Resolving Galaxy Rotation Curves via a Relativistic Separation of Gravitational Force and Potential

J. Rogers, SE Ohio

Abstract:
The observed flat rotation curves of spiral galaxies are a cornerstone piece of evidence for the existence of dark matter. The standard cosmological model (ΛCDM) posits that large, non-baryonic halos are required to explain the discrepancy between the observed velocities of stars and the gravitational force predicted by their visible mass. This paper presents an alternative model that resolves this discrepancy without invoking dark matter. The model is founded on a re-evaluation of how gravitational effects are calculated within a mass distribution. Standard cosmology, based on an application of Birkhoff's theorem, assumes all gravitational effects from an external mass shell cancel inside it. We propose a model where only the gravitational force cancels, while the gravitational potential, which governs time dilation, remains a linear superposition of all mass in the system. When this physically motivated model is applied to observational data for multiple spiral galaxies (e.g., Andromeda, Milky Way), it yields a mass distribution that reproduces the observed rotation curves with high fidelity (RMSE < 2-7 km/s). Critically, the total mass required by the model is physically plausible and consistent with the total mass attributed to these galaxies in dark matter models. This suggests that the galaxy rotation anomaly is not the result of missing matter, but rather a relativistic consequence of the total gravitational potential, an effect overlooked by the standard application of vacuum-based solutions to the non-vacuum interior of a galaxy.

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1. Introduction

The discrepancy between the observed orbital velocities of stars in spiral galaxies and the velocities predicted by Newtonian dynamics based on their visible matter is one of the most significant problems in modern astrophysics. Observations consistently show that stellar velocities in the outer regions of galaxies remain high or "flat," whereas a Keplerian decline is expected. The prevailing solution to this anomaly is the hypothesis of cold dark matter (CDM), forming the basis of the standard ΛCDM cosmological model. This model proposes that galaxies are embedded in vast, non-luminous halos of a new, weakly interacting substance that provides the additional gravitational force.

While the ΛCDM model has been remarkably successful, the lack of direct detection of a dark matter particle has motivated alternative theories, most notably Modified Newtonian Dynamics (MOND). MOND proposes a fundamental change to the law of gravity or inertia at very low accelerations.

This paper presents a third approach. We do not propose a modification to gravity, but rather a more careful application of the principles of General Relativity. We argue that the standard cosmological assumption—that all gravitational effects from an external mass shell cancel within its interior—is a misapplication of an idealized vacuum solution (Birkhoff's theorem) to the non-vacuum environment of a galaxy. By separating the cancellation of vector forces from the superposition of scalar potentials, we construct a model that fully explains galaxy rotation curves.

2. The Standard Assumption: Cancellation of All Gravitational Effects

The standard model's treatment of gravity within a spherical mass distribution is based on the Shell Theorem and its relativistic counterpart, Birkhoff's theorem. The theorem states that for a spherically symmetric, isolated body in a vacuum, the spacetime curvature inside an empty shell is zero (i.e., flat Minkowski spacetime). This has been broadly interpreted to mean that any mass external to an observer's position has no gravitational effect on them.

This leads to the conclusion that both the gravitational force and the rate of a local clock (time dilation) at a radius r are determined only by the mass enclosed, M_enclosed(r). When this principle is applied to galaxies, the enclosed visible mass is insufficient to explain the observed velocities, thus necessitating the addition of dark matter.

3. Proposed Physical Model: Force Cancels, Potential Sums

We postulate that the application of the vacuum-based Birkhoff's theorem to the interior of a galaxy is incorrect. A galaxy is not an empty vacuum. We instead return to a more fundamental principle of Einstein's weak-field approximation: the linear superposition of gravitational potential.

Our model is founded on the following postulate:
Within a spherical mass distribution, the gravitational 

This principle is demonstrably true in our own solar system. At the Sun-Earth L1 Lagrange point, the gravitational forces from the Sun and Earth cancel, resulting in a region of near-zero net force. However, a clock placed at L1 is deep within the combined potential wells of both bodies and demonstrably ticks slower than a clock in deep space. A region of zero force is not a region of zero time dilation.

Applying this principle to a galaxy, we model the observed velocity of a star as a product of two distinct effects:

1. Local Velocity (v_local): Determined by the net gravitational force, which is a function of only the enclosed mass, M_enclosed(r).
   v_local = sqrt(G * M_enclosed(r) / r)

2. Time Dilation Factor: Determined by the total gravitational potential, Φ_total(r), which is the linear sum of potential contributions from all mass in the galaxy, both internal and external.
   Φ_total(r) = Φ_inner(r) + Φ_outer(r)
   Time Dilation = 1 + |Φ_total(r)| / c^2

The final predicted velocity is then:
v_observed = v_local * Time Dilation

4. Methodology and Results

We constructed a numerical model consisting of N concentric spherical shells (typically N=600). An optimization algorithm was employed to find the mass distribution among these shells that minimizes the Root Mean Square Error (RMSE) between the model's predicted velocity curve and the published observational data for a given galaxy. The model was given no prior assumptions about the shape of the mass distribution or its total value.

The model was tested on multiple spiral galaxies, including Andromeda (M31) and the Milky Way. The results were consistently successful:

• High Fidelity Fit: The model produced velocity curves that closely match the observational data, with a resulting RMSE on the order of 1.5 - 7 km/s, well within observational uncertainties.

• Plausible Total Mass: The total mass required by the model to achieve this fit was consistently found to be physically plausible. For example, the model converged on a total mass of ~7.7 x 10^11 solar masses for Andromeda, a value that aligns with standard estimates for the galaxy's total mass including its dark matter halo.

• Data-Driven Mass Profile: The optimizer independently derived a mass density profile that is highly concentrated at the center and features a long, extended halo, consistent with galactic morphology.

5. Discussion

The success of this model is significant because it is subject to a "Triple Lock" of constraints: it must simultaneously find a single mass distribution that (1) provides the correct enclosed mass to shape the inner rotation curve, (2) provides the correct total potential to lift the outer curve via time dilation, and (3) results in a total mass that is independently verified by other astronomical observations.

The odds of a fundamentally incorrect model satisfying these three independent and demanding constraints for multiple different galaxies are extremely low. The model's success suggests it is not merely a "curve-fitting" exercise but is capturing a real physical phenomenon. The fact that the model's performance improves up to a certain shell resolution (~600 shells) and then worsens ("overfitting") indicates that it is correctly separating the physical signal from observational noise.

6. Conclusion

The galaxy rotation curve anomaly can be fully resolved without invoking hypothetical dark matter particles. The model presented here, based on the principle that gravitational force and potential behave differently inside a mass distribution, successfully reproduces the observed flat rotation curves of spiral galaxies.

We conclude that the standard cosmological model's assumption—that all gravitational effects cancel inside an external shell—is a flawed idealization based on a misapplication of a vacuum solution to a non-vacuum environment. By returning to the more fundamental principle of linear superposition for gravitational potential, as dictated by Einstein's theory, the discrepancy vanishes. The "missing mass" was never missing; its effect was present in the total gravitational potential and its influence on the rate of time, an effect that our model now correctly accounts for.