Deriving the Dark Matter Fraction from First Principles: An Inquiry into Relativistic Energy Misinterpretation

J. Rogers, SE Ohio
(For discussion)

Abstract:
The standard cosmological model requires a vast halo of non-baryonic dark matter to account for an enormous energy discrepancy in rotating galaxies. This paper posits that this energy discrepancy is not the result of a missing substance, but of a fundamental misinterpretation of kinetic energy in a relativistic framework. We demonstrate that the "extra" energy attributed to dark matter is an illusionary artifact, calculable as the difference between the observed kinetic energy (KE_obs) and the actual kinetic energy (KE_actual) of a star. This difference is shown to be a direct function of the star's real kinetic energy, amplified by a factor related to the gravitational time dilation at its location. This framework provides a clear, analytical path to deriving the dark matter fraction (Ω_c) from first principles, without invoking any new physics or exotic particles.

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1. The Energy Discrepancy Problem

Modern cosmology is built upon a significant energy imbalance. When observing a star in the outer regions of a spiral galaxy, its observed velocity (v_obs) is far greater than the velocity predicted by the gravitational influence of the visible, baryonic matter (v_actual). This leads to a direct and massive discrepancy in kinetic energy:

• The Observed Kinetic Energy: KE_obs = (1/2) * m * v_obs²

• The Actual Kinetic Energy (from visible matter): KE_actual = (1/2) * m * v_actual²

To balance the energy books, the standard model is forced to invent a new source of mass-energy: the dark matter halo. The energy of this halo is precisely defined as the difference between the observed and actual kinetic energies:

KE_dark_matter = KE_obs - KE_actual

This approach treats KE_dark_matter as a real, physical substance. This paper will demonstrate that this term is not a new substance, but a calculable artifact of a relativistic measurement error.

2. The Relativistic Framework: A Review

Our framework is built on a single, unified principle demonstrated by the successful operation of the GPS network: the rate of a clock is determined by the total gravitational potential at its location, a potential that is a linear sum of all m/r contributions in the system.

For a star in a galaxy, this leads to two conclusions:

1. Its true, local velocity (v_actual) is determined by the gravitational force from the enclosed mass M_enclosed.

2. Our observation of its velocity (v_obs) is scaled by the time dilation factor (γ) at its location, which is determined by the potential of the total mass of the galaxy, M_total.

v_obs = v_actual * γ
where γ = (1 + |Φ_total| / c^2)

3. The Energy of the Illusion

We can now use this relativistic relationship to solve for the "dark matter energy." We start with its definition and substitute our expression for v_obs:

KE_dark_matter = (1/2) * m * (v_actual * γ)² - (1/2) * m * v_actual²

By factoring out the actual kinetic energy term, KE_actual, we arrive at the central result:

KE_dark_matter = KE_actual * (γ² - 1)

This is a profound statement. It reveals that the energy we attribute to dark matter is not a new source of energy. It is the star's actual, baryonic kinetic energy, amplified by a specific relativistic factor. The "dark matter effect" is a function of the real matter's motion and the geometry of spacetime it inhabits.

The term (γ² - 1) represents the energy of the illusion. It is the potential energy stored in the local "time field," which our remote measurement incorrectly perceives as additional kinetic energy.

4. A Direct Path to Deriving the Dark Matter Fraction (Ω_c)

This framework provides a clear, falsifiable method to derive the cosmological "magic number" Ω_c (the dark matter density fraction, measured to be ~0.27) from first principles.

We define Ω_c as the ratio of the total illusory energy to the total energy of the system. For a given galaxy:

1. Calculate the Total Actual Energy ( This is the sum of the rest mass energy of all baryonic matter (M_baryon * c²) plus the sum of the actual kinetic energy of all its stars (Σ KE_actual).

2. Calculate the Total Illusory Energy ( This is the sum of the "dark matter energy" for every star in the galaxy: Σ KE_dark_matter = Σ [KE_actual * (γ² - 1)].

3. The Dark Matter Fraction is the Ratio:

   Ω_c = E_illusion / (E_actual + E_illusion)

Hypothesis: We postulate that this calculated fraction, when averaged over a representative sample of spiral galaxies, will converge to the observed cosmological value of approximately 0.27.

5. Conclusion

The energy discrepancy in galactic rotation is real, but its source has been misidentified. It is not an unknown substance, but a known, though unappreciated, relativistic effect. The energy we call "dark matter" is a calculable illusion, born from the misinterpretation of a star's real kinetic energy as seen through the distorting lens of gravitational time dilation.

This model provides a direct path to proving this case. By calculating the ratio of the "energy of the illusion" to the total energy of a galaxy, we can derive the dark matter fraction from known physics. The need for a universe filled with invisible, exotic particles vanishes, replaced by a more complete understanding of relativity.