yet keep criticizing. You couldn't pass a bare minimum requirement - read presented and still have an arrogance to teach me basics of GR. This is truly shameful!

29 minutes ago, Anton Rize said:

What are you trying to achieve?

I’m trying to show you that your formalism requires some of the very things you reject. Orthogonality requires some notion of inner product. Gradients require derivatives and a metric, as do lengths. And so on. These are things you can easily research yourself, I don’t think it is always necessary to typeset LaTeX for stuff I would consider basic and easily found with a simple search, after one has been made aware of them. If you presume to be simplifying GR, you are operating at a level where you can be reasonably presumed to be able to do this.

The problem is that you are so sure that your idea must be right, that you are no longer receptive to feedback.

BTW, there already exists an almost purely algebraic formulation of GR - check out the papers by Geroch and Heller on Einstein algebra. Note though this still requires the concept of smoothness as an additional ingredient.

15 hours ago, Anton Rize said:

It is about the underlying 4D manifold - the "arena" or "stage" on which the metric is defined in the first place. My thesis is that this manifold itself is the "background" that GR postulates (assumes a priori), but does not derive. Therefore, my question to you as a GR expert is:
Do you agree that the 4D manifold itself is a non-dynamic, a priori presupposition in the standard formulation of GR, separate from the metric that it supports?

As I see it, the mathematics of general relativity doesn't describe the "underlying 4D manifold" itself. Even the overlayed coordinate system, though referenced by some of the equations, is not explicitly defined. I believe you see this as a bad thing... I do not. Over the coordinate system are two types of fields: (1), arbitrary fields of various types whose purpose is to provide a framework for the equations, for example the coordinate transformation law for tensors, where the tensor field is an arbitrary field; and (2), fields that implicitly define the coordinate system and the "underlying 4D manifold" itself, providing structure.

However, particular metrics such as the Schwarzschild metric are expressed using coordinate variables that bear some resemblance to physical variables.

In the Einstein equation, only the specification of the source term (the energy-momentum tensor field) provides any definition of the coordinate system specified by the metric tensor solution. So, if one has two distinct physical energy-momentum density distributions that have the same mathematical source term in terms of their respective coordinate system variables, then the metric tensor solutions in terms of their respective coordinate system variables will also be the same, differing only in the background metric specified by the auxiliary conditions. I regard this as a form of Mach's principle within general relativity. An example of this is frame-dragging, where the rotation of a star or planet tends to drag the surrounding spacetime around with it.

15 hours ago, Anton Rize said:

  On 11/6/2025 at 9:11 PM, KJW said:

What you are essentially saying is that some observations are invalid because they don't conform to some theoretical framework. However, it's important to note that any geocentric coordinate system needs to be constructed in accordance with general relativity, and in such a case will never disagree with observation in any way that a heliocentric coordinate system agrees with observation. So, it isn't a case of heliocentrism verses geocentrism because general relativity isn't choosing one over the other.

Yes I agree geocentric coordinate system can be constructed within GR and will remain coherent. Similar way how Geocentric model was agreeing with observations thruogh epicycles adding new levels of mathematical complexity. This example clearly shows the value of ontological transparency and mathematical simplicity in physics. It leads directly to the "Operational differences" I pointed above..

While physicists will typically choose the simplest coordinate system in which to investigate a problem, being the simplest coordinate system does not in any way make this coordinate system any more valid than any other coordinate system. Simplicity or complexity are not criteria for determining the validity of a coordinate system. All coordinate systems are equally valid. As you know, reality doesn't come with a coordinate system. Coordinate systems are imposed by humans. But in fact, coordinate systems are also a theoretical notion and as such, we can speak of "all possible coordinate systems". And because reality doesn't come with a coordinate system, there is no basis for a preferred coordinate system. General relativity is about having equations that are the same in all coordinate systems. Thus, the equations of general relativity do not select a preferred coordinate system, or indeed any coordinate system. But it is not trivial that the equations of general relativity are the same in all coordinate systems. I mentioned earlier that tensor equations are the same in all coordinate systems. However, not all quantities are tensors (not all quantities obey the coordinate transformation law for tensors).

15 hours ago, Anton Rize said:

[math]G[/math] is not an axiom. It is a translation constant. I use it only to connect my dimensionless model to our historical, "cultural" unit: the kilogram [math](m_0)[/math].

