- #1
physics4all
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- 2
- TL;DR
- Analyze a 1/r² scalar accumulation field with direction set by plane imbalance. Spheres reproduce Newtonian scaling; disks show logarithmic edge enhancement, giving flat rotation curves and Tully–Fisher–like scaling.
Can a scalar accumulation rule yield flat rotation curves and Tully–Fisher–like behavior?
Summary.
I’m exploring mathematic construction in which gravitational influence is treated as a scalar accumulation with kernel (1/r^2), and direction is determined by maximizing imbalance across a plane through the field point.
For spherically symmetric systems, this construction reproduces the same radial scalings as Newtonian gravity (though with different coefficients). For thin disk geometries, it instead produces a logarithmic enhancement near the edge, leading to behavior qualitatively similar to flattened rotation curves.
Scope note. This is a deliberately simplified geometry aimed at isolating scaling behavior rather than providing a complete physical theory.
I’d appreciate feedback on:
the spherical reduction to the logarithmic kernel,
the validity of the thin-disk reduction,
the near-edge asymptotics in the disk case.
1. Definition of the construction
Let the field point be ##q=(r,0,0)##, and a source point ##\mathbf x'=(x,y,z)## with distance
$$
d = \lvert \mathbf r_q - \mathbf x' \rvert.
$$
Define a scalar accumulation:
$$
W(\mathbf r_q) = G m \int \frac{\rho(\mathbf x')}{d^2}\, dV.
$$
To extract a direction, introduce the following rule:
Directional rule:
The acceleration direction is the one that maximizes the difference in accumulated scalar (##W##) across a plane through the field point.
This rule is ad hoc and not derived from a potential; the goal here is to explore the scalings it produces.
For symmetric configurations this selects the radial direction uniquely. The corresponding “force magnitude” can be written as the imbalance across the plane (##x=r##):
$$
F_{NV}(r) = Gm\left(
\iiint_{x<r}\frac{\rho(\mathbf x')}{d^2}\,dV
-
\iiint_{x>r}\frac{\rho(\mathbf x')}{d^2}\,dV
\right).
$$
2. Spherical mass distribution
Here I consider a two-layer sphere: a core of radius ##R_1## with constant density ##\rho_1##, and an outer shell ##R_1<r'\le R## with constant density ##\rho_2##. This keeps the integrals analytically tractable while still allowing a nontrivial radial structure.
Let ##\rho(r')## be spherically symmetric. Using
$$
d^2 = r^2 + r'^2 - 2 r r' \cos\theta,\quad dV = r'^2 \sin\theta\, dr'\, d\theta\, d\phi,
$$
the angular integral gives
$$
\int_0^\pi \frac{\sin\theta\, d\theta}{r^2+r'^2-2rr'\cos\theta}
= \frac{1}{rr'} \ln\frac{r+r'}{\lvert r-r'\rvert}.
$$
Hence the scalar imbalance reduces exactly to
$$
F_{NV}(r)
=
\frac{2\pi G m}{r}
\int_0^R \rho(r')\, r'\,
\ln\frac{r+r'}{\lvert r-r'\rvert}\, dr'.
$$
2.1 Exterior region ##(r>R)##
Using
$$
\int s \ln\frac{r+s}{r-s}\, ds
=
\frac{s^2-r^2}{2}\ln\frac{r+s}{r-s}+rs,
$$
one obtains a closed form. Expanding for ##r\gg R##,
$$
\ln\frac{1+x}{1-x} = 2\left(x+\frac{x^3}{3}+\cdots\right),
$$
gives
$$
F_{NV}(r) = \frac{GMm}{r^2} + O(r^{-4}).
$$
Same leading scaling as Newtonian gravity.
Far outside the sphere, the force behaves just like standard gravity, falling off as ##(1/r^2)## and depending only on the total mass.
