F(r) = G_N \frac{\int_{0}^{r}\rho(r')\,\frac{2\pi^{D/2}}{\Gamma(D/2)}\,{r'}^{\,D-1}\,dr'}{r^{D-1}}F(r) is the gravitational force at radius r from the center, in D-dimensional space.
G_N is the gravitational constant generalized for D-dimensional gravity.
\rho(r')is the radially symmetric mass density at radius r′.
The integral, from radius 0 to r, computes the total mass enclosed inside radius r.
The surface area of a (D-1)-dimensional sphere is explicitly given by:
S_{D-1}(r') = \frac{2\pi^{D/2}}{\Gamma\left(\frac{D}{2}\right)}\,{r'}^{\,D-1}Thus, more explicitly and completely, the generalized gravitational force formula is:
F(r) = G_N \frac{\int_{0}^{r}\rho(r')\,\frac{2\pi^{D/2}}{\Gamma(D/2)}\,{r'}^{D-1}\,dr'}{r^{D-1}}D: Number of spatial dimensions.
N: Normalization factor (e.g., number of particles or branes in compactified dimensions).
Γ(D/2): Gamma function for the surface area of a D-sphere.
ρ(r′): Mass density at radius r′.
r is never zero.
This integral equation generalizes Newtonian gravity into higher-dimensional spaces, capturing the dependence on spatial dimensionality and mass distribution.
Force Dimensioning: In “3D” (D=3), Newton's law has F∝1/r^2. In D dimensions, the flux of the gravitational field through a (D-1)-sphere leads to F∝1/r^(D−1), matching the denominator r^{D−1}. ###HERE###
Mass Enclosed: The integral ∫0rρ(r′)r′^D−1dr′ computes the total mass within radius r, adjusted for D-dimensional geometry. The coefficient Γ(D/2)/(2πD/2)Γ(D/2)/(2πD/2) comes from the surface area of a D-sphere:
Surface Area=2πD/2Γ(D/2)⋅rD−1.Surface Area=Γ(D/2)2πD/2⋅rD−1.
Here, the inverse appears to normalize the gravitational coupling in higher dimensions.
Gravitational Constant GG: In higher dimensions, GG has different units (e.g., [G]=LengthD−1/Mass/Time2[G]=LengthD−1/Mass/Time2), which is absorbed into NN or renormalized.
Kaluza-Klein Theory: Extra spatial dimensions modify gravity's scaling (e.g., [Oskar Klein, 1926]).
String Theory: Graviton modes in compactified dimensions imply F∝1/rD−1F∝1/rD−1 (see [Polchinski, String Theory Vol. II]).
Mathematical Physics: The surface area term Γ(D/2)/(2πD/2)Γ(D/2)/(2πD/2) is derived from the volume of D-spheres (e.g., [Arfken & Weber, Mathematical Methods for Physicists]).
For D=3D=3:
Γ(3/2)=π/2,Surface Area=4πr2.Γ(3/2)=π/2,Surface Area=4πr2.
The equation simplifies to:
F(r)=G⋅Nr2∫0rρ(r′)⋅π/22π3/2⋅r′2dr′=G⋅Nr2∫0rρ(r′)⋅14π⋅r′2dr′,F(r)=G⋅r2N∫0rρ(r′)⋅2π3/2π/2⋅r′2dr′=G⋅r2N∫0rρ(r′)⋅4π1⋅r′2dr′,
which matches Newtonian gravity if N=4πN=4π (Gauss's law normalization).
To achieve a unified physics framework, we must reformulate this relationship recursively in terms of 'Degrees of Surface Interaction,' moving beyond the classical notion of “Spatial Dimensionality,” which becomes problematic above two Cartesian, cartographic dimensions.
