Inertial Time Dilation Predicts Observed Cosmic Redshift Patterns without the Need for Cosmic Expansion

Abstract

The incorporation of inertial scale into General Relativity demonstrates that what we observe as curvature of space-time arises from a relationship between inertia and time. This slightly more profound understanding of General Relativity is distinct from how (GTD) is currently described, where the presence of a mass alone results in curvature of space-time. and not from mass influence alone, and gravitational time dilation (GTD) is derived directly from the principle of Time Dilaton of Inertial Scale (TDS), which states that for static mass scale transformations, time scales by the square root of the inverse of the moments of inertia.

t₁/t₂=(i₂/i₁)^(1/2)

However, since GTD-attributed time dilation holds primary significance within a few orders of magnitude of the Schwartzchild radius, at most, GTD would likely only represent a small portion of the total TDS (and this redshift) observable at any portion of the universe. The remainder is represented by classical Newtonian motion, which we perceive not as time dilation, but simply as “the way things of different sizes move” relative to our largely static inertial scale frame of reference. Large celestial systems that 4are not of sufficient density to produce significant GTD but are massive and expansive enough to exhibit substantial time dilation on an inertial scale when observed from a vastly different inertial scale frame of reference. Objects like nebulae, galaxies, galactic clusters, filaments, especially if observed perpendicular to their longest dimension, would exhibit significant "non-gravitational” redshift. If confirmed, the relationship between universal expansion and redshift would garner reinterpretation.

Introduction

Inertia represents the pace of time for a system:

Gravitational time dilation is a phenomenon that extends from the Law of Space-Time Equivalence, which is derived directly from isometric scaling laws. This law states that for two systems, the relative pace of time is proportional to the relative moment of inertia about the axis of observation between the observer and the observed, or between two systems interacting. Where “system” is defined as any static region of space (and the particles within it), or any collection of particles (and the space between them):

t₁/t₂=i₂/i₁

From this, and by definition, a system's moment of inertia about an axis is its pace of time about that axis. For linear motion, Inertia becomes mass alone. Therefore, inertia is the sole determinant of the pace at which a system evolves.

The law also states that when any isolated system is scaled isometrically by a factor of k, or scaled in all three linear dimensions by k, and scaled mass by k^3, thus maintaining static density over the transformation (at least until we near the Schwartzchild radius), its moment of inertia scales by a factor of k⁵. This also holds when comparing any two objects that are isometric scales of each other:

t₁/t₂=(i₂/i₁)^(1/5)

The following facts arise from isometric scaling laws:

Isometric scaling represents “natural” proportional scaling

Physics is scale invariant - the universe observes the same physics from every observer’s scale frame of reference. Systems of different isometric scales always exhibit indistinguishable physics from the system scale frame of reference.

A system's pace of time emerges from inertia. From an external scale frame of reference, a system transforming isometrically reduces in relative spatial scale by a factor of k, and its relative temporal pace increases by a factor of k.

Static density corresponds to an unchanged pace of time.

A moment of inertia scale factor of k⁵ corresponds to an unchanged pace of time.

Gravitational Scaling

Homothetic scaling, conversely, represents the behavior of static mass transformations:

t₁/t₂=(i₂/i₁)^(1/2)

For static mass transformations, this simplifies to:

t₁/t₂=R₂/R₁

This equivalence describes gravitational time dilation in relativistic settings as well as the relative motion of objects in classical (low-density) settings. Conceptually, imagine what happens as a molecular structure approaches its Schwarzschild radius. It is spherical and of a consistent density gradient with greatest density at the core, [Footnote: This is because, with a spherical system, the lowest moment of inertia state is a neatly organized gradient from most dense at the core to least dense at infinity radius.] and we haven’t yet really compromised intermolecular forces.

Significant gravitation effect really starts to express itself at scales that correspond to our smallest coalesced objects, which would correspond to the scale of the smallest spherical planetary objects we observe. A sphere will continue to increase in density and mass, and eventually, gravitational effects begin to be significant. The gradient of density eventually reaches equilibrium. Once the object approaches its Schwarzschild radius, gravitational time dilation increases in significance, and at the Schwarzschild radius, time stops from an external observer’s perspective. Until this final stretch, the density of the system has remained relatively consistent; however, approaching critical mass, intermolecular forces compress, causing density to grow quickly as radius reduces. When a spherical system changes density as bonds become compromised and eventually break, that change in density represents a change in relative pace of time, and objects that are part of that gravitational system (i.e., in the field of gravity) will experience time dilation compared to their prior state of density. Density is proportional to moment of inertia; greater mass, same radius = greater density and greater moment of inertia.

