Shape of Energy: A Narrative Exposition of WILL Geometry

What is This Page?

This interactive page was created to explain the essential meaning and logic of the WILL Geometry model — but without the heavy mathematical machinery.

Here, the focus is on the core ideas, intuitive explanations, and geometric visualizations, making this new approach to fundamental physics accessible to a much wider audience.

If you want to see the full mathematical derivations, download the complete documents here: Full WILL Geometry Papers

Unified Framework

Section 1: The Universe from a Single Principle

Modern theoretical physics is a magnificent edifice built on the foundation of astonishing predictive power. Theories of relativity and quantum mechanics describe the world with incredible precision, from subatomic particles to cosmic horizons. Yet, in this very foundation, there remain enigmatic elements that cast a shadow over the completeness of our understanding: gravitational singularities where known laws break down; arbitrary constants we can measure but cannot explain; and the necessity to introduce “dark” entities such as dark matter and dark energy simply to align theory with observation. These phenomena indicate that our description of reality may be incomplete.

Against this backdrop, a fundamental question arises: What if we could construct the Universe without any “extraneous details”? What if all observable phenomena—space, time, mass, energy—could be derived from a single, simple, and self-evident statement, without the need for free parameters or inexplicable constants? This approach—what might be called the principle of “epistemological hygiene”—demands that we reject all assumptions not strictly necessary, and build the theory solely on the basis of logical sequence.

Methodological Pillars

It is on this path of uncompromising minimalism that the central, unifying postulate of WILL Geometry is born:

SPACETIME ≡ ENERGY EVOLUTION

At first glance, this statement appears simple, but it carries a revolutionary meaning. It asserts that space and time are not some pre-existing “container” or stage on which physical processes unfold. On the contrary, the very fabric of spacetime is identical to the full structure of all possible transitions and interconnections between energetic states. Spacetime is not a stage; it is the dance itself. Energy does not exist in spacetime; rather, it defines it through its own projections and relations.

This work is not merely a reformulation of existing theories, but rather a narrative account of this new perspective—born from a personal quest to understand the fundamental nature of reality based on the most elementary geometric principles. It is an intellectual journey motivated by the conviction that the Universe’s structure must, at its core, be simple, elegant, and logically closed. The aim of this narrative is not to refute established physics, but to offer a deeper perspective—showing how the known laws of special and general relativity can be not merely postulated, but derived as inevitable consequences of a single, more fundamental principle. We will trace the logical chain from this lone postulate to its ultimate conclusions, demonstrating how it gives rise to all the complexity and beauty of the relativistic world.

Section 2: The Rules of the Game — A Foundation of Pure Logic

If we accept the fundamental postulate—that spacetime is identical to energy evolution—we are immediately confronted with its strict logical consequences. First and foremost: if all of reality is a self-sufficient system of energetic relations, then by definition nothing can exist “outside” this system. There is no external observer, no absolute reference frame or background on which events unfold. The Universe of WILL Geometry must be entirely closed and self-sufficient.

This restriction, philosophical at first glance, generates concrete physical and geometric “rules of the game”:

SPACETIME-ENERGY-EVOLUTION.png

This immediately raises the question: What geometric structures satisfy these strict requirements—being both closed and maximally symmetric? Mathematics gives a definitive answer. Among all possible manifolds, only certain forms possess these properties to perfection.

These geometries are not arbitrary choices or convenient models. They are forced upon us by the logic of the postulate itself. If the Universe is a closed and symmetric system of energetic relations, then any projections of those relations must “live” on these unique surfaces. The circle and the sphere become the only possible “canvases” on which the picture of physical reality can be drawn.

Finally, we must rethink one of the fundamental constants of nature—the speed of light, c. In WILL Geometry, c is not merely the speed at which photons travel. It is the universal speed of evolution, the fundamental tempo of change in the Universe itself. Every interaction, every transformation of energy, occurs at this single, invariant rate. This universal “speed of evolution” is like a vector of constant length. The various physical processes we observe are merely projections of this single vector onto different axes, just as a car’s velocity can be decomposed into northward and eastward motion. The sum of the squares of the projections always remains constant, naturally returning us to the geometry of the circle, where the radius (c) is invariant.

