56 minutes ago, studiot said:

This is exactly what I mean when I say you have not started in the beginning.

Please define mathematically what you mean by both energy and metric.

Note you definition/derivations should be clean of depracatory comparisons with something else.

Otherwise you are not starting with an agreed basis.

Completely agree. Due to forum format I cant just lay on you guys pages and pages of derivations so Im posting shorted versions and by doing so losing rigor and

logical sequence. Maybe if Ill portion it in to digestible peace's?... Lets try and start from the top. This first part is mainly philosophy but we cant move forward without establishing foundations:

This document must be read literally.

All terms are defined within the relational framework of WILL RG.

Any attempt to reinterpret them through conventional notions

absolute energies, external backgrounds, hidden containers

will produce distortions and misreading.

Just like responsibility of formulating lies with the author,

the responsibility of interpretation lies with the reader:

take the words as written, not as filtered through prior formalisms.

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.

Edited November 2Nov 2 by Anton Rize

13 hours ago, Markus Hanke said:

Well, the standard formula is derived from the Schwarzschild metric, so it is an approximation to the extent that the SS metric itself is an approximation to the actual physical situation in the solar system.

That's not necessarily true. Bear in mind that Einstein did not have the Schwarzschild metric when he published general relativity and the anomalous precession of the perihelion of Mercury. I have read that the value of the precession was obtained as the non-conservation of the Laplace-Runge-Lenz vector, although it is not clear to me how the non-Newtonian potential ¹ was obtained without the Schwarzschild metric.

I'm currently working on a problem that involves the Schwarzschild metric, and that problem turned out to be mathematically rather complicated rather quickly. It seems to me that the mathematics could be simplified by using a weak-field approximation. I wasn't suggesting that the Schwarzschild metric was an approximation of the physical solar system, even though the anomalous precession of the perihelion of Mercury is only a part (about 7.5%) of the total precession of the perihelion of Mercury.

¹ The Laplace-Runge-Lenz vector is a constant of motion under an exact inverse-square force law.

13 hours ago, Markus Hanke said:

It should be noted though that the standard formula matches the observed precession value to ~0.1%, so the approximation error is quite small when compared to actual observation.

That not a big achievement. The curvature field of the solar solar system is quite small and only a first-order GR effect would be significant. The approximation I was suggesting would be fine for Mercury around the Sun but may be inadequate for a close orbit around a black hole or neutron star.

@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.

Yes, one often considers trajectories of test masses in spacetime. That's because the definition of a test mass is a point-like particle whose mass is small enough to have negligible effect on the spacetime. This allows the trajectory of the test mass to be determined against the background spacetime. But that is a mere simplification of the mathematics. It works adequately for planets orbiting a star. But the mutual orbits of black holes around each other cannot be treated in this manner and can only be determined as the dynamics of the spacetime itself. Don't confuse the mathematical simplifications that physicists may make with the inadequacy of the theory.

You also mentioned asymptotically flat spacetime. It should be noted that general relativity must reduce to Newtonian theory in the weak-field limit as indeed it does. Also, the Einstein equation is a relationship between the mathematical realm of Ricci calculus and the physical realm. 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.

Also, the foundations of general relativity are more fundamental than you seem to think, and indeed more fundamental than you seem to indicate concerning your theory. For example, you treat energy as foundational, whereas the Einstein tensor field that is equivalent to the energy-momentum density field is derived.

Edited November 2Nov 2 by KJW

32 minutes ago, KJW said:

it seems to me that you do not have a proper understanding of general relativity.

GR is incredibly complex theory and there's so much remains for me to learn. I don't claim that I have a "proper" understanding of it. In fact Im not even sure that I ever will.

1 hour ago, KJW said:

you talk about spacetime being an independent background that you reject, whereas in general relativity, spacetime is not an independent background.

Thanks great question! it’s really helpful in focusing the discussion on what the Relational Geometry model is actually doing.

Let me clarify my position and outline the core logical chain in a concise way.

1. I’m not questioning background independence in GR itself. (though I'm not fully convinced that it is...)

What I’m questioning is the implicit separation between structure and dynamics - between geometry as a pre-existing container and energy as something evolving inside it.

In RG, that distinction is removed entirely. Geometry is the relational manifestation of energy.

2. Once this split is removed, the formulation becomes both simpler and fully self-consistent.

No differential equations are postulated at the base level (so far works without). Any differential rewriting implicitly commits to a specific parameterisation (and thus extra structure). RG derives relations as coordinate-free algebraic identities; the differential expressions shown are translations into the GR language after a chart is chosen.

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

Here [math]\kappa^2[/math] represents the normalized curvature–energy projection, and [math]R_s = 2Gm_0/c^2[/math] defines the mass scale factor.

