Theorem: Uniqueness of $n = 2$ in Spherically Symmetric Density Distributions
Rigorous Mathematical Proof of the Uniformity Condition for the Shell Mass Growth Rate in 3-Dimensional Euclidean Space
§1. Formulation of the Problem
Consider the following entities:
- The space is the 3-dimensional Euclidean space $\R^3$.
- The density distribution is spherically symmetric: $\rho : (0, \infty) \to (0, \infty)$, where $\rho(\mathbf{x}) = \rho(r)$ with $r = |\mathbf{x}|$.
- The density distribution is power-law: $\rho(r) = r^{-n}$ ($n \in \R$, $r > 0$).
- The volume element of a spherical shell is $dV = 4\pi r^2\,dr$.
We define the mass contained within a sphere of radius $r$ as:
$$m(r) \;=\; \int_0^{r} \rho(r’)\cdot 4\pi r’^{\,2}\,dr’$$
In this case, the shell mass growth rate (the derivative of mass per shell) is defined as:
$$\frac{dm}{dr}(r) \;=\; 4\pi\,r^2\,\rho(r)$$
The shell mass growth rate is said to be uniform (independent of the radius $r$) if there exists some constant $C > 0$ such that:
$$\frac{dm}{dr}(r) = C \qquad \text{for all } r \in (0, \infty)$$
holds.
§2. Main Theorem
Let $\rho(r) = r^{-n}$ ($n \in \R$). The following are equivalent:
$\text{(i)}$ $\dfrac{dm}{dr}(r) = 4\pi\,r^{2-n}$ is constant for all $r > 0$.
$\text{(ii)}$ $n = 2$.
That is, $n = 2$ is the unique real number that satisfies the uniformity condition.
§3. Lemmas Required for the Proof
Let $\alpha \in \R$. Define a function $f : (0, \infty) \to \R$ by $f(r) = r^\alpha$.
Then, $f$ is constant on $(0, \infty)$ if and only if $\alpha = 0$.
(Sufficiency: $\alpha = 0 \Rightarrow f$ is constant)
When $\alpha = 0$, $f(r) = r^0 = 1$ for all $r > 0$. Hence, $f$ is a constant function (with value $1$).
(Necessity: $f$ is constant $\Rightarrow \alpha = 0$)
We prove this by contraposition. That is, we show that “$\alpha \neq 0 \Rightarrow f$ is not constant.”
Assume $\alpha \neq 0$. Compare the values of $f$ at $r = 1$ and $r = 2$:
If $f$ is constant, we must have $f(1) = f(2)$, which implies $1 = 2^\alpha$.
Here, we utilize the properties of the exponential function $g(\alpha) = 2^\alpha = e^{\alpha \ln 2}$. Since $\ln 2 > 0$:
- $g(0) = 2^0 = 1$
- $g'(\alpha) = 2^\alpha \ln 2 > 0 \quad (\forall\,\alpha \in \R)$
Thus, $g$ is strictly monotonically increasing on $\R$. Therefore, if $g(\alpha) = 1 = g(0)$, then $\alpha = 0$.
By contraposition, if $\alpha \neq 0$, then $g(\alpha) = 2^\alpha \neq 1$, which means $f(2) \neq f(1)$.
Hence, $f$ is not constant.
Thus, both sufficiency and necessity have been proven, establishing that $f(r) = r^\alpha$ is constant on $(0,\infty)$ $\iff$ $\alpha = 0$.
In $\R^3$, the volume element of a spherical shell of radius $r$ and thickness $dr$ is
$$dV = 4\pi\,r^2\,dr$$
is established.
Representing the volume element of the 3-dimensional Euclidean space $\R^3$ in spherical coordinates $(r, \theta, \varphi)$ ($r \geq 0$, $0 \leq \theta \leq \pi$, $0 \leq \varphi < 2\pi$) yields: $$dV = r^2 \sin\theta\;\,dr\,d\theta\,d\varphi$$
Integrating the angular parts for a spherically symmetric integrand:
Thus, the volume element of the spherical shell is $dV = 4\pi\,r^2\,dr$.
§4. Proof of the Main Theorem
-
Derivation of the Mass Growth Rate
From the density $\rho(r) = r^{-n}$ and the volume element $dV = 4\pi\,r^2\,dr$ (Lemma 3.2), the shell mass growth rate is:$$\frac{dm}{dr}(r) \;=\; 4\pi\,r^2\,\rho(r) \;=\; 4\pi\,r^2 \cdot r^{-n} \;=\; 4\pi\,r^{2-n}$$ -
Equivalent Transformation of the Uniformity Condition
For $\dmdr(r)$ to be constant, $4\pi\,r^{2-n}$ must be independent of $r$.