Thus, from a single postulate, we derive the entire set of rules: conservation, symmetry, and specific geometric structures (the circle and the sphere) that will serve as the arena for all subsequent physical phenomena. Geometry is not imposed upon physics; it is its inevitable logical consequence.

Section 3: Act I – Motion as a Shadow on the Circle (Special Relativity)

Now that the rules of the game have been established, let’s consider the simplest case: one-dimensional motion. According to our logic, this process must be described on the only possible closed and symmetric 1D geometry—the circle. This act of our narrative is devoted to reconstructing special relativity (SR) using nothing but these geometric tools. The main character here is the kinetic projection, denoted by the Greek letter β (beta).

Imagine a unit circle whose radius symbolizes the universal speed of evolution, c, which for simplicity we set to 1. Any state of motion of an object can be represented as a point on this circle. The total “speed of evolution” is a vector drawn from the center to this point. This vector can be decomposed into horizontal (X axis) and vertical (Y axis) projections.

Kinetic-Energy-The-Spatial-Projection

Show Interactive Graph: Motion/Time on the Unit Circle (Desmos)

How Motion and Time Are Just Two Sides of the Same Thing

Imagine all possible states of movement as points around a perfect circle. The radius of this circle is the fastest possible speed in the universe — the universal “speed of change.” Every moving object is like a pointer from the center of the circle to some spot on its edge.

Think of it like sharing a fixed budget: If you spend more of your “change” moving through space, you have less left for moving through time.

Show the math The relationship between space and time projections is just Pythagoras’ theorem for a unit circle:
$ \beta = \frac{v}{c} = cos(\theta_S) $ Kinetic projection
$ L_c = sin(θ_S) = \sqrt{1 − \beta^2} $ Length contraction factor

Why This Explains All the Weirdness of Relativity

When you look at it this way, all the “strange” effects of special relativity — like time slowing down as you go faster — are just the result of keeping the total “speed of change” fixed, but sharing it differently between space and time. There’s nothing mysterious:

Show the math The time slowdown (Lorentz factor) is:
$ T_{d} = \frac{1}{L_{c}} = \frac{1}{\sqrt{1- \beta^2}} = 1/sin(θ_S) $ Time dialation or $ \gamma = \frac{1}{\sqrt{1 - v²/c²}}$

How E = mc² Falls Out Naturally

Here’s the coolest part: That famous equation, E = mc², isn’t some magic law — it’s just what you get when you realize energy and mass are two sides of the same coin. At rest, all of an object’s energy points along the time direction. When it moves, some energy “tilts” into space — but the total stays balanced, so that the piece along time remains constant.

Show the math Total energy: E = γmc²
Rest energy: E₀ = mc²
Momentum: p = γmv

The Energy-Momentum Triangle

A Guide to Interpreting This Combined Diagram:

This diagram intentionally superimposes two different geometric views to illustrate their deep connection. To avoid confusion, please read the following guide:

  1. The Background (Unit Circle): The white unit circle and its associated labels (like L_c, θ_S, and “Time evolution”) represent the ‘Spacetime Projection’ model discussed previously. It is shown here as a reference to illustrate where the fundamental velocity parameter β originates. In this background view, the hypotenuse is always constant (equal to 1).

  2. The Foreground (Energy Triangle): The shaded triangle is the main subject of this section. For this Energy-Momentum Triangle, the axes take on a new, physical meaning:

    • The vertical axis represents the constant Invariant Rest Energy ($E_0$).
    • The horizontal axis represents the growing Momentum ($p$).

The hypotenuse of this main triangle represents the growing Total Energy ($E$), which clearly extends beyond the bounds of the background unit circle.

The key takeaway is to see how the parameter β from the reference circle is used to construct the much larger energy-momentum triangle, visually linking the geometry of spacetime to the energy of an object.