4. From this relation, the local energy density follows directly from geometric normalization over the spherical surface due to [math] \kappa [/math] l "lives" on 2-sphere [math] S^2 [/math]:

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

At the bound condition [math]\kappa^2 = 1[/math] (the horizon limit for non rotation systems [math]r = R_s[/math]) the maximal energy density is reached:

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

Hence the relational normalization identity is immediate:

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

5. This naturally leads to the expression for geometric pressure, which in this framework arises not as a thermodynamic assumption but from closure constrains as the curvature-tension of the surface itself:

[math]P = \frac{c^4}{8\pi G}\frac{1}{r}\frac{d\kappa^2}{dr} = -\,\frac{\kappa^2 c^4}{8\pi G r^2} = -\,\rho c^2.[/math]

Thus the equation of state [math]P = -\rho c^2[/math] is not inserted - it’s a necessary condition of curvature balance.

The “vacuum pressure” of GR therefore appears as an intrinsic property of energy–geometry itself.

6. Substituting these relations yields a purely algebraic form equivalent to the GR mass function identity:

[math]\frac{d}{dr}(r\,\kappa^2) = \frac{8\pi G}{c^2} r^2 \rho(r).[/math]

This is not an additional equation - it’s simply the differential translation of the same algebraic closure into GR notation.

7. The same closure produces the scale–curvature relation:

[math]\Lambda(r) = \frac{1}{r^2}.[/math]

This quantity is not a “cosmological constant” added by hand; it’s the inherent geometric normalization that replaces the need for an external Λ–term.

In GR, Λ must be postulated to restore balance between geometry and energy - a consequence of treating them as distinct entities.

In RG, the balance is intrinsic, so Λ emerges automatically and scale–dependently.

8. Because the entire construction is scale invariant, the usual distinction between “local” and “global” regimes loses meaning.

All scales obey the same algebraic identity

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

Hence cosmological and microscopic domains are connected by the same relational geometry, not separated by arbitrary boundaries.

9. The singularity problem also dissolves naturally.

Since [math]\rho \le \rho_{\max}[/math] by definition, the energy density and curvature can never diverge;

[math]\kappa^2 \le 1[/math] sets an intrinsic upper bound.

Therefore no infinite curvature or energy density can appear - the geometry self-limits by construction.

10. To summarize the closure loop:

[math]\boxed{\kappa^2 = \frac{R_s}{r} = \frac{\rho}{\rho_{\max}}}, \qquad P = -\rho c^2, \qquad \Lambda(r) = \frac{1}{r^2}.[/math]

This single algebraic structure captures the full equivalence of geometry and energy without any external parameters, shows scale invariance, eliminates singularities, and matches the empirical curvature–mass–pressure relations observed in both astrophysical and cosmological systems.

P. S. All of this remains open for independent verification - that’s precisely why I’m sharing these derivations here.

Edited November 3Nov 3 by Anton Rize

You're waxing esoteric now.

Honestly, I can't get past your spacetime \( \equiv \) energy mathematical nonsense.

2 hours ago, 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.

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.

Among other things already pointed out to you, there are robust quantum mechanical reasons why E=ST cannot make sense. Such variables are complementary in quantum mechanics (canonically conjugate). Meaning: when one of them is precisely determined, the other becomes "infinitely fuzzy".

Only that consideration should be enough to make you cease and desist on the whole thing.

So..., again, no.

11 minutes ago, joigus said:

You're waxing esoteric now.

Hahaha! Yeh I agree the "Ouroboros" does sound isosteric. But it also accurately depicts the idea. What can do? I just cant stop my self from being poetic sometimes 🙃. Physics is beautiful and I cant resist it charms ☺️.

18 minutes ago, joigus said:

Honestly, I can't get past your spacetime ≡ energy mathematical nonsense.

I understand its unusual and im not asking anyone to believe me. In fact im finding it hard to believe myself. All I did is questioned this assumption about there separation and now im sharing results.

8 minutes ago, joigus said:

Among other things already pointed out to you, there are robust quantum mechanical reasons why E=ST cannot make sense. Such variables are complementary in quantum mechanics (canonically conjugate). Meaning: when one of them is precisely determined, the other becomes "infinitely fuzzy".

Only that consideration should be enough to make you cease and desist on the whole thing.

So..., again, no.

Its grate that you brought QM. When this approach applied in QM results are even more unbelievable. I will create a separate topic in QM forum so we can discuss it properly. But for now a little teaser about "infinitely fuzzy":

In WILL RG the uncertainty principle isn’t postulated; it arises directly from the closure of energy projection on the unit circle [math]S^1[/math]. When you require that the energy projection returns to its phase after [math]n[/math] full rotations (topological winding number), the minimal definable phase increment is [math]\Delta\theta \ge 2\pi/n[/math].

Two orthogonal projections on [math]S^1[/math],

[math]\beta_X = \cos\theta[/math] (momentum-like) and [math]\beta_Y = \sin\theta[/math] (spacetime-like),

then satisfy the purely geometric relation

[math]\Delta\beta_X \, \Delta\beta_Y \ge \tfrac{1}{2}|\sin 2\theta|\,(\Delta\theta)^2 \;\ge\; \tfrac{1}{2}|\sin 2\theta|\,(2\pi/n)^2.[/math]

When mapped to physical quantities through

[math]\Delta p \propto \Delta\beta_X[/math] and [math]\Delta x \propto \lambda \propto \Delta\beta_Y[/math],

this becomes the familiar Heisenberg bound:

[math]\boxed{\Delta x \, \Delta p \gtrsim \hbar.}[/math]

So the uncertainty principle appears not as a statistical rule or operator property,

but as a geometric consequence of closure on [math]S^1[/math].