That is no different from general relativity. However, when physicists refer to mass, they are referring to it in units of kilograms (or some other comparable units). And it is desirable to determine the spacetime curvature associated with mass specified in kilograms. To do this, it is necessary to use the measured value of G from the Cavendish experiment and apply Newtonian gravitation. Current technology is not sufficiently accurate to directly measure the spacetime curvature associated with kilogram masses. It is worth noting that for astronomical work, the quantity GM (m³ s^–2) is far more accurately known than either G or M alone. And using the Schwarzschild radius to specify the mass of a star or planet does allow one to avoid G because the mass is no longer specified in kilograms.

15 hours ago, Anton Rize said:

Gravitational time dilation ([math]\tau[/math]) is a pure, measurable number (e.g., from redshift). In my model (for a stationary observer), [math]\tau = \kappa_X = \sqrt{1 - \kappa^2}[/math]. An observer can measure [math]\tau[/math] and algebraically find [math]\kappa[/math] without ever knowing [math]G[/math].

Time dilation relates to acceleration and therefore doesn't involve mass.

12 hours ago, Anton Rize said:

  On 11/6/2025 at 11:50 PM, KJW said:

As I said, general relativity must reduce to Newtonian theory in the weak-field limit. So, comparing general relativity in the domain where the relationship between geometry and energy has already been empirically established allows one to extend this relationship to the domain where general relativity is on its own. It is not about accepting Newtonian theory as an ontologically correct theory but accepting Newtonian theory as an empirically accurate theory within its domain of applicability.

This argument only proves my point.

I don't see how. By comparing general relativity in the weak-field limit with Newtonian theory, I'm not in any way weakening general relativity or making it an approximation.

12 hours ago, Anton Rize said:

Classical mechanics, Special Relativity, and General Relativity all operate within this corrupted approximation.

You seem to be suggesting that general relativity is an approximation, and that your theory deviates from general relativity in a way that better agrees with empirical observation. This is a claim that needs justification.

Edited November 8Nov 8 by KJW

Thank you for your detailed response @KJW . Its amazing how different our views are. Pretty much on every point discussed we maintain opposite views. Its a great opportunity for us to shake each others foundational views and reconsider our owns. Im exited! Lets go step by step:

3 hours ago, KJW said:

Over the coordinate system are two types of fields: (1), arbitrary fields of various types whose purpose is to provide a framework for the equations, for example the coordinate transformation law for tensors, where the tensor field is an arbitrary field; and (2), fields that implicitly define the coordinate system and the "underlying 4D manifold" itself, providing structure.

You have put forward a very strong and classic interpretation: that the manifold is "implicitly defined" by the tensor solution ([math]g_{\mu\nu}[/math]), and you even call this "a form of Mach's principle" in action, citing frame-dragging. This gets to the very heart of our disagreement, because this argument appears to be a case of circular logic. My question remains: How can the tensor solution ([math]g_{\mu\nu}[/math]) "implicitly define" the manifold, when the manifold itself must be presupposed (assumed a priori) just to be able to write down the tensor [math]G_{\mu\nu}[/math] in the first place? The entire machinery of tensor calculus requires a smooth manifold on which to operate. You are describing what Einstein wanted GR to be, not what it is. Einstein himself was famously dissatisfied with this very point. In his later years, he wrote about his failure to fully incorporate Mach's principle into GR. He was frustrated that his theory still required this a priori "arena" (the manifold), which had an existence independent of the "players" (the fields/matter). So, when I ask if the manifold is an "a priori presupposition", I am actually agreeing with Einstein's own critique of his theory. When you claim it is not a presupposition and is Machian, you are (respectfully) describing a philosophical goal that GR, by its creator's own admission, never achieved:

On 11/7/2025 at 9:46 PM, Anton Rize said:

Quote: "It is contrary to the spirit of relativity theory to treat space as an independent thing."^*1

On 11/7/2025 at 9:46 PM, Anton Rize said:

Context: Einstein was responding to Mach's ideas on inertia and relativity. He praised Mach's critique of absolute space (from Newton's mechanics) and emphasized that relativity theory views space not as an absolute, independent entity but as relational, influenced by matter. This aligns with Einstein's push against "opportunism" in physics that avoids deeper thinking about space's nature. The letter reflects Einstein's early enthusiasm for incorporating Mach's principles into general relativity, though he later distanced himself somewhat from "Mach's principle."

