📌 Update — May 2026
This post was an early exploration of gravitational geometry that pointed in an interesting direction. The intuition behind it has since been substantially rethought and rebuilt from the ground up.
The current framework — Gravitational Flux Area and the Self-Reinforcing Feedback Principle — replaces the bowl geometry with a more rigorous and physically grounded set of principles.
Readers are encouraged to go directly to the updated post: → [here]
For a long time, the standard response to fast motion in galaxies and galaxy clusters has been simple: if gravity looks too strong, then there must be more mass. That is the basic motivation behind dark matter.
I want to explore a different possibility.
My proposal is this: the problem may not primarily be hidden mass. The problem may be how motion is generated, and how gravity is transmitted.
My two basic assumptions are simple.
Gravity does not directly create speed. Gravity changes angle.
The true source of motion is the speed of light.
In this view, the observed speed is a projection of a deeper motion. I write it as:
v = c * sin(θ)
Here, θ is the angle between the underlying time-directed motion and the space we observe. If θ changes, the observed speed changes. Gravity is then understood not as “a force that manufactures speed from nothing,” but as the mechanism that changes θ.
This is the central philosophical difference.
The usual interpretation is:
large speed -> large gravity -> large hidden mass
My interpretation is:
large speed -> examine the projection mechanism and the transmission path of gravity first
1. The parachute picture
The image that helped me most is a parachute.
The canopy corresponds to the spatial surface we observe.
The lines correspond to the true transmission paths of gravity.
The person below the parachute corresponds to the true central source.
The points where the lines meet the canopy correspond to what we identify as mass locations.
In this picture, the visible center on the canopy is not necessarily the true dynamical center. The true source is deeper, below the canopy, and its influence is transmitted through the lines.
This matters because what we observe in ordinary 3D geometry may only be a projection of the true path.
2. Galaxy rotation curves without dark matter
For galaxy rotation curves, the simplest expression that worked best in my tests is:
v(r) = v_inf * sqrt( r / (r + r0) )
Here:
v_inf is the asymptotic or outer flat speed,
r0 is a core scale.
This equation does two important things naturally:
it rises in the inner region,
it flattens in the outer region.
That is exactly the observed shape of many galaxy rotation curves.
In my interpretation, this is not evidence for an invisible halo of matter. It is evidence that the true transmission of gravity is not governed only by the visible 3D radius. It is governed by a deeper projected structure, like the geometry of the parachute and its lines.
So the flat outer rotation speed does not necessarily mean “more unseen mass at large radius.” It may mean that the central tension-like source is being projected in such a way that the effective angular control falls off much more slowly than Newton’s inverse-square intuition would suggest.
3. Time depth
The second key idea is time depth.
Suppose a star is 50,000 light-years from the center of a galaxy. Then the outer region does not respond to the center “now.” It responds to the center with roughly 50,000 years of time depth.
That means distance already contains time.
t <-> r / c
This is extremely important. It means gravitational transmission is not merely a spatial relation. It is a spacetime relation.
At small scales, the difference in spatial distance strongly matters.
At very large scales, however, something changes.
As the scale becomes very large, the transmission structure begins to look less like a sharp cone and more like a long cylinder. In that regime, differences in ordinary spatial radius become less important than they appear in a simple 3D picture.
That gives a possible way to understand why galaxy clusters can maintain large velocity dispersions without immediately forcing us to assume huge amounts of hidden matter.
4. Velocity versus velocity dispersion
For a galaxy, what matters is speed: how fast stars rotate around the center.
For a galaxy cluster, what matters is velocity dispersion: how widely the galaxy velocities are spread.
These are not the same thing.
In my framework:
galaxy speed comes from the value of θ,
cluster velocity dispersion comes from the spread of θ across the cluster.
So for a cluster, I write, schematically:
σ_cl ~ c * Δ(θ)
This means the cluster problem becomes:
why is the angular distribution so broad?
Again, my answer is not “because there must be more mass.”
My answer is “because the transmission structure is different.”
5. Galaxy clusters: time depth plus expansion history
For clusters, my current view is that the dominant factor is:
time depth + expansion history
The idea is this:
As the universe expands, the effective surface becomes flatter.
As it becomes flatter, the transmission becomes more cylindrical and less conical.
As that happens, spatial differences matter less.
A common transmission structure can therefore produce a large cluster-wide velocity dispersion.
A minimal form I used is:
σ_cl = A0 * F_age * E_exp * ln(R_cl / r0)
where:
A0 is a common scale,
F_age describes the maturity or age factor,
E_exp describes the expansion-history factor,
R_cl is the cluster scale,
r0 is a core scale.
In simple tests, the expansion-history factor appeared to contribute more strongly than the crude network factor I tried first. That suggests that the dominant large-scale effect may come from how long the structure has been stretched and flattened by cosmic expansion.