Planck’s constant [math]\hbar[/math] then plays the role of a conversion factor between

topological winding and dimensional scale - not a “fundamental mystery,” but a calibration invariant.

I’ll open a topic in the QM forum to show the derivation step-by-step - it’s quite beautiful, and surprisingly simple once you see it geometrically.

12 minutes ago, Anton Rize said:

E=ST cannot make sense

A small clarification: my statement [math]\text{SPACETIME} \equiv \text{ENERGY}[/math] isn’t a numerical or dimensional equation like [math]E = S T[/math].

It denotes ontological equivalence - the inseparability of structure and dynamics.

In this sense, energy is not something that exists within spacetime, but what constitutes it.

I’m using “≡” in the logic/philosophy sense of equivalence (non-separability / mutual definability), not as a numeric equality. In logic, “≡” commonly denotes logical or definitional equivalence (see e.g. ‘Triple bar’ and ‘Logical equivalence’ https://en.wikipedia.org/wiki/Triple_bar), whereas “=” is plain equality. So “SPACETIME ≡ ENERGY” is an ontological equivalence, not a formula like E = S·T.

The best way to understand is through category theory - a structural equivalence.

Just as two categories can be equivalent (mutually definable but not identical),

in RG spacetime and energy are two fully intertranslatable aspects of the same relational structure.

So “SPACETIME ≡ ENERGY” means categorical equivalence, not metric equality.

Edited November 3Nov 3 by Anton Rize

I have read that the value of the precession was obtained as the non-conservation of the Laplace-Runge-Lenz vector, although it is not clear to me how the non-Newtonian potential ¹ was obtained without the Schwarzschild metric.

That’s interesting, I didn’t know that. Question is, how do you do this without already knowing the metric…? As you pointed out yourself.

No differential equations are postulated at the base level (so far works without).

But when I asked you that question about the physical scenario earlier, you used the gradient in your answer, which is a differential operator.

25 minutes ago, Anton Rize said:

In WILL RG the uncertainty principle isn’t postulated; it arises directly from the closure of energy projection on the unit circle S1. When you require that the energy projection returns to its phase after n full rotations (topological winding number), the minimal definable phase increment is Δθ≥2π/n.

Two orthogonal projections on S1,

βX=cosθ (momentum-like) and βY=sinθ (spacetime-like),

then satisfy the purely geometric relation

ΔβXΔβY≥12|sin2θ|(Δθ)2≥12|sin2θ|(2π/n)2.

When mapped to physical quantities through

Δp∝ΔβX and Δx∝λ∝ΔβY,

this becomes the familiar Heisenberg bound:

ΔxΔp≳ℏ.

So the uncertainty principle appears not as a statistical rule or operator property,

but as a geometric consequence of closure on S1.

Planck’s constant ℏ then plays the role of a conversion factor between

topological winding and dimensional scale - not a “fundamental mystery,” but a calibration invariant.

I’ll open a topic in the QM forum to show the derivation step-by-step - it’s quite beautiful, and surprisingly simple once you see it geometrically.

In none of the standard formulations of QM is the uncertainty principle presented as a postulate. It can be proved as a theorem from the relation between observables, states, and probabilities defined in the postulates. It is also confirmed in its statistical sense from experiments. Furthermore, it has explanatory power in that particular sense in a wide range of phenomena, from Josephson junctions to polarisers, etc. It's nothing to do with a winding number, but indeed refers to statistical frequencies.

If space-time coordinates are maximally determined, energy-momentum is minimally determined, how could space-time be energy?

You seem to be using some kind of LLM or chatbot to generate what feels like a linguistic version of the shell game.

On 11/3/2025 at 10:44 AM, Anton Rize said:

  On 11/3/2025 at 8:40 AM, KJW said:

it seems to me that you do not have a proper understanding of general relativity.

GR is incredibly complex theory and there's so much remains for me to learn. I don't claim that I have a "proper" understanding of it. In fact Im not even sure that I ever will.

Are you aware of the two chain rules in multivariable calculus?

[math]\dfrac{d\bar{x}^\mu}{d\xi} = \displaystyle \sum_{\nu=1}^{n} \dfrac{dx^\nu}{d\xi} \dfrac{\partial \bar{x}^\mu}{\partial x^\nu}\ \ \equiv\ \ \dfrac{d\bar{x}^\mu}{d\xi} = \dfrac{dx^\nu}{d\xi} \dfrac{\partial \bar{x}^\mu}{\partial x^\nu}[/math]

[math]\dfrac{\partial \psi}{\partial \bar{x}^\mu} = \displaystyle \sum_{\nu=1}^{n} \dfrac{\partial \psi }{\partial x^\nu} \dfrac{\partial x^\nu}{\partial \bar{x}^\mu}\ \ \equiv\ \ \dfrac{\partial \psi}{\partial \bar{x}^\mu} = \dfrac{\partial \psi }{\partial x^\nu} \dfrac{\partial x^\nu}{\partial \bar{x}^\mu}[/math]

where the second expression on each line uses the "Einstein summation convention" (repeated indices, called "dummy indices", [math]\nu[/math] in this case, indicate an implicit summation over those indices).