Your comment about "simplicity" I think isolates the source of our misunderstanding.

When you argue that:

3 hours ago, KJW said:

Simplicity or complexity are not criteria for determining the validity of a coordinate system.

It strongly suggests that you are (perhaps unintentionally) evaluating RG as if it were another coordinate system within GR. This is the very "categorical error" I have been trying to highlight.
RG is not a new coordinate system; it is an entirely separate theory built on a different foundation. It seems we are stuck in a classic "paradigm" problem. The axioms of one system (like GR) can become so ingrained that they become invisible, like "the water a fish can't see". From within that system, it's difficult to analyze the ontological basis of another system. This is why I was so persistent about the importance of the "Operational distinction between SB and RL", as it defines the two different "worlds" we are arguing from: 

3 hours ago, KJW said:

In Substance-Based (SB):

• space-time is a pre-existing metric manifold.

• A single object can be assigned coordinates on this manifold, and physical quantities (such as position, velocity, and distance) are defined with respect to that dynamical but still background.

• The origin (0,0) is arbitrary - a conventional point on an independent grid.

In Relational (RL):

• space and time have no existence apart from the relations among physical entities.

• An isolated object has no meaningful coordinates at all; every observer defines its own (0,0) relational origin.

• Only relations between such self-defined frames are physically meaningful.

In order for me to bridge this gap between our two perspectives, could you tell me how you are seeing the difference between RG and GR at this point?

5 hours ago, Anton Rize said:

How can the tensor solution ([math]g_{\mu\nu}[/math]) "implicitly define" the manifold, when the manifold itself must be presupposed (assumed a priori) just to be able to write down the tensor [math]G_{\mu\nu}[/math] in the first place? The entire machinery of tensor calculus requires a smooth manifold on which to operate.

Ok, here's how I personally see it: We start from an n-tuple of real-numbered (or perhaps complex-numbered) variables, typically [math](x^1,...,x^n)[/math]. This is a coordinate system. Over this coordinate system are various functions of the coordinate system variables. These functions can be scalar fields, components of vector fields, or object fields in general. Also, we can set up another n-tuple of variables, say [math](y^1,...,y^n)[/math] such that each of the [math]y^p[/math] are differentiable functions of all of the [math]x^q[/math] and this system of functions can be inverted so that each of the [math]x^r[/math] are differentiable functions of all of the [math]y^s[/math]. The functions of one set of coordinate variables in terms of another set of coordinate variables is a coordinate transformation. Under a coordinate transformation, the object field functions of the [math](x^1,...,x^n)[/math] variables are transformed to the corresponding object field functions of the [math](y^1,...,y^n)[/math] by the composition of functions, for example:

[math]f(x^1,...,x^n) = f(x^1(y^1,...,y^n),...,x^n(y^1,...,y^n)) = \bar{f}(y^1,...,y^n)[/math]

as well as some coordinate transformation law in accordance with the mathematical properties of the object field. Implicit in the notion of a coordinate transformation is that [math](y^1(x^1,...,x^n),...,y^n(x^1,...,x^n))[/math] is the same location as [math](x^1,...,x^n)[/math] even though [math](y^1,...,y^n)[/math] may be numerically different to [math](x^1,...,x^n)[/math]. Also implicit in the notion of a coordinate transformation is that an object field in one coordinate system is the same object field in the other coordinate system, even though they may numerically differ.

The idea is that I can use the above to create a mathematical description of physical reality, and as a mathematical description, it has mathematical properties that can be determined mathematically and can be correlated back to the physical reality. While one might wish to create a mathematical description of the actual physical reality, I'm more interested in mathematical descriptions of hypothetical arbitrary realities, logically deriving "laws of physics". It is only natural that because physical reality looks like a space over which a coordinate system can be applied, that the above is the appropriate formalism to mathematically describe physical reality.

At this point, there is no metric.

