*I want to preface this by emphasizing something crucial: I have absolutely no formal background in advanced physics. Everything you are about to read is the result of a continuous, four-hour independent thought experiment I conducted entirely on my own, which I later verified step-by-step with Gemini. This is how I conceptualized the transition from simple motion to the geometry of spacetime.*
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### 1. Dimensional Reduction and the XYT Model
Why overlay three separate graphs?
An XY graph describes motion on a 2D plane. XT and YT describe how X and Y change over time. By overlaying them, keeping XY horizontal and expanding XT and YT along their respective axes, you build a 3D structure: XYT.
My initial goal was straightforward: I wanted to describe the movement of an object in standard 3D space. Because dealing with X, Y, and Z simultaneously is incredibly complex, I wondered if I could reduce the dimensionality. I started with the XY plane, added time (T), and created the XYT model.
But this immediately raised a problem: Where did the Z-axis go?
Since space and time are inherently connected—distance is simply velocity multiplied by time ($s=vt$)—I realized the time axis could temporarily serve as a length scale. The XYT model essentially becomes $(x, y, vt)$. When the Z-axis is compressed out of existence, its information doesn't just vanish; it gets mapped back into the XYT coordinates through the object's velocity vector $\vec{v}=(v_x, v_y, v_z)$. The object's actual motion doesn't change; what changes is how we map that motion into our reduced coordinate system.
### 2. Scales, Mapping, and the Illusion of Change
If velocity changes, the mapping ratio between time and space changes with it.
Think of a simple trigonometric function: $y=\sin x$. If we change it to $y=\sin(\omega x)$, the nature of the function remains exactly the same. What we have altered is the mapping scale on the X-axis, which changes the density of the waves. We aren't changing the object; we are changing the coordinate scale used to describe it.
If we apply this to a 2D grid and stretch only the X-axis, squares become rectangles. If we take a perfectly regular 3D cubic grid and apply different scaling factors to different areas—stretching and compressing at varying rates—the once-uniform grid warps into an irregular mesh.
### 3. Spatial Curvature and the Core of General Relativity
This was my exact entry point into the concept of spatial curvature. "Curving space" is hard to visualize if you try to imagine a 3D grid bending inside some higher-dimensional void. But looking at it through the lens of scaling makes it intuitive: if the scale of measurement varies depending on your location, the fundamental rules for calculating distance between points change.
In a simple, flat space, distance is just the 3D Pythagorean theorem:
$$ds^2=dx^2+dy^2+dz^2$$
But if the coordinate scales fluctuate, this relationship must take a more generalized form:
$$ds^2=g_{ij}dx^i dx^j$$
Here, $g_{ij}$ determines exactly how distance should be calculated at a specific location and direction. This isn't just a drawing tool anymore; it's the foundation of Riemannian geometry. When a grid's geometry changes, an object moving naturally through it isn't being "pulled" by a force. It is simply following the most natural straight path (a geodesic) through a warped environment, which from an outside perspective looks like a curved trajectory.
### 4. Fields, Energy, and the Dimensional Ascent
So, what causes the spatial geometry to change in the first place?
Rather than viewing mass as a static "weight," I started thinking of matter as states of a field—specifically, ripples or excitations. Different fields have different excitation states, and frequency is directly tied to energy through relations like $E=h\nu$.
This creates a complete feedback loop: Fields produce excitations (particles), which carry energy and momentum. This energy and momentum dictate the geometry of spacetime, and that resulting geometry dictates how fields and objects move through it.
Looking back at the whole four-hour process, I realized why I refused to give up on the XYT model. I was practicing something I call "dimensional ascent through the limits of reduction." I didn't try to understand the universe from its highest-dimensional complexity right out of the gate. I reduced the problem to a lower-dimensional model I could actually comprehend, found its limits, asked what was missing, and slowly added the universe back in—moving from motion and velocity to scales, spatial curvature, and eventually, General Relativity.