Visualize energy, mass, and momentum as a simple right triangle:

Energy-Momentum-Triangle

When you slide the velocity, you can see how “momentum” grows, and total energy stretches to keep the triangle in perfect balance, but rest energy always stays the same.

What’s the point?

Show the math The Energy-Momentum Relation: $$ E^2 = (pc)^2 + (m_0 c^2)^2 $$ Where: - $E$: Total energy - $p$: Momentum - $m_0 c^2$: Rest energy
Show Interactive Graph: The Energy-Momentum Triangle (Desmos)

Simply saying:

What This Means

Special relativity stops being a bunch of rules about “strange time effects” or “postulates about light.” Instead, it’s just a simple story of how all things must share a fixed “budget” of change between motion and time. Mass, energy, and momentum are simply three perspectives on the same underlying geometric fact.

Section 4: Act II – Gravity as a Shadow on the Sphere

Having cracked motion using a simple circle, let’s tackle gravity. Gravity is different: instead of working in one direction, like motion, it pulls equally from every side — like being surrounded in all directions. So our “canvas” has to be not a line or circle, but a whole sphere.

How Gravity is Just a Different Kind of Projection

Imagine standing at the center of a big sphere — gravity “spreads out” the same in every direction from the mass at the center.

What does κ mean?

You can think of κ as “how much of the universe’s speed limit you’d need to escape gravity right here.”

How Gravity Warps Time

Here’s where the sphere magic happens:

Potential-Energy-The-Temporal-Projection

Gravity isn’t just pulling you in — it’s literally changing how fast your clock ticks, depending on where you are.

Show the math Time dilation near a massive object is: $$ \sqrt{1−\kappa^2} = \sqrt{1−\frac{R_S}{r}} = \sqrt{1−\frac{2GM}{rc^2}} $$ ---
Show Interactive Graph: Gravity as a Shadow on the Sphere (Desmos)

The Hidden Symmetry

Here’s the coolest part:

Situation Geometry Time Slowdown Formula
Moving fast (SR) Circle √(1−β²)
Near mass (GR) Sphere √(1−κ²)

So, what looks like two separate “mysteries” in physics are actually just two projections of the same fundamental story.

Section 5: Energy as a Relation — What κ and β Actually Mean

Key Principle:

Energy isn’t something objects “have”—it’s a measure of differences between states.

When we drop anthropocentric distortions, a clear and intuitive picture emerges:

In this relational view, your perspective is always the reference frame. You are always at zero. Everything else is described by how it differs from your state:

Think of κ and β as your own relational measuring tools:

Energy thus emerges naturally:

Here’s a simple analogy:

Imagine standing on a train platform. A train passes by rapidly: to you, it has significant kinetic energy. But if you jump onto the train, it instantly becomes stationary relative to you. Its kinetic energy is now zero—because your frame of reference shifted. The energy didn’t vanish; your perspective changed.

Bottom line:

Section 6: Unification – When the Circle Meets the Sphere

Now comes the punchline: All this time, we’ve seen motion (special relativity) and gravity (general relativity) as separate “shadows” of the same underlying process, just playing out on different shapes — the circle and the sphere.

But here’s the twist: these two worlds aren’t really separate. In fact, there’s a deep, built‑in connection between them. They’re just different “faces” of one and the same thing.

The Universal “Budget” Principle

Imagine you have a bucket of paint. You can use it to draw a line around a circle (one‑dimensional) —or— you can use it to cover the surface of a sphere (two‑dimensional).

No matter how you use it, your total paint doesn’t change — but the way it “spreads” over these shapes is fundamentally different. This is exactly what happens with energy in the universe:

But both are just different ways of splitting up the same “energy budget.”

Show the key connection The “paint” covering for both shapes leads to a simple rule: $$ \kappa^2 = 2\beta^2 $$ or $$ \frac{\kappa^2}{\beta^2} = 2 $$ where - β² = “share” spent on motion - κ² = “share” spent on gravity

The-Fundamental-Conservation-Law-kb-2

Why does it matter?

