On the Distinction Between Gravitational Force and Potential: A Model of Evolving Temporal Metrology as an Alternative to Dark Energy

J. Rogers, SE Ohio  

Abstract

The ΛCDM model, while highly successful, requires the postulation of Dark Energy to account for the observed apparent acceleration of cosmic expansion. This paper proposes that the effect arises instead from the evolution of our temporal metric, a direct consequence of General Relativity. We argue for a critical distinction between the vector force field of gravity, which cancels under cosmological symmetries (Birkhoff’s Theorem), and the scalar gravitational potential, which is strictly accumulative. We posit that the integrated potential from all mass-energy in the Hubble volume establishes a scalar baseline for the g_00 component of the local metric tensor. As the universe expands, this background potential diminishes, causing a secular acceleration in the proper time of comoving observers. This “metrological acceleration” provides a physical mechanism that reinterprets the Type Ia supernova data without requiring a new energy component, suggesting that Dark Energy may be a consequence of assuming a static temporal metric in a dynamic universe.

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1. Introduction

The discovery of cosmic acceleration from Type Ia supernovae (Riess et al., 1998; Perlmutter et al., 1999) profoundly challenged cosmology. The prevailing resolution is to introduce a cosmological constant Λ or an exotic component called Dark Energy. Yet these explanations add unseen entities rather than revisiting assumptions in our measurement framework.

We explore an alternative: that the apparent acceleration is not a new force of nature but a consequence of evolving temporal metrology. Specifically, we argue that the cumulative gravitational potential of the universe sets the baseline for the rate of time itself, and as this potential evolves with cosmic expansion, so too does our measurement of temporal intervals.

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2. Distinction Between Vector Forces and Scalar Potentials

General Relativity links gravity to spacetime curvature, producing two main observable effects:

1. Force-like behavior: test masses accelerate along geodesics, governed by the gradient of the potential (a vector field).

2. Metric modification: the potential itself alters spacetime intervals, most directly through g_00, which governs clock rates (a scalar effect).

Birkhoff’s Theorem is often invoked cosmologically to argue that the surrounding mass distribution cancels out, leaving no net gravitational effect. This interpretation is correct for the vector force field but not for the scalar potential.

An observer inside a spherical shell of mass feels no net acceleration—vector forces cancel—but still resides in a constant, non-zero potential well. Their clock ticks differently relative to infinity. Conflating the cancellation of vectors with the irrelevance of scalars is a category error.

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3. Empirical Anchor: GPS as Proof of Scalar Accumulation

This principle is not abstract—it underpins engineering practice. The Global Positioning System requires corrections for both Earth’s and the Sun’s gravitational potentials.

• Earth’s field: a satellite clock runs faster than one on the ground, as expected.

• Sun’s field: both clocks share the Sun’s potential baseline, which cancels in differential corrections but exists physically as a non-zero offset.

That global baseline is real, measurable, and strictly accumulative. GPS functions only because engineers account for this fact daily. Thus, scalar potentials demonstrably sum even when vector forces cancel.

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4. A Toy Model for Cosmic Potential

To extend this principle cosmologically, we estimate the cumulative potential from the mass-energy within the Hubble volume:

$$Φ_U(t) \approx - \int_0^{R_H(t)} \frac{G (4\pi r^2 \rho(t))}{r}\,dr = -2\pi G \rho(t) R_H(t)^2 ,$$

where $R_H(t) = c/H(t)$.

Substituting $\rho(t) = \rho_0 a(t)^{-3}$, we find that as the universe expands ($a(t)$ grows), the density drops and the effective potential weakens.

In the weak-field limit, the metric component is

$$g_{00}(t) \approx -(1 + 2(Φ_{\text{local}} + Φ_U(t))/c^2).$$

As $|Φ_U|$ decreases with expansion, $g_{00}$ asymptotically approaches −1, meaning the local rate of proper time $d\tau$ accelerates relative to any fixed coordinate time $dt$. In other words: cosmic time is speeding up.

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5. Reinterpreting FLRW and Supernova Observations

The FLRW metric assumes a universal cosmic time $t$ for comoving observers. Our model preserves the metric but reinterprets its temporal coordinate: atomic clocks measure proper time $d\tau$, not coordinate time $dt$. The relationship between them evolves with the cosmic potential $Φ_U(t)$.

Supernovae at redshift $z$ exploded in epochs when clocks ticked slower. Comparing their durations and photon arrival times against our modern, faster clocks makes them appear farther away, yielding an illusion of accelerating expansion. What ΛCDM attributes to Dark Energy is here explained as a metrological effect of evolving $g_{00}$.

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6. Broader Observational Consequences

This reinterpretation has testable implications beyond supernovae:

• Baryon Acoustic Oscillations (BAO): The standard ruler length may yield distance-redshift relations subtly different from ΛCDM.

• Quasar Variability: Time-dilation signals in quasars should encode both cosmological redshift and evolving clock rates, possibly clarifying current anomalies.

• CMB/ISW Effect: Since the integrated Sachs–Wolfe effect depends on evolving potentials, detailed comparison may reveal discrepancies with ΛCDM predictions.

These provide falsifiable avenues for distinguishing models.

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7. Conclusion

Dark Energy remains an unobserved hypothesis introduced to preserve ΛCDM against supernova evidence. We propose instead that the data reflects a deeper principle: time itself evolves because the cosmic scalar potential evolves.

By correctly distinguishing between cancellable vector forces and accumulative scalar potentials, and by grounding the principle in empirical systems such as GPS, we identify a mechanism—metrological acceleration—that naturally accounts for observed cosmic acceleration.

The result is not an expansion driven by mysterious energy, but a universe whose ticking clocks are gradually accelerating as the cosmic potential well shallows. This reinterpretation replaces exotic new physics with a reexamination of how we measure time itself.