# Poisson Equation for a Scalar Field

1. Mar 7, 2012

### 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?