Bottom line: Motion and gravity are two sides of the same coin. Their relationship isn’t an accident — it’s a built-in, geometric law of how the universe splits up its “energy resources.” This is why they can’t ever be truly separated, and why geometry is the hidden glue in everything.

Where Does the “2” Come From?

All this talk about “budgeting” between circles and spheres leads to a natural question: Why exactly “2”? Where does it come from?

Here’s the simple geometric truth:

So when we ask, “How many times does a circle fit into a sphere?” the answer is:

Show the calculation $$ \frac{\text{Surface area of sphere}}{\text{Circumference of circle}} = \frac{4\pi}{2\pi} = 2 $$

That’s the source of the mysterious “2” in our key equation. It’s not an arbitrary fudge factor — it’s pure geometry.

How It Ties Together

So when energy splits between motion (on the circle) and gravity (on the sphere), the “budget” gets shared according to this built-in, topological ratio. The key relationship:

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

just says: “Gravity’s share is always twice the motion’s share, because a sphere’s surface area is twice the length of a circle’s perimeter, at the most fundamental geometric level.”

In short: The “2” isn’t magic, it’s the deep fingerprint of geometry itself, forever tying together movement and gravity in the universe.

The Photon Sphere: A Point of Perfect Balance

Show Interactive Graph: Q Circle (Desmos)

A remarkable consequence of this unification happens at a special “balance point” — where the two projections, kinetic and potential, are perfectly matched. This occurs when:

\[\kappa^2 + \beta^2 = 1\]

and the two angles are exactly equal.

At this moment, something amazing appears: it matches the “photon sphere” — the special zone around a massive object where light itself can orbit in a perfect circle. (For a black hole, that’s at a distance of 1.5 times its radius.)

What happens to light at the photon sphere? At the photon sphere, a photon moving exactly along the perfect circle could orbit forever — but this path is incredibly unstable. If the photon’s path points even a tiny bit outward, it escapes to infinity. If it’s angled a bit inward, it falls into the black hole. So, the photon sphere is not a prison — it’s a tightrope. Light can still escape if it’s not traveling exactly along the circle.

Section 7: Energy–Symmetry Law (Why No Free Lunch)

The Big Idea

Whenever you compare two observers (say, someone on the ground and someone in orbit), the energy differences they see will always perfectly balance out. No matter how you swap perspectives — nobody ever gets “extra” energy for free. This is the universe’s built-in law of energetic fairness.

How it Works (Without Math)

Imagine:

If an object moves from A (the ground) up to B (orbit):

From B’s point of view, if the object comes down:

But if you add up both “energy stories,” the total change is always zero.

The universe never “creates” or “loses” energy when you swap perspectives — just moves it around.

Show the math $$ \Delta E_{A \to B} + \Delta E_{B \to A} = 0 $$ $$ E_{A \to B}=\frac{1}{2}((\kappa_{A}^{2}-\kappa_{B}^{2})+\beta_{B}^{2}) $$ $$ E_{B \to A}=\frac{1}{2}((\kappa_{B}^{2}-\kappa_{A}^{2})-\beta_{B}^{2}) $$ $$ E_{SYMETRY}=E_{A \to B}+E_{B \to A}=0 $$ Each transfer includes both gravitational ($κ^2$) and kinetic ($β^2$) parts, always balancing out.

Energy-Symmetry-Law

Universal Speed Limit: Why Nothing Goes Faster than Light

This “energy symmetry” is so strict that it naturally sets the universe’s ultimate speed limit.

Math explanation Assume an object could exceed the speed of light ($ \beta > 1$). In that scenario: - The kinetic component ($\beta$) surpasses unity excessively, causing an irreversible imbalance in energy transfer. - No reciprocal transfer could balance this energy, breaking the fundamental symmetry: \begin{equation} \Delta E_{A \to B} + \Delta E_{B \to A} \neq 0 \end{equation} Thus, $ \beta \leq 1$ ($v \leq c$) is required intrinsically to preserve causal and energetic consistency.

In plain English:

Section 8: The Whole Universe in a Single Line

After all the geometric juggling, here’s the punchline: Everything — gravity, motion, energy, even the rules for black holes — can be summed up in just one line.