STE reveals that inertia bears the fundamental relationship with time, and this time dilation manifests as gravity and gravitational time dilation at relativistic scales, as well as changes in motion at classical scales.

How Density, Mass, Volume, and Inertia Interrelate

Systems that are both relatively massive and relatively expansive can achieve extreme moments of inertia. For example, a black hole can have an extreme mass, but because it is very dense, interpreting it as a point mass is a reasonable approximation. However, on average, a galactic cluster contains a variety of extremes in density and distribution, resulting in a greater moment of inertia about the axis of observation, and the observed redshift would be considerably greater than what is predicted by general relativity alone, especially in the vicinity of highly variant systems.

The most extreme moments of inertia are achieved through systems with little mass at the axis, and almost all matter at the radius. A soap bubble, for example, is a sphere with a very high moment of inertia relative to its mass. A sphere shell is the highest moment of inertia that can be achieved with a uniform spherical configuration. The highest density distributed uniform system, on the other hand, is looked at slightly differently. A cloud is essentially a low-density object, and low-density objects (objects that display less gravitational effects at the densities of our scale than larger-scale objects like planets, etc), of course, can be virtually any shape. These systems are at liberty to take on configurations of extreme relative moments of inertia at our scale. A tree has a pretty big moment of inertia, but that same tree at a mass of 1e27kg would be a sphere with a density gradient representing the constituents of the tree. A few orders of magnitude more massive, at the tree’s Schwarzschild radius, the tree is a uniform sphere. So the highest inertia objects are large regions of space that amount to a low-density core and a high-density shell. A big celestial gas cloud with stars around the edge.

A big void of near-scale matter (a “vacuum” of space as we know it) bounded by high-density objects. (Footnote, under scale relativity, an object can be defined as any static region of space or as any static collection of matter.) A nebula comes to mind as an example of a massive object that appears to consist often of a core of “transparent” sub-atomic matter (matter too small to absorb visible light energy), with lobed outer portions or appendages that are translucent and cloudy, eventually turning to “solid-looking” portions at the outer edges. Nebulae concentrate stars in the outer, cloudy regions. As such, A nebula is likely a relatively high moment of inertia celestial object from our perspective, and thus is a good candidate for observing extreme inertial red-shift. A galactic cluster is as well. These systems very likely represent the kinds of distributions responsible for observed LRDs (little red dots).

Rules of thumb regarding inertial system transformation

Given a static mass, the highest moment of inertia is the highest radius = lowest density

Given a static radius, the highest moment of inertia is infinite mass = infinite density

Given a density, the highest moment of inertia is achieved by configuring that mass density away from the axis. Therefore, a bubble represents the highest moment of inertia for objects relative to their mass.

Premise of this Study

While GTD describes a portion of the redshift observed between systems, the complete picture can only be understood through TDS, which states that space-time has a more comprehensive mathematical relationship with the relative moment of inertia (or relative density) of any given system. TDS intrinsically incorporates the time dilation attributed to GTD under general relativity as a component of the overarching “inertial time dilation." This phenomenon is well explored by our team, but not yet recognized by mainstream physics, and the inclusion of this principle into the standard model will greatly simplify its interpretation, strengthen its accuracy, and extend its coverage far beyond our current understanding.

The observed redshift is known to be in excess of that which should be observed from GTD; after all, that is the reason for the universal expansion hypothesis, and therefore, there is an inconsistency between expectation and observation. Under the Standard Model, this has been explained through the Big Bang Theory, which characterizes the universe as expanding outward from a spatial origin. Many new inconsistencies arise from this hypothesis of explanation, and among them, there have been significant observed localized variations in redshift measurements that are inconsistent with the predictions associated with universal expansion. Further, the James Webb telescope has, to date, struggled to detect the predicted primordial universe as hypothesized.

Hypothesis

These deviations in observed and expected redshift are very likely to be the effects of TDS, or “inertial time dilation”. If so, the standard model would be simplified considerably by eliminating the need for additional explanation via galactic redshift. This would also eliminate the mathematical need for constructs such as singularities, nonexistence, the beginning or end of time and space, or any other implications of the Big Bang theory.

