Poisson Equation for a Scalar Field

[matteo86bo]

We all know that for the gravitational field we can write the Poisson Equation:
$\nabla^2\phi=-4\pi G\rho$
But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as $r^{-2}$.
Here is the thing. When you deal with gravity, the Poisson equation is derived from the Gauss's law for gravity:
$\int_{\partial V}\dfrac{GM}{r^2}\cdot d\vec{S}=4\pi G M$
Then we apply the Gauss's law and we get the differential form of the Poisson equation:
$\nabla\cdot\vec{f}=4\pi G\rho$

My question is: suppose that we have a scalar field
$p=\dfrac{L}{4\pi r^2}$
Can we make an analogy between this field and the gravitational force and write a Poisson equation for this field in the following form?
$\nabla\cdot \vec p=l$
where $L=\int l dV$

My question might also be interpreted as: can we apply the Gauss's theorem to a scalar field?

 

[gtoguy97]

The answer to this question is yes, you can indeed apply the Gauss's theorem to a scalar field. For example, consider the electric field of a point charge:\vec E = \frac{Q}{4\pi \epsilon_0 r^2} \hat rwhere Q is the charge of the point source and $\epsilon_0$ is the permittivity of free space. Applying the Gauss's theorem to this scalar field yields:\oint_{\partial V} \vec E\cdot d\vec S = \frac{Q}{\epsilon_0}So, to answer your question, yes, you can write a Poisson equation for a scalar field that scales as $r^{-2}$. The equation would be:\nabla \cdot \vec p = l where $L = \int l dV$.