Note that:

[math]\dfrac{\partial \bar{x}^\sigma}{\partial x^\nu} \dfrac{\partial x^\nu}{\partial \bar{x}^\tau} = \dfrac{\partial \bar{x}^\sigma}{\partial \bar{x}^\tau} = \delta^\sigma_\tau[/math]

[math]\dfrac{\partial x^\varrho}{\partial \bar{x}^\mu} \dfrac{\partial \bar{x}^\mu}{\partial x^\gamma} = \dfrac{\partial x^\varrho}{\partial x^\gamma} = \delta^\varrho_\gamma[/math]

where [math]\delta^\sigma_\tau[/math] is called the "Kronecker delta" and is the identity matrix, and therefore [math]\dfrac{\partial \bar{x}^\sigma}{\partial x^\nu}[/math] and [math]\dfrac{\partial x^\nu}{\partial \bar{x}^\tau}[/math] are the inverse matrix of each other.

So, what is called a "coordinate transformation" in physics is a "change of variable" in calculus.

The chain rule for [math]\dfrac{d\bar{x}^\mu}{d\xi}[/math] is the archetype for the coordinate transformation law for contravariant vectors in general:

[math]\bar{V}^\mu = V^\nu \dfrac{\partial \bar{x}^\mu}{\partial x^\nu}[/math]

The chain rule for [math]\dfrac{\partial \psi}{\partial \bar{x}^\mu}[/math] is the archetype for the coordinate transformation law for covariant vectors in general:

[math]\bar{V}_\mu = V_\nu \dfrac{\partial x^\nu}{\partial \bar{x}^\mu}[/math]

This leads to the coordinate transformation law for the general absolute¹ tensor:

[math]\bar{T}^{\sigma...\varrho}_{\tau...\gamma} = T^{\alpha...\beta}_{\mu...\nu}\ \dfrac{\partial \bar{x}^\sigma}{\partial x^\alpha}...\dfrac{\partial \bar{x}^\varrho}{\partial x^\beta}\ \dfrac{\partial x^\mu}{\partial \bar{x}^\tau}...\dfrac{\partial x^\nu}{\partial \bar{x}^\gamma}[/math]

The coordinate transformation law for tensors guarantee that tensors have the following two important properties:

1: A tensor that is zero in any coordinate system is zero in every coordinate system.

2: A tensor equation that is true in any coordinate system is true in every coordinate system.

¹ There are also relative tensors of various weights for which the coordinate transformation law has an extra factor, emerging naturally as a result of the nature of determinants.

Edited November 4Nov 4 by KJW

On 11/3/2025 at 2:08 AM, Anton Rize said:

The best way to understand is through category theory - a structural equivalence.

Just as two categories can be equivalent (mutually definable but not identical),

in RG spacetime and energy are two fully intertranslatable aspects of the same relational structure.

So “SPACETIME ≡ ENERGY” means categorical equivalence, not metric equality.

Wow! I'm impressed! Last week, when I casually remarked that your ideas reminded me somewhat of category theory, you admitted you knew very little about the subject. And now you have become sufficiently proficient in the subject to lecture us on one of its more subtle points. Well done you!

However, although it is obvious that spacetime is an object in the category of smooth manifolds, it is not obvious to me in which category you think energy belongs. Is it the category of natural numbers, or some other category?

And can you please describe in detail the functors between these categories?

Edited November 4Nov 4 by Xerxes

22 hours ago, KJW said:

Are you aware of the two chain rules in multivariable calculus?

Now I am. It took me a while to decipher your message... And I'm not sure that I understand it correctly. Problem is that I can't see the actual physics behind it. Can you?

What I understood in lame terms. Correct me if I'm wrong please:

1. You start by postulating some background structure with the symbols [math]x^\nu[/math], and then introduce another mapping [math]\bar{x}^\mu[/math] Already at this step I have questions:

a. On what basis are we justified in postulating any background at all?
b. Were all other possible backgrounds proven incompatible before choosing this one?
c. Why must a background exist in the first place?
d. Does it introduce inconsistencies?
e. Is it directly measurable, or purely a descriptive assumption?

2. Then, in order to keep physics “the same” (covariant) within this background, you introduce dummy indices and transformation rules. But again - are these operationally real quantities, or formal devices? Can we measure it?

3. Finally, the “coordinate transformation law for contravariant vectors.” At that point it begins to look like mathematical bookkeeping designed to preserve internal consistency of an unjustified background assumption, rather than expressing physical content.

In short, what I see is scaffolding built to stabilise a non-observable, unmeasured, metaphysical claim that the Universe possesses an underlying background structure. There’s no physics in it - only a scaffolding to prevent contradiction within its own premises.

