From GPS to Mach's Principle: On the Unity of Motion, Time, and Inertia

J. Rogers, SE Ohio

Abstract: The conventional separation of Special (SR) and General Relativity (GR) is a cornerstone of modern physics, yet it is challenged by a simple re-analysis of the GPS time dilation problem. By acknowledging that an orbiting object's velocity is constrained by the gravitational potential, the seemingly separate SR and GR effects are shown to be two facets of a single, unified potential. This paper argues that this is not a special case, but the universal condition, as truly "free motion" does not exist in a universe governed by gravity. All motion is, in fact, a state of nested orbits. This leads to the conclusion that our local rate of time is a dynamic property set by the sum of all gravitational potentials in the cosmos. This, in turn, provides a direct, physical mechanism for Mach's Principle, where inertia emerges as a consequence of this cosmically-defined local time. We conclude that a "force" is the experience of being displaced from our natural geodesic, or "straight line" motion, within this universal web of nested orbits.

---

1. The Foundational Insight: The False Separation in the GPS Problem

The canonical example of relativistic effects in action is the time correction for the Global Positioning System. The standard model treats this as a linear sum of two distinct phenomena: a gravitational time dilation (a GR effect) and a velocity time dilation (an SR effect). While numerically correct, this approach contains a profound conceptual flaw: it treats the satellite's velocity v as an independent variable.

However, the velocity required for a stable orbit is entirely determined by the gravitational potential at that orbit (v² = GM/r). As demonstrated in "The Unified Relativistic Potential" (Rogers, 2025), substituting this constraint into the standard equations causes the two "separate" effects to collapse into a single, unified expression. The total time dilation is shown to be a singular phenomenon, governed entirely by the mass of the central body and the radii of the observer and the satellite.

This is not a mere mathematical convenience. It is a statement that for any object whose motion is governed by gravity, the kinematic effects we attribute to Special Relativity are not independent of, but are entirely subsumed by, the potential effects of General Relativity.

2. The Universalization of the Principle: All Motion is Nested Orbits

The next logical step is to question whether the GPS satellite is a special case or the universal rule. The foundational concept of an "inertial frame"—an object in "free motion" far from any gravitational influence—is the domain of Special Relativity. However, in a universe where gravity has infinite range, such an inertial frame is a mathematical fiction that does not physically exist. Every particle, at every moment, is under the gravitational influence of every other particle in the universe.

Therefore, there is no "free motion." There is only motion constrained by a hierarchy of gravitational potentials. All motion is orbital. An object on Earth is in orbit around the Sun; the Sun is in orbit around the galactic center; our galaxy is in a gravitational dance with the Local Group, and so on.

The concept of a "straight line" is simply an orbit with a radius of curvature so vast that it is locally imperceptible. This forces the conclusion that the unified potential seen in the GPS problem is not a special case. It is the only case. The separation between SR and GR is an artifact of applying an idealized, non-existent state of motion (the inertial frame) to a real, interconnected universe.

3. The Cosmic Clock: Local Time as a Function of the Universal Potential

If all motion is a state of nested orbits, then our local experience of reality must be a function of this total orbital state. The most fundamental aspect of our local experience is the rate at which time passes. Extending the logic from the unified potential, our local rate of time is not a universal constant, but is determined by the sum of all gravitational potentials throughout the cosmos.

The total potential at any point in spacetime can be expressed as Φ_total = Σ GM/r, where the sum is over all other masses in the universe. The rate of a local clock is therefore a dynamic function of this cosmic potential. As the r to every other galaxy, star, and planet is constantly changing, our local "tick" of the clock is subtly fluctuating, set at every moment by our relationship to the entire cosmic web. Time is not absolute; it is a relational property that emerges from the global structure of the universe.

4. A Physical Mechanism for Mach's Principle

This view provides the long-sought physical mechanism for Mach's Principle, which posits that inertia is not an intrinsic property of a body but is determined by the total mass distribution of the universe. The missing link has always been the how.

The mechanism is time itself.

1. Inertia is Defined by Time: Force and acceleration are defined with respect to time (a = dv/dt, F = m * dv/dt). An object's inertia (m) is its resistance to a change in velocity over a given time interval (dt).

2. Time is Set by the Cosmos: As established above, the local time interval dt is determined by the total cosmic potential Σ GM/r.

3. Therefore, Inertia is Set by the Cosmos: If the dt in the very definition of inertia is being set by the sum of all mass in the universe, then inertia itself must be a direct consequence of the sum of all mass in the universe.

In a universe devoid of other matter, Σ GM/r would be zero, the rate of time would be radically different, and inertia would be nonexistent. In our universe, the vast cosmic potential sets our local time to the rate we experience, which in turn gives inertia the value we measure.

5. Redefining Force: The Experience of Displacement

This unified framework provides a final, elegant clarification on the nature of force. If our "natural" state of motion is to follow a geodesic—a straight line—through the 4D spacetime defined by the total cosmic web of nested orbits, then what is a force?

A force is the experience of being pushed off our natural, cosmically-defined worldline.

Inertia is the resistance to this displacement. We do not "feel" our immense orbital velocity around the Sun because we are following our natural path. We do feel the force of a car seat pushing us forward because it is actively displacing us from the geodesic we would otherwise be following. Force is the measure of our interaction with an agent that seeks to alter our state of motion within the universal gravitational potential.

6. Conclusion

A simple algebraic substitution in the GPS time dilation problem, when its logical consequences are followed without regard for historical divisions, leads to a profound re-imagining of physics. It reveals that the separation of SR and GR is an illusion, that all motion is nested orbit, that our local experience of time is a function of the entire cosmos, and that inertia is an emergent property of that relationship. This provides a clear, physical mechanism for Mach's Principle. No new physics was required to reach these conclusions; we only needed to dissolve the artificial boundaries we had placed on our own successful theories and see the unified, geometric reality they have described all along.

Appendix A- A testable prediction is that inertia may be subtly changing as our experience of time accelerates gradually. 

The idea that inertia might be subtly influenced or scaled by gravitational potential is gaining theoretical consideration. Several models and hypotheses suggest that inertia could be linked to the scalar gravitational potential contributed by all masses in the universe, which implies that local inertial properties could vary slightly depending on the surrounding cosmic mass distribution.

Key points:

• Inertia as a gravitational effect: Some theories propose that inertia arises from the gravitational interaction with the entire universe's mass, in line with Mach's principle, meaning that inertia could be influenced by the total gravitational potential, not just local matter.

• Modified inertia models: Ideas such as Sciama's early work and later modern modifications suggest that inertia is an inductive effect, built up through the gravitational potential contributed by distant matter.

• Potential subtle variations: If inertia depends on the aggregate gravitational potential (scalar and additive), then as cosmic conditions evolve—such as in an expanding universe—the effective inertial mass could experience minute shifts.

Summary:

While mainstream physics generally treats inertia as a fundamental property, there is ongoing research and speculative theoretical models examining whether it might be subtly modifiable by the universe's gravitational potential field. This remains an active area of foundational physics with many open questions and potential experimental implications.

References:
1. J. Rogers, The Unified Relativistic Potential: A Re-examination of the GPS Time Dilation Problem

2. J. Rogers, All Motion as Nested Orbits: A Hierarchical Model of Time Fields