Because physical reality has no coordinate system, and because the above coordinate systems are arbitrary, the laws of physics must be independent of any coordinate system and therefore be expressed as tensor equations. However, the partial derivative of a tensor is not a tensor. Thus, one must introduce a connection object field, also not a tensor, that converts a partial derivative to a covariant derivative, which is a tensor. The coordinate transformation law of the connection object field compensates for the non-tensor coordinate transformation law of the partial derivative. If one attempts to coordinate-transform the connection object field to zero, so that the covariant derivative is equal to the partial derivative, then this is only possible if a particular expression in terms of the connection object field is zero. This particular expression defines the Riemann curvature tensor field. What we have now is the existence of non-equivalent mathematical descriptions, identifiable by the connection object field as well as the Riemann curvature tensor field.

If one makes a couple of assumptions (which I won't discuss because they go beyond standard general relativity), the metric tensor can be introduced such that the connection object field can be expressed in terms of the metric tensor and its first-order partial derivatives. The metric tensor introduces the notion of magnitude to the coordinate system. It also identifies non-equivalent mathematical descriptions. It is in this sense that one can say that the metric "implicitly defines the coordinate system and the 'underlying 4D manifold' itself, providing structure". That is, although the metric is expressed in terms of the coordinate system, in doing so it implicitly defines the coordinate system, as well as the "underlying 4D manifold" due to its identification of non-equivalent mathematical descriptions.

Edited November 9Nov 9 by KJW

I see RG as a somewhat abstract theory that seems to coincidently exploit a particular simplicity of the Schwarzschild metric in coordinates for which the radial coordinate maintains a Euclidean spherical surface area (there is a tendency for particular formulae to agree with Newtonian theory as a result). It's not clear to me if general relativity was used during the derivation of RG, even though you do use the Schwarzschild radius. I also note that according to the virial theorem under Newtonian gravitation:

2<T> = –<V>

where <T> is the time average of the total kinetic energy of all the particles, and <V> is the time average of the total potential energy of all the particles. Apart from the minus sign, it seems to correspond with your

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

noting that [math]\beta^2[/math] is kinetic and [math]\kappa^2[/math] is gravitational.

I would like a detailed explanation of:

Gravitational Lensing [math]\alpha = 2\kappa^2[/math]

Overall, it seems to me that your theory would be too simple to handle more difficult problems in general relativity.

However, I am still unable to figure out the rationale behind your formulae. I feel you need to ground your theory on a more well-known basis instead of what seems to me to be a little too esoteric.

Thank you @KJW for your detailed answer.

You’re absolutely right that GR starts from a differentiable manifold: an [math]n[/math]-tuple on which tensor fields live, transforming covariantly under coordinate changes.

RG begins one step before that assumption - it asks what minimal relational structure must exist for such coordinate transformations to even make sense.

In GR you have a split:

[math]\text{structure as} (\text{manifold + metric}) + \text{dynamics as} (\text{fields + constants}).[/math]

RG removes that split by identifying structure and dynamics as two projections of the same relational invariant:

[math]\textbf{Spacetime} \equiv \textbf{Energy}.[/math]

Once you do that, two closed, maximally symmetric carriers appear automatically - [math]S^1[/math] (kinematic) and [math]S^2[/math] (gravitational) - with closure relations

[math]\beta_X^2+\beta_Y^2=1,\quad \kappa_X^2+\kappa_Y^2=1.[/math]

Their ratio of degrees of freedom gives the exchange law

[math]\kappa^2 = 2\beta^2,[/math]

which algebraically reproduces all stationary GR results when [math]\kappa^2=2GM/(rc^2)[/math].

The “virial-like” look isn’t coincidental - it’s geometric closure rather than time-averaged dynamics.

Lensing [math]\alpha = 2\kappa^2[/math] follows because light occupies the single-axis state [math]\beta=1[/math]; its full energy budget lies on the [math]\kappa[/math] projection, producing the factor of two that GR attributes to curvature + time dilation.

GR therefore reappears as the differential expansion of this algebraic identity:

[math]\kappa^2 = R_s/r = \rho/\rho_{\max}.[/math]

RG doesn’t replace GR - it compresses it to its pre-metric core.

---

Here's the logical flow diagram. Ich step is a logical necessity of the previous so no fitting or borrowing taking place. I'm happy to provide rigours derivation for any of this steps if you want. Please let me know if it helps to get the idea:

Edited November 10Nov 10 by Anton Rize

10 hours ago, Anton Rize said:

You’re absolutely right that GR starts from a differentiable manifold: an [math]n[/math]-tuple on which tensor fields live, transforming covariantly under coordinate changes.