Whether or not this proves to be true, we predict no different behavior than what has been observed. If the hypothesis proves true, it would call into question the hypothesis of universal expansion, which would be a departure from established hypotheses of the origins of the universe. But one that would greatly simplify, expand, and explain our understanding of our universe.

Confirming the Hypothesis

The expected observations would be very distinct between universal expansion and gravitational time dilation vs. inertial time dilation. As gravitational time dilation only represents the “high-density” aspect of the broader inertial time dilation, predictions of time dilation under low-density conditions will therefore always be exceeded by observations. Extremely massive but relatively low-density and simultaneously extremely expansive systems, systems with a more extreme surface area to mass ratio, the opposite of a condensed sphere, a highly distributed mass, characterize high-inertia systems.

Validation 1: Redshift

Universal expansion would exhibit a redshift consistent with universal expansion.

TDS would exhibit a different redshift depending on the granularity of the measurement and the total moment of inertia of a system.

A nebula or galactic cluster would exhibit considerably more redshift than expected compared to what we would see around a black hole, neutron star, or massive Star, where gravitational redshift is expected. A planetary system might exhibit more redshift than expected relative to the massive star itself.

A galaxy would be observed to exhibit more redshift when viewed along its axis of rotation, and as little as i^(1/2) if observed perpendicular to its axis of rotation. So galaxies and objects observed along their axis of rotation will exhibit the maximum redshift possible for that system, and therefore would exhibit considerably more redshift than expected from GTD alone.

Systems with a lower moment of inertia relative to our inertial frame of reference will exhibit a blue shift, generally have a smaller spatial scale, and exhibit faster average motion.

Massive galactic clusters and similar structures would exhibit a similar degree of inertia to extremely high-mass objects. Thus, we would be able to attribute all observed redshifts, including Little Red Dots and other anomalous observations, to TDS.

Systems at or below our localized moment of inertia, calculated for the Earth at the average surface radius, will show no shift or blueshift.

Visually, redshift would not look like the map of GTD, which would be focused on individual gravitational points (stars, neutron stars, black holes). But with inertial time dilation, the map would generally reflect the density of what is being observed, with distortion at the extreme relative redshift due to a cubic scaling relationship for density vs. a square relationship for inertia.

Validation 2: The Primordial Universe

Universal expansion is explained by a Big Bang event, which, if observed at locations close to the calculated origin of the Big Bang, would reveal a primordial universe. In this universe, we would see only very young structures forming and expect to observe no older structures anywhere.

TDS predicts no Big Bang, and, rather, says time, space, and precision are unbounded. It predicts that no matter where one observes, or how far back in time one looks, we will see roughly the same structures, distributed across roughly the same age distribution. We would reveal older structures, no matter where we look in the universe. No matter how far back in time we look, we will see roughly the same structures consistent with the scale being observed. We would never observe evidence of the primordial universe absent evidence to the contrary. For example, we might observe a structure that seems primordial, but there will also be what appear to be far older objects present.

Verification 3: Observed Singularities

Universal expansion implies the existence of singularities. The Big Bang and the center of a black hole are considered singularities under the Standard Model. We have not observed singularities in practice, and to the contrary, we have observed where we expect singularities, and no evidence of singularity has been identified.

TDS predicts the non-existence of tangible singularities, infinities, or pre- or post-existence. Properties would extend in extent and precision without tangible limits, and we would continue to expand the extent and precision of observation.

Verification 4: The Cosmic Microwave Background

The interpretation of the CMB is intimately tied to the interpretation of universal expansion. The standard model says that the CMB indicates that the universe has a limited scale range.\

TDS predicts the observed CMB as an artifact of energy detected from particles just beyond the limits of practical observability (i.e., smaller than subatomic and larger than the observable universe). The CMB would be observed to be identical at every scale of section at every point in space. TDS does not rely upon its conclusion as a premise, and because it predicts both observed redshift and observed background energy, neither established interpretation is positioned to serve as a logical counterpoint.

Verification 5 - Lensing

The current interpretation of the Standard \Model predicts lensing around or between significant gravitational sources, but does not predict similar behavior in relation to high inertial systems such as galactic clusters or galaxies.

TDS predicts lensing between or around large, high-inertia systems. Lensing would be observed to be more distributed than predicted with GTD alone, and this difference is inertial lensing. It's identical to gravitational lensing because it's the same time dilation in a different context. It would be observed as significant lensing between large distributed systems, not unlike that observed in exclusively high-gravity settings.