See the Universe’s one-line “code” $$ \kappa^2 = \frac{R_s}{r_d} = \frac{\rho}{\rho_{max}} $$

What does this actually mean?

In plain words:

Critical radius / current radius = current density / critical density

Unified-Field-Equation

No matter how you look at it — as a distance, or as an amount of energy — the universe always keeps you within the same “safe fraction” of the maximum possible.

The “Critical Density” Explained

How is the density limit set? $$ \rho_{max} = \frac{c^2}{8\pi G r_d^2} $$ — “The smaller the radius, the greater the allowed density — but there’s always a cap.”

What this solves

Instead of singularities, black holes become places where everything is packed to the maximum allowed, but always stays finite and well-behaved.

Table: “Old School” GR vs. WILL Geometry

  General Relativity (GR) WILL Geometry
Geometry & Energy Equated by differential equations Identified algebraically (one-to-one)
Math Formalism Complicated tensors, calculus Simple, projection-based logic
Singularities Yes, possible (bad news!) Impossible
Density Limit Undefined (needs quantum gravity, still a mystery) Always set, changes with position, never infinite

Bottom line: This single “line of code” is the ultimate summary of WILL Geometry:

Theoretical-Ouroboros

Closure of the Theoretical Framework

The unified field equation completes the ab initio derivation begun with the fundamental postulate:

\[\text{SPACETIME} \equiv \text{ENERGY EVOLUTION}\]

This is the unified geometric field equation we derived. It expresses the complete equivalence:

\[\text{GEOMETRY} \equiv \text{ENERGY DISTRIBUTION}\]

We have shown that this single postulate, through pure geometric reasoning, necessarily leads to an equation which mathematically expresses the very same equivalence we began with. We started with a single fundamental statement about energy and its evolution, from which geometry and physical laws are logically derived, and these derived laws then loop back to intrinsically define and limit the very nature of energy and space, proving the self-consistency of the initial postulate. From a philosophical and epistemological point of view, this can be considered the crown achievement of any theoretical framework—the “Theoretical Ouroboros”. But let’s remain skeptical. We are here for Physics!

Theoretical Ouroboros

The WILL framework exhibits perfect logical closure: the fundamental postulate about the nature of spacetime and energy is proven as the inevitable consequence of geometric consistency.

Section 9: Grounding the Vision – From Abstraction to Reality

A theory, no matter how beautiful, is just a clever idea until it passes real-world tests. Science is all about matching what we think to what we actually see in nature. A beautiful theory is useless if it does not align with what we observe. This section is devoted to grounding the abstract vision of WILL Geometry by demonstrating that it makes precise, testable predictions about real physical phenomena.

Let’s examine these two key examples, which serve as rigorous empirical tests of the entire theoretical construction.

Empirical-Validation

1. Time Correction in the GPS System

The Global Positioning System (GPS) is perhaps the ideal laboratory for testing relativistic effects in everyday life. For the system to function with high accuracy, GPS satellites must account for two relativistic effects that influence the rate of their onboard clocks compared to clocks on Earth:

The standard approach is to calculate these two corrections separately and sum them. WILL Geometry offers a more fundamental approach. It asserts that these two effects are not independent, but are linked by the unbreakable relation κ² = 2β² and can be combined into a single unified energetic parameter Q, where Q² = κ² + β². Using this single parameter to calculate the overall relativistic time shift for the Earth–satellite system, WILL Geometry predicts that the clocks on GPS satellites should run ahead of ground-based clocks by about 38 microseconds per day.

This result exactly matches the empirically measured value that must be continually added to the GPS system for it to function correctly. The success of this prediction is a powerful confirmation not only of the individual parts of the theory, but—more importantly—of the very principle of unification. It shows that treating kinetic and gravitational effects as unified geometric projections is not just a theoretical sophistication but a practical necessity for arriving at the correct answer.