---
Earlier you were asking me about [math]\frac{2\pi Q_{\mathrm{orbit}}^{2}}{1-e^{2}},\qquad Q_{\mathrm{orbit}}^{2}:= \beta^2(a)+\kappa^2(a)[/math] and interpretation of the Q parameter. I think I finally got it. And it's so simple:


Relational Displacement as Q Parameter

Each observer places itself at the relational origin [math](\beta,\kappa)=(0,0)[/math].

When I (as an observer) look at another system, I measure its relational projections [math](\beta,\kappa)[/math] in my own frame. The scalar

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

is the radius at which that other system’s centre appears in my relational plane.

If the other system now looks back at me, it again puts itself at [math](0,0)[/math] and applies the same rule. It measures my [math](\beta,\kappa)[/math] and again obtains

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

This expression represents the total deviation of the object from the observer’s relational origin [math](\beta,\kappa)=(0,0)[/math].

Geometrically, the observer is always located at the centre of its own relational circle [math]S^{1}[/math] (or sphere [math]S^{2}[/math]), and any external object appears as a point [math](\beta,\kappa)[/math] on that plane.

The scalar [math]Q[/math] therefore measures the radius of relational displacement between two systems.

---

Covariance doesn't need to be scaffolded from some arbitrary rules. It's a fundamental property of Relational Geometry.

17 minutes ago, Anton Rize said:

Now I am. It took me a while to decipher your message... And I'm not sure that I understand it correctly. Problem is that I can't see the actual physics behind it. Can you?

What I understood in lame terms. Correct me if I'm wrong please:

1. You start by postulating some background structure with the symbols xν, and then introduce another mapping x¯μ Already at this step I have questions:

a. On what basis are we justified in postulating any background at all?
b. Were all other possible backgrounds proven incompatible before choosing this one?
c. Why must a background exist in the first place?
d. Does it introduce inconsistencies?
e. Is it directly measurable, or purely a descriptive assumption?

2. Then, in order to keep physics “the same” (covariant) within this background, you introduce dummy indices and transformation rules. But again - are these operationally real quantities, or formal devices? Can we measure it?

3. Finally, the “coordinate transformation law for contravariant vectors.” At that point it begins to look like mathematical bookkeeping designed to preserve internal consistency of an unjustified background assumption, rather than expressing physical content.

In short, what I see is scaffolding built to stabilise a non-observable, unmeasured, metaphysical claim that the Universe possesses an underlying background structure. There’s no physics in it - only a scaffolding to prevent contradiction within its own premises.

---
Earlier you were asking me about 2πQ2orbit1−e2,Q2orbit:=β2(a)+κ2(a) and interpretation of the Q parameter. I think I finally got it. And it's so simple:


Relational Displacement as Q Parameter

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

When I (as an observer) look at another system, I measure its relational projections (β,κ) in my own frame. The scalar

Q2=β2+κ2

is the radius at which that other system’s centre appears in my relational plane.

If the other system now looks back at me, it again puts itself at (0,0) and applies the same rule. It measures my (β,κ) and again obtains

Q2=β2+κ2

This expression represents the total deviation of the object from the observer’s relational origin (β,κ)=(0,0).

Geometrically, the observer is always located at the centre of its own relational circle S1 (or sphere S2), and any external object appears as a point (β,κ) on that plane.

The scalar Q therefore measures the radius of relational displacement between two systems.

---

Covariance doesn't need to be scaffolded from some arbitrary rules. It's a fundamental property of Relational Geometry.

I don't see how I can contribute further if you will not answer my key questions.

Like it or not

Your post above has a metric.

Please also tell me how you can arive at

(anything)², without a metric ?

It is quite possible to do some geometry, and even more topology, without a metric.

Such geometry uses similarity transformations, which do not need one.

4 minutes ago, studiot said:

I don't see how I can contribute further if you will not answer my key questions.

Did I miss your question? Last one you asked was this and I roll out the most detailed unswear so far:

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

  On 11/2/2025 at 8:41 PM, studiot said:

This is exactly what I mean when I say you have not started in the beginning.

Please define mathematically what you mean by both energy and metric.

Note you definition/derivations should be clean of depracatory comparisons with something else.

Otherwise you are not starting with an agreed basis.

Completely agree. Due to forum format I cant just lay on you guys pages and pages of derivations so Im posting shorted versions and by doing so losing rigor and

logical sequence. Maybe if Ill portion it in to digestible peace's?... Lets try and start from the top. This first part is mainly philosophy but we cant move forward without establishing foundations:

This document must be read literally.

All terms are defined within the relational framework of WILL RG.

Any attempt to reinterpret them through conventional notions

... and so on...

Was there another one that I missed?

18 hours ago, Xerxes said:

Wow! I'm impressed! Last week, when I casually remarked that your ideas reminded me somewhat of category theory, you admitted you knew very little about the subject. And now you have become sufficiently proficient in the subject to lecture us on one of its more subtle points. Well done you!

However, although it is obvious that spacetime is an object in the category of smooth manifolds, it is not obvious to me in which category you think energy belongs. Is it the category of natural numbers, or some other category?

And can you please describe in detail the functors between these categories?