I deliberately chose not to use the term "differentiable manifold". That term gives the subject a topological viewpoint. No, this is about analysis. It is an extension of the mathematics that is taught at school, so it is at the foundations of mathematics. You might argue that pedagogical foundations are different to logical foundations, but is it really? I'm starting from the Cartesian [math]n[/math]-th power of the real number line (or of the complex number plane) [math]\mathbb{R}^n[/math] (or [math]\mathbb{C}^n[/math]), defining functions over that domain similar to the way a school student would recognise [math]f(x)[/math], except that I'm considering [math]f(x^1,...,x^n)[/math], a function of [math]n[/math] variables instead of one, identifying each of the [math]x^p[/math] with each of the real number lines of [math]\mathbb{R}^n[/math] (or each of the complex number planes of [math]\mathbb{C}^n[/math]) using superscript indices [math]p = 1\ \text{to}\ n[/math]. And just as function [math]f(x)[/math] returns a single number from the domain of [math]\mathbb{R}[/math] (or [math]\mathbb{C}[/math]), a function [math]T^{p_1,...,p_r}_{q_1,...,q_s}(x^1,...,x^n)[/math] can return [math]n^{r+s}[/math] values from the domain of [math]\mathbb{R}^{n^{r+s}}[/math] (or [math]\mathbb{C}^{n^{r+s}}[/math]). With the introduction of the notion of limits, calculus can be performed. But note that by considering the [math]n[/math] partial derivatives of [math]f(x^1,...,x^n)[/math]:

[math]\dfrac{\partial f(x^1,...,x^n)}{\partial x^p}[/math] for [math]p = 1\ \text{to}\ n[/math]

one has naturally introduced a function that returns a vector. So really, I could've introduced limits (calculus) at the [math]f(x^1,...,x^n)[/math] stage and allow functions that return higher level objects to define themselves naturally.

So, by using the term "differentiable manifold", you've made what I said above seem more advanced and esoteric than it really is.

10 hours ago, Anton Rize said:

RG begins one step before that assumption

I don't think there really is "one step before that assumption". You have introduced a "Topological Requirement" in your logical flow diagram, suggesting to me that you believe Topology is a foundational subject in mathematics, and that because your topology has less structure than a differentiable manifold, that your theory is closer to first principles than the way I described the foundations of general relativity. But I personally regard mathematics as the manipulation of symbols that are defined by symbolic expressions called axioms, and that a theorem can only be considered to be rigorously proven if it can be done mechanically. This is an ideal that I have yet to achieve, though I have learnt a lot from my attempts. Thus, to me, mathematics is like a tree where the theories that are higher up rely on theories that are below it but not on the theories that are above it. But because of the symbolic nature of mathematics, the foundations of mathematics (the trunk of the tree) is based on the logic of symbol manipulation rather than what the symbols may represent.

Please explain how you get from RC to a "differentiable manifold".

10 hours ago, Anton Rize said:

it asks what minimal relational structure must exist for such coordinate transformations to even make sense.

The way I see it, the foundations of reality are not purely mathematical. I ask what can be assumed to be true without any prior knowledge of how reality behaves? The fundamental answer to this question is that reality can be mathematically described and that any knowledge gained about the properties of mathematical descriptions translate to the properties of reality. And because reality looks like a "differentiable manifold", this can form the basis of a mathematical description of reality.

10 hours ago, Anton Rize said:

In GR you have a split:

[math]\text{structure as} (\text{manifold + metric}) + \text{dynamics as} (\text{fields + constants}).[/math]

I don't think I do that. Please explain why you think I do that.

10 hours ago, Anton Rize said:

Lensing [math]\alpha = 2\kappa^2[/math] follows because light occupies the single-axis state [math]\beta=1[/math]; its full energy budget lies on the [math]\kappa[/math] projection, producing the factor of two that GR attributes to curvature + time dilation.

Doesn't that disagree with the photon sphere [math]\dfrac{3}{2}\kappa^2[/math] ?