Identifying the Moment of Inertia of a Massive Distributed Celestial System

To calculate the moment of inertia, one must first consider the angle of observation or influence. If one or both of the systems being compared (observed and observer, or system a and system b) are not spherical, inertia is dimensional, so observations of moment of inertia must take into account the moment of inertia about the axis of observation. As long as you are observing dimensionally, you can abstract a high moment of inertia by using average radius and total mass (or average density and total volume - conceptually not unlike the concept behind the Schwartzchild radius).

To identify the moment of inertia of a high mass distributed system, whether the system is defined as a large cloud peppered with extremely dense regions of stars or a pair of high-density objects, one must determine the average radius and mass about the axis of observation. For example, if the system were defined as two neutron stars separated by a few light years, the system would have a very high moment of inertia. Massive objects that are not sufficiently dense to form spherical structures, odd-shaped clouds, or filaments will have extremely high moments of inertia when observed perpendicular to their largest dimension.

From another of our articles [degerla 2025], the relative moment of inertia of a system about the axis of observation can be abstracted as a spherical system by selecting spherical systems or by selecting spherical regions within a system over which the average moment of inertia can be calculated.

Table 1: Comparison of Three Example Systems with Properties Calculated via Python Script

Property	Earth	Galactic Cluster	Neutron Star
Radius	6.37814 × 10⁶ m	7.100000 × 10²² m	1.980000 × 10³ m
Mass	5.97217 × 10²⁴ kg	3.980000 × 10⁴⁵ kg	1.000000 × 10³⁰ kg
Volume	1.08685 × 10²¹ m³	1.49921 × 10⁶⁹ m³	3.25150 × 10¹⁰ m³
Density	5.49493 × 10³ kg/m³	2.65472 × 10⁻²⁴ kg/m³	3.07550 × 10¹⁹ kg/m³
Moment of Inertia	9.71806 × 10³⁷ kg·m²	8.02527 × 10⁹⁰ kg·m²	1.56816 × 10³⁶ kg·m²
Schwarzschild Radius `r_s`	8.87006 × 10⁻³ m	5.91122 × 10¹⁸ m	1.48523 × 10³ m
GTD per GR formula	9.9999999930465 × 10⁻¹	9.9995837079848 × 10⁻¹	4.9988227873603 × 10⁻¹
Surface Gravity	9.79829 × 10⁰ m/s²	5.26953 × 10⁻¹¹ m/s²	1.70245 × 10¹³ m/s²
Escape Velocity	1.11799 × 10⁴ m/s	2.73546 × 10⁶ m/s	2.59648 × 10⁸ m/s

Note that for the systems above, the moments of inertia for the neutron star and Earth are comparable, while surface gravity on the neutron star is eleven orders of magnitude greater. For moments of inertia, despite its relatively low density, due to the distribution of the mass, the system is 50+ orders of magnitude greater in moments of inertia than the others. Only at r_s does gravity dominate time dilation; even at 4/3 r_s, gravitational time dilation has diminished to 0.5 (half-time). This means that the majority of the universe is not under significant gravitational time dilation, but IS under significant relative time dilation, depending on the inertial scale of the observer and the observed.

Implications

If TDS, STE, and the principles described herein are validated through peer review and further observational and experimental testing, it would garner revision of the standard model. The formal inclusion of inertial time dilation into the standard model and general relativity would greatly expand our tangible understanding of how space, time, and matter interrelate. It would also afford physics a long-needed opportunity to grow beyond the limitations and mysteries imposed by our current models, free of the significant obstacles of presumption and potential for systemic misinterpretation of observed data.

That said, validating TDS would not, to my knowledge, change even a single measurement or observation ever recorded. It only affects the interpretation of such observations.

References

[goldstein1980] Goldstein, H. (1980). Classical Mechanics.

[schwarzschild1916radius] Schwarzschild, K. (1916). On the Gravitational Field of a Mass Point.

[crc2019] CRC Handbook of Chemistry and Physics (2019).

[einstein1915] Einstein, A. (1915). The Field Equations of Gravitation.

[degerlia2025] DeGerlia, T. D. (2025). Introducing Inertial Density and Characterizing the Schwarzschild Condition as a Constant Threshold of Mass over Radius (m/r)
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Preprint

Inertial Time Dilation Predicts Observed Cosmic Redshift Patterns without the Need for Cosmic Expansion