Show Interactive Graph: Earth GPS (Desmos)

2. Precession of Mercury’s Orbit

One of the earliest triumphs of Einstein’s general relativity was its explanation of the anomalous precession (slow rotation) of Mercury’s elliptical orbit. Observations showed that the perihelion (the point closest to the Sun) of Mercury’s orbit shifts by an additional 43 arcseconds per century, above what Newtonian gravity predicted.

WILL Geometry also faces this classic test. Using its fundamental equations and parameters (β and κ, calculated for Mercury in the gravitational field of the Sun), the theory allows us to compute the expected value of this relativistic precession. The result obtained within WILL Geometry matches both the GR prediction and the observed data with machine-level precision.

Detailed calculations for both examples are provided in this document, but the narrative conclusion is clear: WILL Geometry is not merely a philosophical construction. It passes the most rigorous experimental and observational tests. It not only reproduces the successes of standard relativistic physics but does so from deeper, more unified first principles, lending its predictions extra weight and elegance. A theory born of pure logic finds its exact reflection in the workings of the real world. WILL Geometry stands up to every challenge that nature throws at it, not just matching standard physics, but explaining it from the ground up, using nothing but logic and geometry.

Show Interactive Graph: Sun Mercury (Desmos)

Section 10: A New Reality of Change — Dynamics Without Time

Let’s get to the wildest part of WILL Geometry: What if time isn’t fundamental at all?

Rethinking Change and Time

In “classic” physics, you always start with:

Here, time is like a river: everything happens inside this flow.

But in WILL Geometry, it’s the other way around:

So what is “dynamics” now?

Instead of motion playing out in time, you have a web of “allowed” states, all rigidly connected. Any change in one parameter instantly forces all others to adjust, so the system stays in balance.

Imagine a black hole gaining mass:

What, then, is time?

Time is just our way of describing the difference between one balanced state and the next. It’s not some external clock ticking in the background — it’s a label for the sequence of changes.

Time does not drive change — instead, change defines time.

Why does this matter?

Bottom line: In WILL Geometry, the universe is not a machine running inside time — it’s a perfectly balanced structure, forever reshaping itself. What we call “time” is simply our story for that ongoing process of rebalancing.

Section 11: Conclusion — The World as a Projection

Our journey through WILL Geometry draws to a close. But remember: in this short exploration, we’re only scratching the surface. The model itself has already been extended to cover cosmology and quantum mechanics — with results and detailed applications available here:

WILL Geometry — Results & Predictions

We started with one simple idea:

SpaceTime is just energy in motion.

From this, logic alone led us to:

This journey uncovered a hidden unity behind the laws of physics, replacing arbitrary rules with pure geometry and energy flow. Even the “weird” effects of black holes and GPS satellites fall naturally out of this approach.

At the core is one bold claim:

Energy doesn’t just exist in space — it creates space, by its projection.

All of physics becomes a story about “the projective curvature of the energy flow.” And the essence of it all is captured in a single, dimensionless invariant:

See the WILL invariant $$ W_{ill} = \frac{E \cdot T^2}{M \cdot L^2} = \frac{L_d\,E_0\,T_c\,t_{d}^{2}}{T_d\,m_0\,L_c\,r_{d}^{2}}= \frac{\frac{1}{\sqrt{1-\kappa^{2}}}m_{0}c^{2}\cdot\sqrt{1-\kappa^{2}}\left(\frac{2Gm_{0}}{\kappa^{2}c^{3}}\right)^{2}}{\frac{1}{\sqrt{1-\beta^{2}}}m_{0}\cdot\sqrt{1-\beta^{2}}\left(\frac{2Gm_{0}}{\kappa^{2}c^{2}}\right)^{2}}=1 $$

Energy, mass, time, and length — not disconnected ideas, but tightly bound faces of a single self-consistent structure.

The-WILL-Invariant

The Name “Will”

The name Will reflects both the harmonious unity of the equation and a subtle irony towards the anthropic principle, which often intertwines human existence with the causality of the universe. The equation stands as a testament to the universal laws of physics, transcending any anthropocentric framework.

This is not the end, but just a beginning. For the full mathematical theory, and its extensions to cosmology and quantum mechanics, see: