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Flat Galactic Rotation Curves Interpreted as Evidence for the Mach's Principle

This paper proposes that flat galactic rotation curves and the Baryonic Tully-Fisher Relation are manifestations of Mach's Principle, where the MOND acceleration scale arises from cosmological boundaries, a hypothesis supported by high-redshift observations that favor a baryon-only, $\Lambda$ -dominated universe over standard $\Lambda$ CDM or MOND models.

Original authors: Andres Escala

Published 2026-08-12

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CC BY 4.0

Original authors: Andres Escala

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a tiny ant walking on a giant, spinning merry-go-round. If you walk near the center, you feel a strong pull toward the middle, and you have to hold on tight. But if you walk all the way to the very edge, something strange happens: the pull doesn't get weaker like you'd expect. Instead, it stays just as strong, as if an invisible hand is still holding you. In our real universe, stars in the outer edges of galaxies act exactly like that ant. They spin so fast that, according to the laws of gravity we learned in school, they should fly off into space. Yet, they stay put. For decades, scientists have tried to explain this "missing" grip. The most popular idea is that there is a huge, invisible cloud of "dark matter" holding the stars in, like a ghostly net. Another idea suggests that the rules of gravity themselves change when things move very slowly. But there is a third, older, and more philosophical idea called "Mach's Principle." It suggests that your motion isn't just about you; it's about how you relate to everything else in the universe. This paper asks a big question: What if the reason stars don't fly off isn't because of invisible ghosts or new rules, but because the stars can "feel" the edge of the entire universe?

The author, Andr´es Escala, takes a fresh look at the mystery of these fast-spinning stars. He proposes that the strange behavior isn't caused by a new, mysterious force or a hidden particle, but by a connection between a galaxy and the rest of the cosmos. He suggests that the "acceleration scale" (a specific speed limit where things start acting weird) isn't a random number made up by scientists. Instead, it's a direct message from the size and mass of the entire observable universe. Think of it like this: if you are in a small room, you only feel the walls if you bump into them. But if you are in a giant, empty hall, you might start to feel the walls even when you are far away, just because the hall is so huge. Escala argues that when a star moves slowly enough, it finally "feels" the edge of the universe, and that feeling changes how it moves.

The paper rewrites a famous rule about galaxies, called the Baryonic Tully-Fisher Relation, into a "cosmic" version. Instead of just looking at the stars and gas in a galaxy, the new formula connects the galaxy's speed directly to the total mass of the universe. The author shows that if you assume the universe is mostly empty space filled with a "dark energy" that pushes things apart (a flat, baryon-only universe), the math works out perfectly to explain why stars spin the way they do. When the author checks this idea against real data from very distant galaxies (looking back in time to when the universe was younger, around redshifts of 2 and 5), the results are surprising. The data seems to favor this "Machian" view over the standard idea that the universe is full of dark matter. In fact, the observations suggest that the universe might be made of almost entirely normal stuff (like stars and gas) and dark energy, with almost no dark matter at all.

However, the paper is careful not to claim this is a solved mystery. The author suggests that flat rotation curves are "compelling evidence" for this idea, but it remains a hypothesis that needs more testing. The paper argues against the standard "Lambda-Cold Dark Matter" model, which relies on invisible particles we haven't found yet, and also against the idea that the strange acceleration scale is just a new, fixed constant of nature. Instead, it suggests that the universe itself is the key. If this is true, it would mean that the laws of physics aren't just about what's happening right here and now, but are deeply connected to the shape and size of the entire cosmos. It transforms the problem of "missing mass" into a story about how the universe's boundaries shape the motion of everything inside it. While the standard model of cosmology is still the favorite of many scientists, this paper offers a playful, bold, and mathematically consistent alternative that makes the universe feel like one giant, interconnected machine.

Technical Summary: Flat Galactic Rotation Curves Interpreted as Evidence for Mach's Principle

Problem Statement
The paper addresses the long-standing discrepancy between observed galactic rotation curves and Newtonian dynamics, traditionally explained by the Lambda-Cold Dark Matter ( $\Lambda$ CDM) paradigm or Modified Newtonian Dynamics (MOND). While $\Lambda$ CDM relies on the existence of non-baryonic dark matter, direct laboratory detection of Weakly Interacting Massive Particles (WIMPs) has remained elusive. Conversely, while MOND successfully predicts galactic dynamics via an empirical acceleration scale $a_0 \approx 1.2 \times 10^{-10} \, \text{m/s}^2$ , it treats $a_0$ as a new fundamental constant and struggles at cluster scales. The author argues that the scientific community has largely ignored the potential link between the MOND acceleration scale and cosmological parameters, viewing it instead as a coincidence. The core problem is whether the flat rotation curves of disk galaxies can be interpreted not as evidence for missing mass or a new fundamental constant, but as a manifestation of Mach's Principle, where local inertia is determined by the global distribution of matter in the Universe.