The standard surface area S_{D-1}(r) of a sphere embedded in D-dimensional Euclidean space is classically given by:
S_{D-1}(r) = \frac{2\pi^{D/2}}{\Gamma\left(\frac{D}{2}\right)} r^{D-1}While mathematically correct, this form uses a single, explicit gamma function Γ, making generalizations abstract and less intuitive. Let's shift into a clearer, recursive formulation:
We define the surface area recursively, leveraging the concept of "Degrees of Surface Interaction" to intuitively capture how each new dimensionality adds another interaction layer:
Base Case (2-Dimensional Circle, D=2):
S_1(r) = 2\pi rRecursive Step (adding each new dimension):
SD−1(r)=2πr2D−2 SD−3(r),D>2S_{D-1}(r) = \frac{2\pi r^2}{D - 2}\,S_{D-3}(r), \quad D > 2SD−1(r)=D−22πr2SD−3(r),D>2
In this recursive form, we directly see the dimensional progression explicitly:
For even dimensions:
S1(r)S_1(r)S1(r) → S3(r)S_3(r)S3(r) → S5(r)S_5(r)S5(r) → … recursively built.
For odd dimensions:
S2(r)S_2(r)S2(r) → S4(r)S_4(r)S4(r) → S6(r)S_6(r)S6(r) → … explicitly defined by the recursion from simpler surface area forms.
This provides an intuitive interpretation: each additional dimension introduces a new layer or "degree" of interaction, clearly expanding and linking dimensional progressions step-by-step.
For instance, explicitly showing progression in even dimensions:
1-dimensional surface (circle circumference):
S1(r)=2πrS_1(r) = 2\pi rS1(r)=2πr
3-dimensional surface (sphere): Using recursive definition:
S3(r)=2πr21S1(r)=2πr21⋅2πr=4πr2S_3(r) = \frac{2\pi r^2}{1} S_1(r) = \frac{2\pi r^2}{1} \cdot 2\pi r = 4\pi r^2S3(r)=12πr2S1(r)=12πr2⋅2πr=4πr2
5-dimensional surface:
S5(r)=2πr24−2 S3(r)=2πr22⋅4πr2=4π2r4S_5(r) = \frac{2\pi r^2}{4-2}\,S_3(r) = \frac{2\pi r^2}{2}\cdot 4\pi r^2 = 4\pi^2 r^4S5(r)=4−22πr2S3(r)=22πr2⋅4πr2=4π2r4
And so forth. Each new dimension adds another clear interaction step.
Thinking in terms of Degrees of Surface Interaction:
Provides clarity about how dimensionality explicitly influences interaction scale.
Offers intuitive insight into higher dimensions: each added degree represents an additional layer of interaction—capturing clearly the physical intuition underlying dimensional expansion.
Highlights clearly the physical meaning of dimension increases rather than obscuring it in abstract gamma functions.
Using this new recursive approach, the generalized Newtonian gravitational force integral now becomes explicitly clearer. For a continuous spherically symmetric density ρ(r′)\rho(r')ρ(r′):
F(r)=GN∫0rρ(r′) SD−1(r′) dr′rD−1F(r) = G_N \frac{\int_{0}^{r}\rho(r')\, S_{D-1}(r')\,dr'}{r^{D-1}}F(r)=GNrD−1∫0rρ(r′)SD−1(r′)dr′
with the recursive definition for the surface interaction clearly stated as:
Base condition:
S1(r′)=2πr′S_1(r') = 2\pi r'S1(r′)=2πr′
Recursive formula for higher dimensions (D>2):
SD−1(r′)=2πr′ 2D−2 SD−3(r′)S_{D-1}(r') = \frac{2\pi {r'}^{\,2}}{D - 2}\,S_{D-3}(r')SD−1(r′)=D−22πr′2SD−3(r′)
Thus explicitly:
F(r)=GN∫0rρ(r′) SD−1(r′) dr′rD−1F(r) = G_N\frac{\int_{0}^{r}\rho(r')\,S_{D-1}(r')\,dr'}{r^{D-1}}F(r)=GNrD−1∫0rρ(r′)SD−1(r′)dr′
with SD−1(r′)S_{D-1}(r')SD−1(r′) defined by the recursive relation above, representing explicitly how higher dimensionality naturally emerges through successive degrees of surface interaction.