Poisson Equation for a Scalar Field

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

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

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