10 hours ago, Anton Rize said:

GR therefore reappears as the differential expansion of this algebraic identity:

[math]\kappa^2 = R_s/r = \rho/\rho_{\max}.[/math]

What is [math]\rho/\rho_{\max}[/math] referring to? Also, GR has the energy-momentum tensor [math]T_{\mu\nu}[/math] (also known as the stress-energy tensor), which has ten independent components. Why are you ignoring nine of the independent components? In what way is [math]r_s/r[/math] representative of GR beyond the Schwarzschild metric?

Edited November 10Nov 10 by KJW

Thank you @KJW this is fascinating!
I believe we getting closer to unveiling the differences in our deep foundational views on reality and role of mathematics in it.

You see Im seeing every mathematical choice or operation as deep philosophical statement. So for me your "Cartesian n-th power of the real number line" is a statement about ontological status of spacetime. Its like you saying "reality has a grid" and I calling it the ontological statement about background structure.
This is operationaly the same as:

"In Substance-Based (SB):

• space-time is a pre-existing metric manifold.

• A single object can be assigned coordinates on this manifold, and physical quantities (such as position, velocity, and distance) are defined with respect to that dynamical but still background.

• The origin (0,0) is arbitrary - a conventional point on an independent grid.
  
  For you, as far as I understood, this is not a statement about the structure of reality but "opening your toolbox" of sort. This reveals the core of our philosophical difference. My methodology assumes that all mathematical choices are ontological statements. 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.

  
  
  

1 hour ago, KJW 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"?

2 hours ago, KJW said:

I don't think there really is "one step before that assumption"

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)

2 hours ago, KJW said:

Please explain how you get from RC to a "differentiable manifold".

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

The manifold (your [math]\mathbb{R}^n[/math] "grid") is the very a priori assumption that RG rejects. In RG, there is no pre-existing "arena" or "stage". Instead, the appearance of a 4D manifold (what you call "reality") is an emergent phenomenon - a "shadow" or projection cast by the underlying algebraic, relational structure of WILL (WILL ≡ SPACE-TIME-ENERGY).

The manifold isn't the foundation of physics; in my model, it is its consequence.

11 hours ago, KJW said:

Doesn't that disagree with the photon sphere 32κ2 ?

No, this appears to be a misunderstanding of the terms in my model. There is no contradiction.

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]).

As an interesting note, this condition [math]\theta_1 = \theta_2 \approx 54.7^\circ[/math], which is known in physics as the "Magic Angle". This is not a coincidence.

11 hours ago, KJW said:

What is ρ/ρmax referring to? Also, GR has the energy-momentum tensor Tμν (also known as the stress-energy tensor), which has ten independent components. Why are you ignoring nine of the independent components?

What is [math]\rho/\rho_{max}[/math]?

Derivation of Density

Translating RG (2D) to Conventional Density (3D).

In RG [math]\kappa^2[/math] is the 2D parameter defined in the relational manifold [math]S^2[/math] . In conventional physics, the source term is volumetric density [math]\rho[/math], a 3D concept defined by the "cultural artifact" (a Newtonian "cannonball" model) of mass-per-volume.

To bridge our 2D theory with 3D empirical data, we must create a "translation interface". We do this by explicitly adopting the conventional (Newtonian) definition of density, [math]\rho \propto m_0/r^3[/math], as our "translation target".

From the projective analysis established in the previous sections:

[math]\kappa^2 = \frac{R_s}{r},[/math]

where [math]\kappa[/math] emerges from the energy projection on the area of unit sphere [math]S^2[/math], and [math]R_s = 2Gm_0/c^2[/math] links to the mass scale factor
[math]m_0 = E_0/c^2[/math].

This leads to mass definition:

[math]m_0 = \frac{\kappa^2 c^2 r}{2G}[/math]

To translate this into a volumetric density, we first adopt the conventional 3D (volumetric) proxy, [math]r^3[/math]. This is not a postulate of RG, but the first step in applying the legacy (3D) definition of density:

[math]\frac{m_0}{r^3} = \frac{\kappa^2 c^2}{2G r^2}[/math]

This expression, however, is incomplete. Our [math]\kappa^2[/math] "lives" on the 2D surface [math]S^2[/math] (which corresponds to [math]4\pi[/math]), while the [math]r^3[/math] proxy implicitly assumes a 3D volume. To correctly normalize the 2D parameter [math]\kappa^2[/math] against the 3D volume, we must apply the geometric normalization factor of the [math]S^2[/math] carrier, which is [math]1/4\pi[/math].