2.2 Interior core ##(r<R_1)##
For constant density,
$$
F_{NV}(r) = 2\pi G m \rho_1 r,
$$
while Newtonian gravity gives
$$
F_N(r) = \frac{4\pi}{3} G m \rho_1 r.
$$
So both have the same linear scaling ##(F\propto r)##, with a different coefficient.
Inside the core, the force increases linearly with distance from the center, just as in Newtonian gravity.
2.3 Intermediate region ##(R_1<r<R)##
Carrying out the split integrals yields
$$
F_{NV}(r)
=
2\pi G m \rho_2 r
+
\frac{2\pi G m (\rho_1-\rho_2)}{r}
\left[
\frac{R_1^2-r^2}{2}\ln\frac{r+R_1}{r-R_1}+rR_1
\right].
$$
Expanding for ##R_1/r<1##,
$$
F_{NV}(r)
=
\frac{3}{2}\frac{GmM(<r)}{r^2}
+
O(r^{-4}),
$$
so again the same radial dependence as Newtonian gravity.
In the shell region, the force still behaves like ##(M(<r)/r^2)##, matching the usual gravitational scaling.
2.4 Behavior near the outer boundary ##(r\to R^-)##
Expanding the exact expression near ##r\to R## (via Taylor expansion of the logarithmic kernel) gives the heuristic asymptotic
$$
F_{NV}(r)
\sim
2\pi G m \rho_2 R
+
\pi G m \rho_2 \ln\frac{R}{R-r}.
$$
Thus the setup develops a logarithmic enhancement near sharp edges, unlike Newtonian gravity.
Near the surface, the force grows logarithmically instead of remaining finite, which is a key qualitative difference from Newtonian gravity.
3. Thin disk
Here I restrict to a single-layer disk with constant density to keep the calculation manageable. Preliminary analytic checks with more complicated density profiles suggest the same qualitative behavior persists, so this simplified setup is sufficient to illustrate the main effect.
To keep the presentation concise, I omit regimes that reproduce standard Newtonian scaling and focus on where the behavior differs.
Consider a uniform disk of radius ##R##, thickness ##h\ll R##, density ##p##, total mass
$$
M = \pi p h R^2.
$$
The exact expression is
$$
F_{NV}(r)
=
G m p
\iiint_{x^2+y^2\le R^2,\; |z|\le h/2}
\frac{dx\,dy\,dz}{(r-x)^2+y^2+z^2}.
$$
3.1 Thin-disk reduction (approximate)
For ##h\ll R##, away from the region where ##(r-x)^2+y^2\sim h^2##,
$$
\int_{-h/2}^{h/2}\frac{dz}{(r-x)^2+y^2+z^2}
\approx
\frac{h}{(r-x)^2+y^2}.
$$
This gives
$$
F_{NV}(r)
\approx
G m p h
\int \frac{dx\,dy}{(r-x)^2+y^2}.
$$
The thin-disk approximation reduces the 3D integral to an effectively 2D one by treating the vertical direction as small.
3.2 Polar coordinates about the field point
Let
$$
x = r + \rho\cos\phi,\quad y = \rho\sin\phi.
$$
Then
$$
(r-x)^2 + y^2 = \rho^2,
$$
and the disk boundary gives
$$
\rho_\pm(\phi)
=
\sqrt{R^2 - r^2 \sin^2\phi} \pm r\cos\phi.
$$
Hence
$$
F_{NV}(r)
\approx
G m p h
\int_{-\pi/2}^{\pi/2}
\ln\left(\frac{\rho_+}{\rho_-}\right)\, d\phi.
$$
Re-centering coordinates at the field point simplifies the geometry and produces a logarithmic integral form.
3.3 Inner region ##(r\ll R)##
Using ##\rho_\pm \approx R \pm r\cos\phi##,
$$
\ln\frac{\rho_+}{\rho_-}
\approx \frac{2r}{R}\cos\phi,
$$
so
$$
F_{NV}(r) \approx \frac{4Gmph}{R} r.
$$
Thus
$$
v(r) \propto r.
$$
Near the center of the disk, rotation speed grows linearly with radius, similar to standard mass distributions.