From Geometry to Transmission Dimensionality
Up to this point, the discussion has been based on intuitive geometric pictures: the parachute, the cone, and the cylindrical propagation.
These are not merely visual metaphors. They represent fundamentally different ways in which influence propagates across spacetime.
- In the parachute picture, the force originates from a central point and spreads outward through structured connections.
- In the conical picture, propagation follows a directed path through spacetime.
- In the cylindrical limit, the propagation becomes nearly uniform across large scales.
This suggests that the effective mode of transmission itself changes with scale.
Instead of always spreading in a fully three-dimensional manner, the propagation behaves as if it is constrained to lower-dimensional structures:
- At small scales, transmission behaves like a 3D spread (similar to inverse-square law).
- At galactic scales, it resembles a 2D-like propagation across a surface.
- At cluster scales, it approaches a quasi-1D or uniform transmission.
This change can be interpreted as a transition in effective transmission dimensionality.
Importantly, this is not a statement about the actual number of spatial dimensions, but about how influence propagates within spacetime.
This interpretation provides a bridge between the intuitive geometric models and the quantitative predictions discussed below.
Phenomenological Transition of Transmission Dimensionality
The geometric interpretation presented in this work suggests that the mode of gravitational transmission is not fixed, but evolves with scale. In particular, as the time-depth of propagation increases, spatial distance differences become progressively less significant, leading to an effective reduction in transmission dimensionality.
To capture this transition in a simple and explicit form, we introduce a phenomenological ansatz for the effective transmission dimensionality:
n(r) = 2 / (1 + r / L*)
where L* is a characteristic length scale associated with the onset of dimensional transition.
This expression reproduces the expected limiting behaviors:
- For r ≪ L*, n(r) ≈ 2, corresponding to standard three-dimensional inverse-square behavior.
- For r ~ L*, n(r) ≈ 1, corresponding to effectively two-dimensional propagation, as suggested by galactic rotation behavior.
- For r ≫ L*, n(r) → 0, corresponding to nearly uniform transmission, consistent with cluster-scale dynamics.
Importantly, this expression is not derived from first principles, but is introduced as a minimal phenomenological model that encodes the scale-dependent transition implied by the geometric picture (parachute, conical, and cylindrical propagation).
Despite its simplicity, this form provides a unified description that connects local gravitational behavior, galactic rotation, and cluster-scale dynamics within a single framework.
Further work is required to determine whether such a transition can be derived from a deeper underlying theory.
Testable Predictions of the Projection-Based Gravity Model
A key strength of any physical model is its ability to produce testable predictions. The framework proposed here, based on projection geometry and scale-dependent transmission dimensionality, leads to several clear observational expectations.
1. Transition Radius in Galaxies
The model predicts the existence of a transition radius where the effective transmission dimensionality changes from 3D (Newtonian regime) to 2D (galactic regime).
At small radii:
- Gravity follows the inverse-square law.
At large radii:
- The effective law becomes closer to 1/r behavior.
- This naturally produces flat rotation curves.
2. Universality of Outer Rotation Curves
Because the dynamics are governed by projection geometry rather than detailed mass distribution, outer rotation curves of galaxies should exhibit universal behavior regardless of morphology.
3. Weak Radial Dependence in Galaxy Clusters
At the scale of galaxy clusters, the effective dimensionality approaches a lower limit (n ≈ 0), implying:
- Velocity dispersion should show weak dependence on radius.
- Large-scale systems behave as if embedded in a nearly uniform transmission field.
4. Saturation-Type Expansion Behavior
The model suggests that cosmic expansion may not be purely linear at all scales, but instead follow a saturation-type relation:
v(r) = c sin(r / Rs)
This implies:
- Linear behavior at small distances (Hubble law)
- Gradual suppression at very large distances
5. Redshift as an Accumulated Projection Effect
Rather than being directly interpreted from recession velocity, redshift may be understood as an accumulated projection effect of the angle θ:
z ≈ tan(θ)
This allows:
- Velocity to saturate
- Redshift to continue increasing
Final Remark
This framework does not claim to be complete. Instead, it provides a consistent geometric interpretation that connects local dynamics (galaxies) and large-scale structure (clusters and cosmic expansion) within a unified projection-based picture.
Future work should focus on confronting these predictions with observational data.
What this proposal is, and what it is not
This is not yet a finished theory.
It is not a claim that every detail has been solved.
It is a structured alternative interpretation.
It says that at least two major dark-matter arguments may be approached differently:
galaxy rotation curves,
galaxy-cluster velocity dispersion.
The central message is not “mass never matters.”
The central message is:
gravity may have been interpreted too much as a mass-counting problem, and not enough as a transmission-and-projection problem.
Final thought
If fast motion can be explained by projection, transmission path, time depth, and expansion history, then dark matter may not be the only serious way to read the data.
That does not prove dark matter is wrong.
But it does mean this:
there is room to think again.
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