Inertial Scaling Laws: Rethinking Time, Space, and the Structure of the Universe

About three years ago, I started exploring something I’d been thinking about since I was a kid: scaling in the universe and how scaling laws work. What began as a fun but serious exercise has developed into a body of research with three published papers, DOI numbers, and implications that extend from quantum mechanics to cosmology.

Where It Started

It started with a simple question: what if we compared the dynamics of a hydrogen ion—the electron and the nucleus—versus the Sun-Saturn system? What came back was remarkable: the mass ratio between those two systems is only off by a factor of two or three. Not even an order of magnitude.

That doesn’t necessarily mean anything on its own. But it got me thinking about the similarity between systems at radically different scales. With a background in chemistry and a solid understanding of quantum mechanics, this felt like an appropriate question to pursue rigorously.

The Scaling Law

Looking at isometric scaling—where you maintain density—I found a direct relationship between the pace of a system’s dynamics and its moment of inertia about any axis. The law holds universally: if you isometrically scale any system, regardless of shape or dynamics, the moment of inertia scales as the fifth power of the scale factor. And the scale factor is always the time scale ratio.

This is significant because it means mass and radius abstract away. We’re left with a universal statement: the fifth root of the moment of inertia is the system’s time scale.

From Scaling to Time

This result has a natural connection to time dilation. We know time dilation exists from general and special relativity. But this appears to be a different kind—an inertial time dilation that affects systems based on their scale rather than their velocity or gravitational field.

Using a concept I call “inertial density”—inertia over volume—the model paints a picture of a universe where time is multidimensional. Every system has its own emergent pace of time. A planet has one pace; its core has another. The law holds at every level you examine.

The Schwarzschild Threshold

One of the more striking results: if you convert the Schwarzschild radius into a static global constant, you get a value of 6.73295 × 10²⁶ kg/m. This becomes a universal threshold. Take the ratio of mass over mean radius for any system and divide by this constant—if the ratio is less than one, you have a black hole. But you still have a positive ratio, no imaginary numbers, all the way to the r=0 asymptote.

This allows evaluation of Kerr black holes (rotating black holes) using a single formula. All three spatial dimensions can have different inertial densities, and the model handles the shape of rotation and dimensional aspects without requiring separate models for different types of black holes.

Cosmic Redshift

The theory offers a natural explanation for cosmic redshift: it’s a regular artifact of inertial time dilation affecting large, broad, low-density systems. General and special relativity emerge as edge cases of a broader inertial relativity—the relativity of inertial scale, operating in both rotational and linear modes.

Implications

If the model is correct, several provocative conclusions follow:

• Singularities cannot exist
• There is no big bang—the universe is infinite in scale and distance
• The Penrose paper on singularities doesn’t correctly reflect what happens with relativity (Einstein himself didn’t go for it)
• Light is not a wave-particle duality any more than any other object—it’s a wave of energy that, like water waves or sound waves, has discrete packet characteristics determined by its scale domain

Transparency and Rigor

Three papers have been written on this work, all with DOI numbers. All data goes into GitHub—everything is transparent and reproducible. The mathematics is rigorous to the point where you can derive general and special relativity under a single principle.

This research represents the intersection of rigorous physics, creative thinking, and the kind of complex problem-solving that defines my consulting practice. Whether the problem is a rotating black hole or a tangled legal dispute, the approach is the same: first principles, mathematical rigor, and a willingness to follow the evidence wherever it leads.