3.4 Near-edge regime ##(h \ll R-r \ll R)##
Let ##r = R - \epsilon##, with ##h \ll \epsilon \ll R##.
A boundary-layer analysis (heuristic) gives
$$
F_{NV}(r)
\approx
\pi G m p h
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right].
$$
This logarithmic behavior arises from the near-singular region of the kernel; in more realistic disks with smooth density profiles, this divergence is expected to be regularized.
Hence
$$
v^2(r) = r F_{NV}(r)
\sim
\pi G p h\, r
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right].
$$
Using ##M = \pi p h R^2## and ##r\sim R##,
$$
v^4(r)
\sim
\pi G^2 p h\, M
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right]^2.
$$
Near the edge, a logarithmic enhancement appears, slowing the decline of rotation speeds and mimicking flat rotation curves.
3.5 Edge limit
At ##r=R##,
$$
F_{NV}(R)
\sim
\frac{\pi G m p}{8}\frac{h^2}{R},
\qquad
v^2(R)\sim \frac{\pi G p}{8}h^2.
$$
At the very edge, the force remains finite and is set by the disk thickness rather than the radius.
4. Discussion
For spherical systems, this scalar accumulation rule reproduces Newtonian radial scalings in all regimes (core, shell, exterior), though with different coefficients.
Near sharp boundaries, it produces a logarithmic enhancement absent in Newtonian gravity.
In disk geometry, this enhancement leads to slowly varying ##v(r)##, qualitatively resembling flattened rotation curves.
The resulting scaling
$$
v^4 \propto M
$$
appears, up to logarithmic corrections, near the disk edge.
5. Questions
Is the spherical reduction to the logarithmic kernel correctly applied here?
Is the thin-disk reduction, especially near the boundary layer, legitimate?
Is the logarithmic near-edge asymptotic derived consistently?
I’m particularly interested in whether the disk asymptotics are mathematically sound, since that is where the logarithmic behavior enters.
More broadly, I’m interested in whether this kind of purely geometric construction and is worth pursuing further, or if there is a fundamental obstruction I’m missing.
Summary.
I’m exploring mathematic construction in which gravitational influence is treated as a scalar accumulation with kernel (1/r^2), and direction is determined by maximizing imbalance across a plane through the field point.
For spherically symmetric systems, this construction reproduces the same radial scalings as Newtonian gravity (though with different coefficients). For thin disk geometries, it instead produces a logarithmic enhancement near the edge, leading to behavior qualitatively similar to flattened rotation curves.
Scope note. This is a deliberately simplified geometry aimed at isolating scaling behavior rather than providing a complete physical theory.
I’d appreciate feedback on:
the spherical reduction to the logarithmic kernel,
the validity of the thin-disk reduction,
the near-edge asymptotics in the disk case.
1. Definition of the construction
Let the field point be ##q=(r,0,0)##, and a source point ##\mathbf x'=(x,y,z)## with distance
$$
d = \lvert \mathbf r_q - \mathbf x' \rvert.
$$
Define a scalar accumulation:
$$
W(\mathbf r_q) = G m \int \frac{\rho(\mathbf x')}{d^2}\, dV.
$$
To extract a direction, introduce the following rule:
Directional rule:
The acceleration direction is the one that maximizes the difference in accumulated scalar (##W##) across a plane through the field point.
This rule is ad hoc and not derived from a potential; the goal here is to explore the scalings it produces.
For symmetric configurations this selects the radial direction uniquely. The corresponding “force magnitude” can be written as the imbalance across the plane (##x=r##):
$$
F_{NV}(r) = Gm\left(
\iiint_{x<r}\frac{\rho(\mathbf x')}{d^2}\,dV
-
\iiint_{x>r}\frac{\rho(\mathbf x')}{d^2}\,dV
\right).
$$
2. Spherical mass distribution
Here I consider a two-layer sphere: a core of radius ##R_1## with constant density ##\rho_1##, and an outer shell ##R_1<r'\le R## with constant density ##\rho_2##. This keeps the integrals analytically tractable while still allowing a nontrivial radial structure.