Methodology
The paper employs a theoretical and comparative approach, utilizing dimensional analysis and cosmological modeling to reinterpret existing empirical relations:

Dimensional Analysis: The author analyzes the dimensions of the MOND acceleration scale $a_0$ alongside the speed of light $c$ , Newton's constant $G$ , and the mass ( $M_u$ ) and radius ( $R_u$ ) of the observable Universe. This is used to test whether $a_0$ functions as a fundamental constant replacing $G$ or as a derived cosmological scale.
Rewriting the Baryonic Tully-Fisher Relation (BTFR): The standard MOND deep-limit relation ( $V^4 = G M_{bar} a_0$ ) is algebraically transformed by substituting the cosmological relation $G a_0 = c^4 / M_u$ . This yields a cosmologically formatted BTFR: $M_{bar}/M_u = (V/c)^4$ .
Cosmological Evolution Modeling: The paper derives a redshift-dependent acceleration scale $a_0(z)$ based on the Friedmann equation for a flat universe, incorporating density parameters for matter ( $\Omega_m$ ) and dark energy ( $\Omega_\Lambda$ ). This predicts that the normalization of the BTFR should evolve with redshift as the total mass of the observable Universe changes.
Observational Comparison: The theoretical predictions for the evolution of the BTFR zero-point are tested against high-redshift observational data ( $z \sim 2$ $z \sim 2$ and $z \sim 5$ $z \sim 5$ ) from recent studies (e.g., Genzel et al., Nestor-Shachar et al., Gómez-Espinoza et al.). These data points are compared against three cosmological models:
- Standard MOND (implying $a_0 \propto (1+z)^{3/2}$ in a matter-only universe).
- Standard $\Lambda$ CDM ( $\Omega_m \approx 0.315, \Omega_\Lambda \approx 0.685$ ).
- The proposed Machian scenario ( $\Omega_m \approx 0.05$ [baryons only], $\Omega_\Lambda \approx 0.95$ ).

Key Contributions

Reinterpretation of $a_0$ : The paper argues that $a_0$ is likely not a new fundamental constant but a scaling factor derived from cosmological boundaries, satisfying the relation $a_0 \sim c^2/R_u \sim c^4/GM_u \sim c/t_u$ . This suggests the transition to non-Newtonian dynamics occurs when local accelerations become small enough to "feel" the global cosmic boundary.
Cosmological Form of the BTFR: By rewriting the BTFR as $M_{bar}/M_u = (V/c)^4$ , the paper establishes a direct link between local galactic dynamics and the total mass of the observable Universe, a relation that holds regardless of the specific Machian derivation mechanism (e.g., Sciama's inertial induction or Unruh/Gibbons-Hawking radiation thresholds).
Prediction of BTFR Evolution: The framework predicts that the normalization of the BTFR must evolve cosmologically as $M_u(z)$ changes. This provides a testable constraint that distinguishes between standard MOND, $\Lambda$ CDM, and Machian interpretations.

Results

Dimensional Consistency: The analysis confirms that treating $a_0$ as a fundamental constant replacing $G$ is difficult due to the lack of a physical role for $G$ in this context. However, the relations $a_0 \approx c^2/R_u$ and $a_0 \approx c^4/GM_u$ align with the observed scales of the Universe.
Observational Fit: When comparing the predicted evolution of the BTFR zero-point against high-redshift data ( $z \sim 2$ $z \sim 2$ and $z \sim 5$ $z \sim 5$ ), the data significantly favor a flat, baryon-only, $\Lambda$ $Λ$ -dominated universe ( $\Omega_m \approx 0.05, \Omega_\Lambda \approx 0.95$ $Ω_{m} \approx 0.05, Ω_{Λ} \approx 0.95$ ).
- The standard $\Lambda$ CDM model ( $\Omega_m \approx 0.315$ ) fails to match the observed offsets.
- The standard MOND prediction (assuming a matter-only universe where $a_0$ scales as $(1+z)^{3/2}$ ) is inconsistent with the data, which show "falling" rotation curves at $z \sim 2$ that imply lower accelerations than the standard matter-only model predicts.
- The Machian model, which predicts a slower evolution of $a_0(z)$ due to the dominance of the cosmological constant, aligns with the observed data points.
High-Redshift Disk Galaxies: The results are consistent with observations of strongly baryon-dominated disk galaxies at high redshifts, which challenge the $\Lambda$ CDM requirement for significant dark matter fractions but fit the Machian interpretation where local dynamics are governed by the global boundary.

Significance and Claims
The paper claims that flat galactic rotation curves may be interpreted as compelling evidence for Mach's Principle. The primary significance lies in shifting the explanation for galactic dynamics from the existence of invisible dark matter halos to a modification of how local inertia relates to the global Universe.

Paradigm Shift: The results suggest that the "missing mass" problem may be a misinterpretation of a global boundary condition. The characteristic acceleration $a_0$ represents the threshold where local Newtonian dynamics interface with the causal boundary of the cosmic horizon.
Resolution of Tensions: This framework offers a parameter-free explanation for the coupling between baryonic matter and cosmic scales, potentially resolving the tension between galactic rotation curves and high-redshift observations without invoking dark matter.
Limitations and Open Questions: The author acknowledges that while the Machian interpretation fits the galactic and high-redshift data well, challenges remain at the scale of galaxy clusters and in reproducing the full spectrum of the Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillations (BAO). The paper posits that the thermodynamic complexity of clusters might mask the local-to-global acceleration transitions, and that the specific Machian process operating in the early Universe remains to be fully defined. The work concludes that while $\Lambda$ CDM relies on dark matter as a free parameter to fit data, the Machian view offers a predictive, structural explanation, though further high-redshift observations are required to fully constrain the evolution of the BTFR zero-point.

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