prelude
We start the previous theory of gravity, that of Newton. The conventional Newtonian statement: \[ \begin{equation} \mathbf{F}=-\frac{GMm}{r^2}\mathbf{e}_{r} \end{equation} \] where \(\mathbf{r}=r\mathbf{e}_r\), equivalently, we could use the gravitational potential \(\Phi\), the potential iss related to the mass density \(\rho\) by Poisson's equation \[ \begin{equation} \nabla^2 \Phi=4 \pi G \rho \end{equation} \]
Physical Interpretation \(\nabla^2 \Phi=4 \pi G \rho\)
The total gravitational flux through a closed surface is proportional to the total mass enclosed by the surface, with a proportionality constant of (-\(4\pi G\)). Which is \[ \oint \mathbf{g} \cdot d \mathbf{A} = -4 \pi G M_{\text{enclosed}} \] represents Gauss's law for gravity in integral form within Newtonian gravity, describing the relationship between the gravitational field and mass distribution. Here is a detailed explanation:
- Negative sign: Indicates that gravity is an attractive force, so the flux points inward (opposite to the outward normal vector of the surface).
- (\(4\pi\)): Arises from the total solid angle of a closed surface in three-dimensional space (e.g., the surface area of a sphere is (4r^2)).
- (\(M_{\text{enclosed}}\)): Total mass enclosed within the surface, defined as \((M_{\text{enclosed}} = \int \rho \, dV)\), where \((\rho)\) is the mass density.
explanation
1) Newtonian Gravitational Field
For a point mass \((M)\), the gravitational field \((\mathbf{g})\) at distance (r) is:
\[
\mathbf{g} = -\frac{GM}{r^2} \mathbf{e}_r,
\] where the negative sign denotes attraction toward the mass.
2) Gravitational Flux Through a Spherical Surface
Consider a spherically symmetric closed surface (e.g., a sphere of radius \((r)\)) enclosing mass \((M)\):
- The field \((\mathbf{g})\) is uniform in magnitude and radial in direction across the surface.
- The flux is the product of the field magnitude and the surface area:
\[
\oint \mathbf{g} \cdot d\mathbf{A} = g \cdot 4\pi r^2 = -\frac{GM}{r^2} \cdot 4\pi r^2 = -4\pi GM.
\] The result depends only on the enclosed mass \((M)\), not on the radius \((r)\).
✏️ note
on the spherical surface at a radial distance \(r\), the direction of the gravitational field \(g\) is orthogonal to the surface and oriented radially inward, as shown in the figure
\[ \begin{aligned} \int \vec{g} \cdot d \vec{A} & =\int \vec{g} \cdot \vec{n} d A \\ & =\int g d A \end{aligned} \] it's readily to see \(=g \int d A=g \cdot 4 \pi r^2\), Alternatively, we can elaborate further, from fig2-4: \[ \begin{aligned} d A & =d s \cdot 2 \pi r \sin \theta \\ & =2 \pi r \sin \theta d s \\ d s & =r d \theta \\ d A & =2 \pi r^2 \sin \theta d \theta \end{aligned} \] Subsequently, \[ \begin{aligned} \int g d A & =g \int 2 \pi r^2 \sin \theta d \theta \\ & =g\left(2 \pi r^2\right)[-\cos \theta]_0^\pi \\ & =g \cdot 2 \pi r^2 \cdot 2 \\ & =4 \pi r^2 g \end{aligned} \]
3) Generalization to Arbitrary Surfaces and Mass Distributions
Using the superposition principle and the divergence theorem:
- For any closed surface, the flux remains \((-4\pi G M_{\text{enclosed}})\).
- External masses contribute no net flux through the surface, as their inward and outward fluxes cancel.
Connection to the Divergence Theorem
By the divergence theorem, the surface integral of flux equals the volume integral of the divergence of \((\mathbf{g})\):
\[
\oint \mathbf{g} \cdot d\mathbf{A} = \int \nabla \cdot \mathbf{g} \, dV.
\] Combining this with Gauss's law:
\[
\int \nabla \cdot \mathbf{g} \, dV = -4\pi G \int \rho \, dV,
\] yields the differential form (Poisson's equation):
\[
\nabla \cdot \mathbf{g} = -4\pi G \rho.
\]