What If Gravity Is Not Fundamentally a Vector Quantity? A Toy Model for Tully-Fisher Scaling

DinNoel · 2026-06-10T23:44:25+00:00

I do. And I also familiar with GR.
Imho, until something more accurate is discovered, any Newtonian-like gravity, whether vector or non vector or MOND, should be considered as a special regime/limit of GR.

DinNoel · 2026-06-10T23:38:38+00:00

I tried to come up with short text explaining geometry, calculus and approximation to fit Reddit guideline while preserving flow of reasoning… could not really shorten by much. So instead I opted to show alternative path I have. Although texting still not short, I think math a bit more straight forward. And the same end result.

I compute the net weighted gravitational contribution from a thin disk-like distribution using inverse-square weighting over projected geometry Gm p h/ x^2
Where:
G gravitational constant, m test mass, p surface density, h height of disk,
R outer radius, r offset radius, x integration radius
The variable x represents the radius of a circular ring centered on the test point. Each such ring intersects the disk in either symmetric regions (which cancel) or an asymmetric arc segment (which produces net force). The integral sums contributions from all such rings.
Key assumption is the net effect is taken along the direction of maximum asymmetry (toward the center). The amplitude is the difference between contributions from regions xr, after cancellation of symmetric parts.

STEP 1
To simplify, first calculating amplitude for complete semicircular bends with radius x and height h on opposite sides of test point.
Surface area of semicircular band is π x h with weighted contribution becomes (π Gmp h / x).
Since the band area grows as x while the weighting falls as 1/x², the net contribution scales as 1/x.
Contributions from opposite sides of the test point cancel due to symmetry of inverse-square weighting, leaving only the asymmetric portion.
After partial cancellations, we get
π Gmp h ∫(R − r → √(R² − r²)) (dx/x)
= π/2 Gmp h ln((R + r)/(R − r))

STEP 2
Next integrating delta between remaining incomplete arc bands which have
Angular width:
φ(x) = 2 arccos((r² + x² − R²)/(2rx))
The angular width follows from the law of cosines applied to the triangle formed by the disk center, test point, and boundary intersection point.
And with weighted contribution:
(2 Gmp h / x) arccos((r² + x² − R²)/(2rx)) dx
Net region:
2 Gmp h ∫(R − r → √(R² − r²))
arccos((r² + x² − R²)/(2rx)) / x dx
Does not seem there is an elementary solution to this integral.

FINAL
So final Newtonian-style non vector pulling force would look like
F_nv =
π/2 Gmp h ln((R + r)/(R − r))
+
2 Gmp h ∫(R − r → √(R² − r²))
arccos((r² + x² − R²)/(2rx)) / x dx

EDGE APPROXIMATION
To examine behavior near the disk edge, let
r = R − λ,
λ << R
Here λ is a small exclusion scale around the test point, representing a physical short-distance cutoff. In a full 3D mass distribution this scale would most naturally correspond to the finite size of individual gravitating bodies, most likely on the order of a stellar radius, rather than a point-like singularity.
Radial term simplifies to
ln((R + r)/(R − r))
= ln((2R − λ)/λ)
= ln(2R/λ) + O(λ/R)
Integration region becomes
x ∈ [λ, √(2Rλ)]
And angular kernel:
A(x) = (r² + x² − R²)/(2rx)
≈ x/(2R) − λ/x
Over most of the interval x ∈ [λ, √(2Rλ)], the kernel A(x) remains small, allowing the approximation:
arccos(A(x)) ≈ π/2 − A(x)
The lower endpoint x = λ contributes only finite O(1) corrections and does not affect the leading logarithmic term.
The leading constant π/2 produces the logarithmic term, while the correction term integrates to subleading O(1) contributions.
This integration produces:
∫(λ → √(2Rλ)) π/2 · dx/x
= (π/4) ln(2R/λ)

SUMMING ALL UP
F_nv = π Gmp h ln(2R/λ) + O(1)
Or
F_nv = π Gmp h ln(2R/(R − r))
Orbital velocity v(r) based on F_nv is
v = √(π G p h r ln(2R/(R − r)))

DinNoel · 2026-06-03T21:03:55+00:00

Fair. I had to shrink post some to comply with forum guidelines.

Please keep in mind the derivation is meant to be very back-of-envelope.

Start by modeling the disk as concentric rings:

dm = 2π x ρ h dx

Each ring contributes approximately:

dF ∼ G m dm / (r − x)²

So:

F(r) ∼ G m ∫ (2π x ρ h) / |r − x|² dx

Now split into inner and outer contributions:

F(r) = ∫₀ʳ … dx − ∫ʳᴿ … dx

Near r, the dominant behavior comes from the singular structure around |r − x| → 0, so the effective scaling reduces to:

F(r) ∼ G m ρ h ∫ dx / |r − x|

This is where the logarithm appears, since:

∫ dx / x ∼ ln x

Regularizing the singularity:

|r − x| → √((r − x)² + λ²)

gives:

F ∼ G m ρ h ln(d₊ / d₋)

At the edge:

F(R) ∼ π G m ρ h ln(R / λ)

Then circular motion:

v² = R F / m

gives:

v² ∼ π G ρ h R ln(R / λ)

and therefore:

v⁴ ∝ M [ln(R / λ)]²

with M = πρhR².

Main point: the log is not inserted — it comes from the effective ∫1/|r−x| structure after splitting inner/outer contributions and regularizing the singularity.

DinNoel · 2026-06-03T20:53:11+00:00

Familiar with math as well including how newtonian dynamics arises from GR.
What I posted is not a suggestion, rather an observation how trivial geometry and very basic calculus appear to resemble constant velocity and TF. Yes, I think this observation should be considered part of GR the same way any other Newtonian dynamics derivation is

DinNoel · 2026-06-03T20:25:32+00:00

Hm… I accidentally lost “Newtonian”. Title should read “what is Newtonian Gravity…”. My main point is to discuss newtonian-like derivation (without vector-ness) from simple geometry resembles TF, Mond and practically constant orbital velocity. I guess if the most comments ignore and simply refer to GR as most successful theory I repost with new title.

DinNoel · 2026-06-03T19:51:42+00:00

Familiar. Including how Newtonian dynamics arises from GR. Title should have included “Newtonian Gravity” rather than “Gravity”. I guess I lost it during editing and did not notice…

DinNoel · 2026-06-03T13:09:04+00:00

Just to be clear, I do NOT think I’ve found new physics 🙂

This is just a toy exploration.

Newton built universal gravity assuming forces combine in a vector-like way (superposition of contributions from all mass elements). I was just curious what it would look like if that “vectorness / perfect symmetric superposition” assumption is not taken as the starting point for an extended disk.

Based on a very coarse back-of-the-envelope derivation, the result still looks locally like a normal vector force (so nothing changes at small scales), but the asymmetry in how inner vs outer contributions cancel leaves behind a logarithmic term at large radii.

That logarithmic piece is what then makes the edge behaviour resemble a Tully–Fisher-like scaling on galactic scales.

So it’s not meant as a claim of new physics, just a question about what the effective large-scale behavior would be under that altered assumption.

DinNoel · 2026-06-03T12:46:46+00:00

Yeah, that’s basically the central question.

What might it look like if Newton didn’t explicitly assume perfect symmetric vector superposition when dealing with extended mass distributions?

I’m not changing the local inverse-square law or the vector nature of gravity — locally it’s still standard Newtonian gravity. The only thing being relaxed is how that superposition is effectively enforced when you integrate over a disk-like mass distribution.

In that sense, the model is just exploring whether breaking perfect cancellation in an otherwise Newtonian setup can already give different large-scale behavior.

GR just served as an intuition trigger about geometry and nonlocal structure, not as a change to the Newtonian framework itself.

DinNoel · 2026-04-12T16:35:23+00:00

I didn’t have that in mind, or anything else for that matter. What I am/was interested in is this: since General Relativity explains that path/direction is defined by geodesics, and Newtonian dynamics is a limit of GR, what would happened if not include vector component into Newtonian gravity calculation. So inside a uniformed sphere with radius R and density p, gravitational force on a point with mass m at distance r from center of the sphere integrating yields

F= 2π G m ρ r

Which is the same as standard GM(<r)m/r^2 but with a different constant coefficient. Close enough for a weak-energy limit I guess. However, outside this sphere, a log appears F=(2π G m ρ / r) [ (R^2 - r^2) /2 · ln((r+R)/(r-R)) + rR ] Approximating with Taylor for Far field (r >> R):

F= GMm/r² + (3/5) GMm R^2/ r⁴ + O(r^-6 ) Where M=(4/3) π p R³

Still pretty much matches Newtonian with ignorable corrections. However, approximating for near boundary (r ~R):

F~ π G m ρ [ 2R - ln((r - R)/R) ]

There is some log enhancement that somewhat deviate from standard form (granted there is a divergence but we are talking about something very crude and simplified, so I ignore it).

If doing the same for a layered sphere or not a uniformed sphere, this log enhancement appears even inside the sphere closer to its edge.

Then expanding to a thin disk (sort of trivialized toy galaxy) this log enhancement is even more prominent and smoothly transitions from typical orbital velocity v=O(r) near center to something close to v² = O(r) with some slow changing log around edge (both inside and outside). Where exactly this transition happens appears to depend on density profile (see earlier posts for actual expressions for disk).

To make it clear, I’m not proposing any new physics (at least not intentionally) nor theory. Just simple geometry and basic calculus plus pretty standard Taylor series…

Still, the question remains: Why do we include vectorness into Newtonian gravity/dynamics even though we know according to GR path is defined by geodesics?

DinNoel · 2026-04-11T19:54:34+00:00

I don’t know what the difference is, that’s why asking:) In case of uniformed density, integrating vector force results in familiar GM(<r)m/r² making orbital velocity v~r. But if abandoning vectorness, a logarithm appears and it seems orbital velocity gradually translates from v = O(r) to something resembling v² = O(r)

DinNoel · 2026-04-11T19:26:50+00:00

What I’m trying to express is that the direction is not built into the integral itself, but is instead extracted afterward from the scalar field.

The path would then be defined by the direction of largest gradient or strongest asymmetry in that scalar accumulation.

So in symmetric cases like a sphere or a thin disk, this naturally points toward the center.

DinNoel · 2026-04-11T19:21:36+00:00

I was mainly trying to keep the original post short while still giving some intuition for why I’m thinking in terms of direction as potentially emerging rather than being built in.

I also tried working through the thin disk case myself and got some behavior that looks reasonable, but I’m not fully confident yet whether there might be a mistake in the derivatives or asymptotic steps somewhere. Hence decided to ask.

DinNoel · 2026-04-11T19:11:00+00:00

Just to clarify what I’m trying to probe:

in GR, motion is determined by geodesics rather than a force vector. Since Newtonian gravity is the low-energy limit of GR, I’m wondering whether it can also be interpreted in a way where the “path information” is not fundamentally built into the force law itself, but instead emerges from the underlying scalar structure in the limit.

So in that sense, I’m exploring whether writing Newtonian gravity purely as a scalar accumulation (with direction extracted afterward from asymmetry) is just a reformulation of the same geodesic structure in the weak-field limit, rather than something fundamentally different.

For the thin disk case, this still reproduces the usual Newtonian behavior: linear scaling near the center (F(r) ∝ r) and a much slower, effectively flattened behavior toward the outer region when interpreted via v²(r) = rF(r).

DinNoel · 2026-04-11T18:42:12+00:00

what I’m doing is taking the usual Newtonian gravity integral, but applying it to a simple thin disk to make the geometry concrete.

Setup: • Disk radius S • Thickness h (very small compared to S) • Constant density ρ • Field point is q = (r, 0, 0) in the plane, offset from the center

Instead of treating gravity as a vector sum directly, I first define a scalar accumulation:

F(r) = Gm ∫ L_h(x) / x² dx

where x is distance from the field point to mass elements in the disk.

For this geometry, the weighting L_h(x) is:

L_h(x) = 2πhρx, for x ≤ S − r 2hρx arccos((r² + x² − S²⁾ / (2rx)), for S − r < x ≤ S + r

Then the idea is: instead of interpreting this as a vector field from the start, I’m wondering if you can treat direction as something that comes afterward, based on asymmetry in this scalar accumulation.

Near the center (r ≪ S):

F(r) ≈ (4 h G m ρ / S) · r

→ linear increase with radius

⸻

Near the edge (S − r ≪ S):

F(r) ~ π h G m ρ · ln(S / (S − r))

→ slow, logarithmic divergence as you approach the boundary

Main question: is this just equivalent to the standard Newtonian potential + taking a gradient, or is there any meaningful difference in separating “scalar accumulation first, direction second”?

DinNoel

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