Thank you, Xerxes. No lectures here. Just using all I got. And yes, I took your hint seriously and tried to formalize the relation in categorical language. It turned out to fit remarkably well. At least it seems to me. I'm only starting to get familiar with category theory.

In this scheme the "category of energy" is not the category of numbers but a symmetric monoidal posetal category that I call [math]\mathbf{Bud}[/math] – short for "projection budgets". Its objects are the dimensionless relational budgets [math](\beta^2,\kappa^2)[/math], and morphisms represent admissible transformations between them. The monoidal product [math]\boxtimes[/math] simply joins independent subsystems, and the unit object [math]e[/math] is the trivial (zero) budget.

On the other side, the 2-category of observers [math]\mathbf{Obs}[/math] has:

- objects – relational states (observers),

- 1-morphisms – causal transitions labeled by [math](\beta,\kappa)[/math],

- 2-morphisms – observer re-parametrizations (self-centering operations).

From this we build the homotopy 1-category [math]\mathcal{W}=\mathrm{Ho}^1(\mathbf{Obs})[/math], carrying a monoidal product [math]\otimes[/math] for composition of processes.

Then two strict monoidal functors arise naturally:

[math]S,E:(\mathcal{W},\otimes)\to(\mathbf{Bud},\boxtimes)[/math]

defined by

[math]S(A)=\kappa_A^2,\quad E(A)=2\beta_A^2[/math].

Their natural isomorphism [math]\eta:S\Rightarrow E[/math] expresses the closure law [math]\kappa^2=2\beta^2[/math], which is just the WILL equivalence

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

So in this sense, energy and spacetime are functorially equivalent descriptions of one relational transformation, not distinct substances or metric entities.

---

This is just my first attempt and its most likley wrong. But I hope it sufficient to see the idea in developing.

2 hours ago, Anton Rize said:

Did I miss your question? Last one you asked was this and I roll out the most detailed unswear so far:

  On 11/2/2025 at 12:45 PM, Anton Rize said:

... and so on...

Was there another one that I missed?

Was there another one that I missed?

You response to my last post is a prime example.

It contains just one question.

2 hours ago, studiot said:

Please also tell me how you can arive at

(anything)², without a metric ?

In English we identify questiony by the question mark symbol ???????????????

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

I also said questions - which in English is a plural.

Each reply of your seems to respond to one easy statement/question only, leaving the hard ones untouched.

2 hours ago, Anton Rize said:

Thank you, Xerxes.
This is just my first attempt and its most likley wrong. But I hope it sufficient to see the idea in developing.

Ha! Serves me right for trying to be sarcastic on an international forum.

But the fact remains that both your theory and category theory are deeply non-constructivist, by which is meant they assume the existence of objects they are not equipped to construct. Category Theory is unashamedly like this, as it is not designed to dispense with what it might regard as "lower level constructions" rather to build upon them as a sort of meta-mathematics

Your theory, on the other hand, arrogantly declares it has no need of the usual sorts gadgets we use in differential geometry, while implicitly using them all.

Hence studiot's very relevent questions.

4 hours ago, Anton Rize said:

I'm not sure that I understand it correctly. Problem is that I can't see the actual physics behind it. Can you?

I was focusing on the mathematics. You said general relativity is an incredibly complex theory and that you're not sure you'll ever have a proper understanding of it. I don't think it is an especially difficult theory, but it does require a level of mathematical skill that I cannot assume you possess. General relativity is based on calculus, so it is necessary to have skills in that mathematical subject. Also, because spacetime is four-dimensional, the calculus one needs is multivariable calculus. The two chain rules of multivariable calculus is at the bare minimum of the skills required as these deal with the notion of coordinate transformations, a key concept in general relativity.

My post so far has only dealt with the concept of tensors. Although modern mathematics seeks to remove the notion of coordinates, they are actually key to the concept of tensors, which is not about going beyond scalars, vectors, and matrices, but about the notion of covariance and how the coordinate transformation law manifests that. I touched upon the machinery of tensors when I mentioned dummy indices and the Einstein summation convention, as well as the distinction between superscript indices and subscript indices with regards to the coordinate transformation law. But I have yet to even start upon general relativity itself. I would say that general relativity doesn't really start until the concept of "covariant derivatives" are introduced. This leads to the notion of a "connection field", and subsequently to the notion of a "curvature field".

Getting back to calculus, suppose one is given a vector field [math]V_j(x^1,...,x^n)[/math], and we enquire: Does there exist a scalar field [math]\phi(x^1,...,x^n)[/math] such that:

[math]V_j = \dfrac{\partial \phi}{\partial x^j}[/math]

is satisfied? How would one determine if [math]V_j(x^1,...,x^n)[/math] is the partial derivative of a scalar field [math]\phi(x^1,...,x^n)[/math]? We know that the partial derivative of [math]\phi(x^1,...,x^n)[/math] satisfies:

[math]\dfrac{\partial^2 \phi}{\partial x^k \partial x^j} - \dfrac{\partial^2 \phi}{\partial x^j \partial x^k} = 0[/math]

and therefore it is necessary for [math]V_j(x^1,...,x^n)[/math] to satisfy:

[math]\dfrac{\partial V_j}{\partial x^k} - \dfrac{\partial V_k}{\partial x^j} = 0[/math]

If [math]\dfrac{\partial V_j}{\partial x^k} - \dfrac{\partial V_k}{\partial x^j} = F_{jk} \ne 0[/math], then this is an obstruction to the existence of scalar field [math]\phi(x^1,...,x^n)[/math]. The non-zero Riemann curvature tensor field is also an obstruction to the existence of a solution to a particular system of partial differential equations corresponding to a coordinate transformation. The necessary condition is straightforward; the sufficient condition is less straightforward.