This normalization is the necessary geometric step to interface the 2D relational carrier ([math]S^2[/math]) with the 3D legacy definition of density:

[math]\rho = \frac{1}{4\pi}\left( \frac{\kappa^2 c^2}{2G r^2} \right)[/math]

[math]\rho = \frac{\kappa^2c^2}{8\pi G r^2}[/math]

[math]\text{Local Density} \equiv \text{Relational Projection}[/math]

Maximal Density.

At [math]\kappa^2 = 1[/math] (the horizon condition (for non rotating systems), [math]r=R_s[/math]), this density reaches its natural bound, [math]\rho_{\max}[/math], which is derived purely from geometry:

[math]\rho_{\max} = \frac{c^2}{8\pi G r^2}[/math]

Normalized Relation.

Thus, our "translation" reveals an identity: the geometric projection [math]\kappa^2[/math] is simply the ratio of density to the maximal density:

[math]\kappa^2 = \frac{\rho}{\rho_{\max}} \;\;\Rightarrow\;\; \kappa^2 \equiv \Omega[/math]

---

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.

For example, I explicitly derive Pressure in my paper as: [math]P(r) = - \rho(r) c^2[/math] My model replaces your 10 independent, descriptive inputs with one fundamental parameter and its derived geometric consequences.

This also answers the question of the Cosmological "Constant".
In RG, [math]\Lambda[/math] is not an ad-hoc addition. It emerges naturally from this same derivation. If we identify the vacuum energy [math]\rho_{\Lambda}[/math] with the maximal density of space itself, [math]\rho_{max}[/math],
then: [math]\Lambda(r) = \frac{8\pi G}{c^2} \rho_{\Lambda} = \frac{8\pi G}{c^2} \left( \frac{c^2}{8\pi G r^2} \right)[/math] [math]\Lambda(r) = 1/r^2[/math]

The cosmological "constant" is not a "constant" at all, but simply the inherent, scale-dependent curvature of the relational geometry itself.

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.

Edited November 11Nov 11 by Anton Rize

11 hours ago, Anton Rize said:

Your 10-component
Tμν is not a fundamental law of nature

The energy-momentum tensor is the conserved quantity associated with spacetime translation-invariance under Noether’s theorem; it isn’t just an ad-hoc invention.

12 hours ago, Anton Rize said:

It does not need 10 independent components as an input. Instead, it uses one parameter (ρ) and derives other necessary properties (like Pressure) as emergent consequences.

But these nine components all themselves form sources of gravity, and are not derivable from just energy-density. If you ignore them, your model cannot recover all the dynamics of GR, which you claimed it is able to do.

Edited November 11Nov 11 by Markus Hanke
Corrected time to spacetime translation symmetry

@Markus Hanke if you want me to continue dialog with you first need to answer my questions:

On 10/27/2025 at 4:19 PM, Anton Rize said:

@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.

then you have to show minimal human decency and admit that you was wrong:

On 10/27/2025 at 11:21 PM, Anton Rize said:

  On 10/27/2025 at 3:14 PM, 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.

then answer this question:

On 10/30/2025 at 5:04 PM, Anton Rize said:

  On 10/30/2025 at 3:09 PM, Markus Hanke said:

What you do with these criticisms we offered is up to yourself, it’s your hypothesis after all.

@Markus Hanke

Your last statement is an assertion, not a criticism. For it to become a criticism, it has to obtain an objective form like an equation or a well defined logical construct.

But let's explore your assertion:

1. Let's assume you are correct, and I am "tacitly using" concepts that require a metric.

2. However, as you've seen, I can't find this "hidden metric" in my algebra. You can't find it. It doesn't appear in the equations.

3. And most importantly, it is not needed to derive the correct precession of Mercury to [math]10^{-7}[/math] precision, as shown by my algebraic formula: [math]\Delta\phi = (2\pi Q^2)/(1-e^2)[/math].

So, if this "hidden metric" is mathematically invisible, algebraically unnecessary, and operationally redundant... what is its physical meaning? It becomes a useless entity.

Sorry, but this is why your assertion not only is not a valid criticism, but also just not valid at all.

then apologise for your yapping, and only after we might continue dialog. Before that all your yapping I will ignore.

Edited November 12Nov 12 by Anton Rize