Since $4\pi > 0$ is a positive constant, this is equivalent to:
$$r^{2-n} \;\text{being constant on}\; (0,\infty).$$ -
Application of Lemma 3.1
Let $\alpha := 2 – n$. Applying Lemma 3.1 to the power function $f(r) = r^\alpha = r^{2-n}$ yields:
$$r^{2-n} \text{ is constant} \;\iff\; \alpha = 0 \;\iff\; 2 – n = 0 \;\iff\; n = 2$$ -
Conclusion
Combining Steps 1–3, we have shown:
$$\frac{dm}{dr}(r) = 4\pi\,r^{2-n} \;\text{is constant} \;\iff\; n = 2$$
When $n = 2$, $\dmdr = 4\pi\,r^{2-2} = 4\pi\,r^0 = 4\pi$ (a positive constant).
§5. Corollaries and Remarks
When $n = 2$, the constant value of the shell mass growth rate is:
$$\frac{dm}{dr} = 4\pi$$
Therefore, the total mass contained inside the sphere of radius $R$ is:
$$m(R) = \int_0^R 4\pi\,dr = 4\pi R$$
and the mass is linearly proportional to the radius.
The condition $n = 2$, namely $\rho(r) = 1/r^2$, is closely related to the fundamental solution $\Phi(r) = -1/r$ of the 3-dimensional Laplace equation:
$$\nabla^2 \Phi = 0 \qquad (r \neq 0)$$
The fact that Gauss’s law
$$\oint_S \nabla\Phi \cdot d\mathbf{S} = -4\pi$$
holds independently of the spherical radius $r$ is mathematically identical in structure to the uniformity condition of $\rho \propto 1/r^2$.
In the $d$-dimensional Euclidean space $\R^d$ ($d \geq 2$), the volume element of a spherical shell is:
$$dV_d = \frac{2\pi^{d/2}}{\Gamma(d/2)}\;r^{d-1}\,dr = S_{d-1}\,r^{d-1}\,dr$$
(where $S_{d-1}$ is the area of the $(d-1)$-dimensional unit sphere).
The mass growth rate with respect to the density $\rho(r) = r^{-n}$ is:
$$\frac{dm}{dr} = S_{d-1}\;r^{d-1-n}$$
and the condition for this to be constant is:
$$\boxed{n = d – 1}$$
This recovers $n = 2$ when $d = 3$.
§6. Verification of the Completeness of the Proof
This proof relies solely on the following logical steps, without utilizing any physical intuition:
- Calculation of the volume element in 3-dimensional spherical coordinates (Analysis: Multiple Integrals)
- Definition of the mass growth rate as the product of density and volume element (Measure Theory)
- Algebraic properties of power functions $r^a \cdot r^b = r^{a+b}$ (Exponent laws of real numbers)
- Strict monotonicity of the exponential function $2^\alpha$ (Analysis: Calculus)
- Derivation of necessary and sufficient conditions via equivalence transformation (Logic)
Each of the four premises of Theorem 2.1 is indispensable for the proof:
- 3 dimensions: If removed, we obtain $n = d – 1$ (Corollary 5.3), demonstrating that the uniqueness of $n = 2$ is specific to 3D.
- Spherically symmetric: If removed, the density would also depend on the angular directions, requiring the shell mass to be defined by the angular average of $\rho$.
- Power-law density $\rho = r^{-n}$: If removed, the general solution becomes $\rho(r) = C/(4\pi r^2)$ (a specific functional form rather than an arbitrary positive function).
- Volume element $4\pi r^2 dr$: This is determined by the Euclidean metric; in curved spaces, modifications are necessary.
The uniqueness of the solution $n = 2$ forms the mathematical foundation of the following physical laws:
- Newton’s law of universal gravitation: $F \propto 1/r^2$ (inverse-square law)
- Coulomb’s law: $F \propto 1/r^2$
- Gauss’s law: The flux passing through a closed surface is independent of the radius of the sphere.
- Bertrand’s theorem: Classification of central force fields with closed orbits.
These are all physical expressions of the present theorem, stating that $1/r^2$ is the unique harmonic spherically symmetric power-law density in 3-dimensional space.
§7. Alternative Proof (Proof by Differentiation)
If $\dmdr(r) = 4\pi\,r^{2-n}$ is constant on $(0,\infty)$, its derivative with respect to $r$ must be zero:
Since $4\pi > 0$ and $r^{1-n} > 0$ ($\forall\,r > 0$):
$$4\pi\,(2-n)\,r^{1-n} = 0 \;\iff\; 2 – n = 0 \;\iff\; n = 2$$
Conversely, when $n = 2$, $\dmdr = 4\pi\,r^0 = 4\pi = \text{const.}$, confirming the equivalence.
Note: The alternative proof uses the fact that “the derivative of a constant function is zero” (a consequence of the fundamental theorem of calculus), replacing the strict monotonicity argument of Lemma 3.1 with a derivative calculation. The two are logically equivalent.