Analytical solution to Poisson's equation for gravity

Question

I am studying Poisson's equation for gravity. $$\nabla^2 \varphi = 4\pi G\rho$$ I have read that it is solved analytically using some Green's function, to give the well known formula of potential $$ \varphi(r) = -\frac{Gm}{r} $$ But I have not studied the Green's function. So, can anybody tell me how to solve this Poisson's equation?

Answer 1

I am studying Poisson's equation for gravity. $$\nabla^2 \varphi = 4\pi G\rho$$ I have read that it is solved analytically using some Green's function

The intuition behind Green's functions is that they act as propagators. One wants to write the potential $\varphi$ as

$$\varphi=\int{G(r,r')\rho(r)}d^3x$$

Plugging this into the Poisson equation one can notice that the Green function has to obey the following differential equation

$$\nabla^2 G(r,r') = 4\pi G\delta (r-r')$$

The above equation can be solved by using the Fourier transform technique. You'll get something like

$$G(r,r') \propto -\frac{1}{|r-r'|}$$

Which then yields the desired result

$$\varphi=\int{G(r,r')\rho(r)}d^3x=-\int{\frac{G\rho(r)}{|r-r'|}}d^3x$$

Answer 2

Modifying the general solution for the Poisson equation, presented in the article of Wikipedia, for the gravitational case where: $f(\mathbf{r}') = 4\pi G\rho$ (the density being in general a function of $\mathbf{r}'$)

$$\varphi(\mathbf{r}) = - \iiint \frac{G\rho}{ |\mathbf{r} - \mathbf{r}'|}\, \mathrm{d}^3\! r', $$

This expression can be understood as the sum of the contribution of all mass elements for the total potential on a given coordinate $\mathbf{r}$.

$G\rho \mathrm{d}^3\! r' = G\rho dV = Gdm$

For certain cases, as the field of a spherical symmetric mass over an external point, the integral can be analytically solved, resulting in a function of the distance to the center of the sphere:

$$\varphi(\mathbf{r}) = -\frac{Gm}{r}$$

The only problem is that if we substitute this solution on left side of the Poisson equation the result is zero, not $4\pi G\rho$. It can be interpreted in my opinion as an anomally, due to fact that the field has the same value if all mass is at a singularity in the center. And in this case, $\rho$ is not well defined. It is zero everywhere and infinite at the singularity.