4 hours ago, Anton Rize said:

On what basis are we justified in postulating any background at all?

Look around you. It is clear that we exist in a space that is at least four-dimensional. Even if we can't derive this space logically from first principles, we can assume its existence on the basis of empirical observation.

Although it is assumed that a differentiable manifold exists and that one can overlay a coordinate grid onto it, the mathematics itself does not indicate a priori the physical nature of the differentiable manifold nor provide any physical interpretation of the coordinate variables. In the mathematics I presented in this and my previous post, I made no mention of a metric, nor did I even specify the dimensionality of the space, simply indicating it as [math]n[/math]. In the mathematics of tensor calculus, most formulae are independent of the dimensionality of the space, but some formulae do depend on the dimensionality of the space, so it's a good idea to work in [math]n[/math]-dimensional space, even if general relativity itself is four-dimensional. And why spacetime is four-dimensional is an interesting question. It turns out that four-dimensional spaces have special properties not possessed by other-dimensional spaces, though whether these answer the question of why spacetime is four-dimensional is debatable.

4 hours ago, Anton Rize said:

Is it directly measurable, or purely a descriptive assumption?

The way I personally see it, the whole of physics is based on the question of whether or not one description of reality is describing the same reality as another description of reality. This involves two ideas: (1), that there are many different ways to describe the same reality; and (2), that there are different possible realities which can be distinguished by their descriptions even though different descriptions are not necessarily describing different realities.

Special relativity deals with the notion that the measured values of various quantities depend on the frame of reference from which it is measured. So, the notion of coordinate transformations does have a physically measurable consequence, although not all coordinate systems correspond to frames of reference.

However, the set of all possible coordinate systems is a theoretical notion that identifies a particular reality and distinguishes it from a different reality.

Edited November 5Nov 5 by KJW

2 hours ago, Xerxes said:

Your theory, on the other hand, arrogantly declares it has no need of the usual sorts gadgets we use in differential geometry, while implicitly using them all

Exactly 👍 That’s what I tried to point out before.

On 11/2/2025 at 12:45 PM, Anton Rize said:

Nothing is assumed. Everything is derived.

Assuming that you actually answer my last question here is my next one.

Is is possible to do without assumptions ?

Even the statement 'There are no prior assumptions' is an assumption itself.

12 hours ago, studiot said:

Like it or not

Your post above has a metric.

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.

10 hours ago, studiot said:

Please also tell me how you can arive at

(anything)², without a metric ?

Please tell me what from this answer remains unclear for you?:

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

This document must be read literally.

All terms are defined within the relational framework of WILL RG.

Any attempt to reinterpret them through conventional notions

absolute energies, external backgrounds, hidden containers

will produce distortions and misreading.

Just like responsibility of formulating lies with the author,

the responsibility of interpretation lies with the reader:

take the words as written, not as filtered through prior formalisms.

Foundational Approach

Guiding Principle

Nothing is assumed. Everything is derived.

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

Mathematical hygiene is the geometry of reason

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:

fixed manifold + metricstructure+fields + constantsdynamics

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 mi=mg 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.

SPACETIME≡ENERGY

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)

SPACETIME≡ENERGY

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.

S1

(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.

S2

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) S1 for directional (one-degree-of-freedom) relational transformation.

(b) S2 for omnidirectional (central, all-directions-equivalent) relational transformation.

Proof

• For one relational degree of freedom, the classification of connected closed 1-manifolds yields S1 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, S2 is the maximally symmetric representative. Nontrivial quotients spoil global isotropy and are excluded.

Corollary (Uniqueness)

Under the Working Principle with Closure, Conservation, and Isotropy, S1 and S2 are necessary relational carriers for, respectively, directional and omnidirectional modes of energy transformation.

Remark (Non-spatial Reading)

Throughout, S1 and S2 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 SPACETIME≡ENERGY. From this we deduce: (i) closure, (ii) conservation, (iii) isotropy, and hence (iv) the unique selection of S1 and S2 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.

5 hours ago, studiot said:

Is is possible to do without assumptions ?

Even the statement 'There are no prior assumptions' is an assumption itself.

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

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?

8 minutes ago, Anton Rize said:

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

Space pre-exists to what exactly? Matter?

11 hours ago, Xerxes said:

But the fact remains that both your theory and category theory are deeply non-constructivist,

You know what I actually find arrogant?

Unjustified declarations stated with full confidence - for example:

"both your theory and category theory are deeply non-constructivist."

Please pinpoint exactly where you see non-constructivism.

Everything in my construction is derived step by step - see "Foundational Approach" above.