Let ##\rho(r')## be spherically symmetric. Using
$$
d^2 = r^2 + r'^2 - 2 r r' \cos\theta,\quad dV = r'^2 \sin\theta\, dr'\, d\theta\, d\phi,
$$
the angular integral gives
$$
\int_0^\pi \frac{\sin\theta\, d\theta}{r^2+r'^2-2rr'\cos\theta}
= \frac{1}{rr'} \ln\frac{r+r'}{\lvert r-r'\rvert}.
$$
Hence the scalar imbalance reduces exactly to
$$
F_{NV}(r)
=
\frac{2\pi G m}{r}
\int_0^R \rho(r')\, r'\,
\ln\frac{r+r'}{\lvert r-r'\rvert}\, dr'.
$$
2.1 Exterior region ##(r>R)##
Using
$$
\int s \ln\frac{r+s}{r-s}\, ds
=
\frac{s^2-r^2}{2}\ln\frac{r+s}{r-s}+rs,
$$
one obtains a closed form. Expanding for ##r\gg R##,
$$
\ln\frac{1+x}{1-x} = 2\left(x+\frac{x^3}{3}+\cdots\right),
$$
gives
$$
F_{NV}(r) = \frac{GMm}{r^2} + O(r^{-4}).
$$
Same leading scaling as Newtonian gravity.
Far outside the sphere, the force behaves just like standard gravity, falling off as ##(1/r^2)## and depending only on the total mass.
2.2 Interior core ##(r<R_1)##
For constant density,
$$
F_{NV}(r) = 2\pi G m \rho_1 r,
$$
while Newtonian gravity gives
$$
F_N(r) = \frac{4\pi}{3} G m \rho_1 r.
$$
So both have the same linear scaling ##(F\propto r)##, with a different coefficient.
Inside the core, the force increases linearly with distance from the center, just as in Newtonian gravity.
2.3 Intermediate region ##(R_1<r<R)##
Carrying out the split integrals yields
$$
F_{NV}(r)
=
2\pi G m \rho_2 r
+
\frac{2\pi G m (\rho_1-\rho_2)}{r}
\left[
\frac{R_1^2-r^2}{2}\ln\frac{r+R_1}{r-R_1}+rR_1
\right].
$$
Expanding for ##R_1/r<1##,
$$
F_{NV}(r)
=
\frac{3}{2}\frac{GmM(<r)}{r^2}
+
O(r^{-4}),
$$
so again the same radial dependence as Newtonian gravity.
In the shell region, the force still behaves like ##(M(<r)/r^2)##, matching the usual gravitational scaling.
2.4 Behavior near the outer boundary ##(r\to R^-)##
Expanding the exact expression near ##r\to R## (via Taylor expansion of the logarithmic kernel) gives the heuristic asymptotic
$$
F_{NV}(r)
\sim
2\pi G m \rho_2 R
+
\pi G m \rho_2 \ln\frac{R}{R-r}.
$$
Thus the setup develops a logarithmic enhancement near sharp edges, unlike Newtonian gravity.
Near the surface, the force grows logarithmically instead of remaining finite, which is a key qualitative difference from Newtonian gravity.
3. Thin disk
Here I restrict to a single-layer disk with constant density to keep the calculation manageable. Preliminary analytic checks with more complicated density profiles suggest the same qualitative behavior persists, so this simplified setup is sufficient to illustrate the main effect.
To keep the presentation concise, I omit regimes that reproduce standard Newtonian scaling and focus on where the behavior differs.