Show the specific element that is assumed rather than derived.

---

As for your second claim:

11 hours ago, Xerxes said:

Your theory, on the other hand, arrogantly declares it has no need of the usual sorts gadgets we use in differential geometry, while implicitly using them all.

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. You can't find it in my mathematics.

Nobody here has found it. It’s invisible.

3. It’s non-operational. It does not affect any outcome.

The equations reproduce observation with high precision.

4. That makes such a "metric" a useless, non-operational artefact - pure ontological inflation.

Would you call that fair reasoning?

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

10 hours ago, KJW said:

Look around you. It is clear that we exist in a space that is at least four-dimensional. Even if we can't derive this space logically from first principles, we can assume its existence on the basis of empirical observation.

Thank you, @KJW . Your explanation helps clarify where our views diverge.

You wrote:

"Look around you. It is clear that we exist in a space that is at least four-dimensional. Even if we can't derive this space logically from first principles, we can assume its existence on the basis of empirical observation."

I understand your point. And its seems reasonable at first. But that line of reasoning is structurally identical to early geocentric arguments:
“Look around you - it is obvious that the Sun goes around the Earth.”
Both rely on perceptual immediacy rather than logical derivation.

The fact that perception appears spatial does not logically justify the postulate of a background. It only proves that our measurements project into four observable parameters - not that the universe is a 4-dimensional container.

With WILL Relational Geometry (RG), I start from the opposite principle:

Nothing is assumed. Everything is derived.

The 4-parameter appearance emerges naturally from relational closure and symmetry - not from a pre-declared background.

The crucial point is that both GR and RG reproduce identical empirical results:
1. Mercury precession [math]\Delta\varphi=\frac{2\pi Q_{\mathrm{orbit}}^{2}}{1-e^{2}}=5.0208724126 \times 10^{-7}[/math]
2. Gravitational Lensing [math]\alpha = 2\kappa^2 = \frac{4Gm_0}{bc^2}[/math] For light [math]\beta = 1[/math] implies [math]\\beta_Y = 0[/math]
The orthogonal projection [math]\beta_Y =0 [/math] disappears; there is no dual axis available for symmetric partition. Therefore, the specific energy potential for light is not halved but complete explaining the factor of 2.
3. The radii [math]1.5R_s[/math] and [math]3R_s[/math] are known from General Relativity, their emergence here from two distinct and fundamental geometric symmetries[math]\theta_1=\theta_2[/math] and [math]Q=Q_t[/math] is not imposed but arises from the internal consistency of the WILL framework.
4. Time correction required for GPS synchronization
[math]\Delta \tau_{{\rm GPS} \rightarrow {\rm Earth}} = \left(1-\frac{\tau_{Earth}}{\tau_{GPS}}\right)\cdot D_{ayS}\cdot M_{icro}= 38.5219216525~\mu\text{s/day}[/math]
where [math]\tau = \kappa_{X}\cdot\beta_{Y}[/math]
5. Energy Symmetry law [math]\Delta E_{GPS \rightarrow Earth}+\Delta E_{Earth \rightarrow GPS} = -6.1265399845\times10^{-10} + 6.1265399845\times10^{-10} = 0[/math]
and in the same time its not just conformation within RG: [math]\frac{U_{EGPS}+T_{EGPS}}{m_{GPS}c^{2}}= \Delta E_{Earth \rightarrow GPS} = 6.1265399845\times10^{-10}[/math]
6. And finally 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.


When two models give the same predictions, the principle of ontological parsimony demands preference for the one with fewer assumptions and free parameters.

That’s exactly what happens here:

1. Mathematics becomes radically simpler (no metric, no connection fields).

2. Paradoxes and self-referential problems disappear.

3. Unique new predictions appear from relational closure alone.

4. Operational content increases - everything used is observable or derivable.

5. Physical meaning and intuition are restored - equations map directly to measurable relations, not to background abstractions.

So I’m not disputing the usefulness of GR calculus; I’m showing that its predictive content can be derived from a cleaner relational base, where “space” and “energy” are two sides of the same invariant.

And I find it hard to believe my self. It literally driving me crazy. And what I've showed its just a small fraction just a tip of the iceberg. Im not asking to believe me. Im asking to find the Truth together.

Edited November 6Nov 6 by Anton Rize

5 hours ago, Anton Rize said:

  16 hours ago, KJW said:

Look around you. It is clear that we exist in a space that is at least four-dimensional. Even if we can't derive this space logically from first principles, we can assume its existence on the basis of empirical observation.

I understand your point. And its seems reasonable at first. But that line of reasoning is structurally identical to early geocentric arguments:
“Look around you - it is obvious that the Sun goes around the Earth.”
Both rely on perceptual immediacy rather than logical derivation.

The fact that perception appears spatial does not logically justify the postulate of a background. It only proves that our measurements project into four observable parameters - not that the universe is a 4-dimensional container.

You say this like you believe the geocentric viewpoint is invalid. In general relativity, one can construct a geocentric coordinate system, and such a coordinate system is just as valid as any other coordinate system, in accordance with the principle of general relativity. 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.

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