Consider a uniform disk of radius ##R##, thickness ##h\ll R##, density ##p##, total mass
$$
M = \pi p h R^2.
$$
The exact expression is
$$
F_{NV}(r)
=
G m p
\iiint_{x^2+y^2\le R^2,\; |z|\le h/2}
\frac{dx\,dy\,dz}{(r-x)^2+y^2+z^2}.
$$
3.1 Thin-disk reduction (approximate)
For ##h\ll R##, away from the region where ##(r-x)^2+y^2\sim h^2##,
$$
\int_{-h/2}^{h/2}\frac{dz}{(r-x)^2+y^2+z^2}
\approx
\frac{h}{(r-x)^2+y^2}.
$$
This gives
$$
F_{NV}(r)
\approx
G m p h
\int \frac{dx\,dy}{(r-x)^2+y^2}.
$$
The thin-disk approximation reduces the 3D integral to an effectively 2D one by treating the vertical direction as small.
3.2 Polar coordinates about the field point
Let
$$
x = r + \rho\cos\phi,\quad y = \rho\sin\phi.
$$
Then
$$
(r-x)^2 + y^2 = \rho^2,
$$
and the disk boundary gives
$$
\rho_\pm(\phi)
=
\sqrt{R^2 - r^2 \sin^2\phi} \pm r\cos\phi.
$$
Hence
$$
F_{NV}(r)
\approx
G m p h
\int_{-\pi/2}^{\pi/2}
\ln\left(\frac{\rho_+}{\rho_-}\right)\, d\phi.
$$
Re-centering coordinates at the field point simplifies the geometry and produces a logarithmic integral form.
3.3 Inner region ##(r\ll R)##
Using ##\rho_\pm \approx R \pm r\cos\phi##,
$$
\ln\frac{\rho_+}{\rho_-}
\approx \frac{2r}{R}\cos\phi,
$$
so
$$
F_{NV}(r) \approx \frac{4Gmph}{R} r.
$$
Thus
$$
v(r) \propto r.
$$
Near the center of the disk, rotation speed grows linearly with radius, similar to standard mass distributions.
3.4 Near-edge regime ##(h \ll R-r \ll R)##
Let ##r = R - \epsilon##, with ##h \ll \epsilon \ll R##.
A boundary-layer analysis (heuristic) gives
$$
F_{NV}(r)
\approx
\pi G m p h
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right].
$$
This logarithmic behavior arises from the near-singular region of the kernel; in more realistic disks with smooth density profiles, this divergence is expected to be regularized.
Hence
$$
v^2(r) = r F_{NV}(r)
\sim
\pi G p h\, r
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right].
$$
Using ##M = \pi p h R^2## and ##r\sim R##,
$$
v^4(r)
\sim
\pi G^2 p h\, M
\left[
1 + \ln\left(\frac{4(R-r)}{h}\right)
\right]^2.
$$
Near the edge, a logarithmic enhancement appears, slowing the decline of rotation speeds and mimicking flat rotation curves.
3.5 Edge limit
At ##r=R##,
$$
F_{NV}(R)
\sim
\frac{\pi G m p}{8}\frac{h^2}{R},
\qquad
v^2(R)\sim \frac{\pi G p}{8}h^2.
$$
At the very edge, the force remains finite and is set by the disk thickness rather than the radius.
4. Discussion
For spherical systems, this scalar accumulation rule reproduces Newtonian radial scalings in all regimes (core, shell, exterior), though with different coefficients.
Near sharp boundaries, it produces a logarithmic enhancement absent in Newtonian gravity.
In disk geometry, this enhancement leads to slowly varying ##v(r)##, qualitatively resembling flattened rotation curves.
The resulting scaling
$$
v^4 \propto M
$$
appears, up to logarithmic corrections, near the disk edge.
5. Questions
Is the spherical reduction to the logarithmic kernel correctly applied here?
Is the thin-disk reduction, especially near the boundary layer, legitimate?
Is the logarithmic near-edge asymptotic derived consistently?
I’m particularly interested in whether the disk asymptotics are mathematically sound, since that is where the logarithmic behavior enters.
More broadly, I’m interested in whether this kind of purely geometric construction and is worth pursuing further, or if there is a fundamental obstruction I’m missing.
Last edited: