Poisson's Equation and Gravitational Boundary Data

Mass density determines the Laplacian of the Newtonian potential, not the potential by itself. A gravitational boundary-value problem combines $\nabla^2\Phi=4\pi G\rho$ with boundary data: Dirichlet data fix $\Phi$, Neumann data fix its outward derivative, and an isolated system imposes decay at infinity. Those choices select one field from the family differing by harmonic functions.

Required background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Ordinary Differential Equations provides initial-value problems, linearization, and asymptotic solution methods. Helpful background. Partial Differential Equations and Green Functions provides characteristics, boundary data, distributions, and Green-function methods.

Poisson’s equation as a boundary-value problem

On a connected region $\Omega\subset\mathbb R^3$,

$$\nabla^2\Phi=4\pi G\rho, \qquad \mathbf g=-\nabla\Phi.$$

Integrating over $V\subset\Omega$ gives

$$\oint_{\partial V}\nabla\Phi\cdot\mathbf n\,dA =4\pi G\int_V\rho\,d^3x,$$

or $\oint\mathbf g\cdot\mathbf n\,dA=-4\pi GM_V$. Positive mass thus produces inward flux. This integral statement remains valid when a point source makes $\Phi$ nonsmooth.

On a bounded domain, typical conditions are

$$\Phi|_{\partial\Omega}=f \quad\text{or}\quad \partial_n\Phi|_{\partial\Omega}=h.$$

Neumann data must obey

$$\oint_{\partial\Omega}h\,dA =4\pi G\int_\Omega\rho\,d^3x,$$

the integrated field equation. They determine $\Phi$ only up to a constant. On a disconnected domain, this compatibility condition and the undetermined constant apply separately to each connected component.

Uniqueness from an energy identity

If $\Phi_1$ and $\Phi_2$ solve the same source problem, set $w=\Phi_1-\Phi_2$. Then $\nabla^2w=0$. Green’s first identity gives

$$\int_\Omega |\nabla w|^2d^3x =\oint_{\partial\Omega}w\,\partial_nw\,dA -\int_\Omega w\nabla^2w\,d^3x.$$

For equal Dirichlet data, $w=0$ on the boundary; for equal Neumann data, $\partial_nw=0$. The right side vanishes, so $\nabla w=0$. Dirichlet data then force $w=0$, while Neumann data leave one constant. This proof shows exactly why sources without boundary behavior are insufficient.

Uniqueness is not existence. The energy argument says that two sufficiently regular solutions cannot differ, but constructing a solution still requires compatible source, domain, and boundary regularity. Corners, discontinuous densities, point sources, and unbounded domains may demand weak or distributional solutions. In physical applications the integral flux law is often the most robust check across such singularities, while the pointwise equation is applied only away from them.

Green function and its sign

In free space,

$$\nabla_{\mathbf x}^2G_0(\mathbf x,\mathbf y) =\delta^{(3)}(\mathbf x-\mathbf y), \qquad G_0=-\frac{1}{4\pi|\mathbf x-\mathbf y|}.$$

The sign is checked by integrating over a small sphere: $\partial_r[-1/(4\pi r)]=1/(4\pi r^2)$, whose outward flux is $+1$. Therefore

$$\Phi(\mathbf x)=-G\int \frac{\rho(\mathbf y)}{|\mathbf x-\mathbf y|}\,d^3y.$$

On a bounded domain, a Green function must be adapted to the chosen boundary data. Skinner 2015, §10.3.3, pp. 136–137, PDF constructs the Dirichlet case and sketches a Neumann analogue, but its literal simultaneous conditions $\nabla^2G=\delta$ and $\partial_nG=0$ are incompatible after integration over $\Omega$. A valid Neumann construction must instead handle the constant zero mode consistently with the compatibility condition derived above. For a compact source at large $r=|\mathbf x|$, define the observation direction $\mathbf n=\mathbf x/r$. Then

$$\Phi=-G\left[ \frac{M}{r}+\frac{\mathbf n\cdot\mathbf D}{r^2}+O(r^{-3}) \right],$$

with $\mathbf D=\int\rho\mathbf y\,d^3y$. Choosing the center of mass as origin sets $\mathbf D=0$. Reall 2022, §2.6, pp. 29–30, PDF derives the same localized-source expansion for electrostatics; replacing charge density by mass density and the Coulomb coupling by $-G$ gives the displayed gravitational formula. Tong, Vector Calculus, §§5.1–5.2.3, pp. 92–106, PDF derives Newtonian Poisson gravity, its boundary conditions and uniqueness result, and the free-space Green-function solution.

The multipole expansion has two separate controls that should not be conflated. The source must fit within a bounded region of characteristic radius $a$, and the observation point must satisfy $r\gt a$; truncating after the dipole then incurs terms controlled by powers of $a/r$. Centering the coordinates removes the mass dipole, but it does not make the quadrupole vanish. Conversely, boundary-induced harmonic fields need not be determined by the source multipoles at all. A numerical solution in a finite box must therefore document its outer boundary condition before its far field can be compared with the isolated-space formula.

Worked example: a uniform sphere

Let a sphere of radius $R$ have constant density $\rho=3M/(4\pi R^3)$ and $\Phi\to0$ at infinity. Spherical symmetry gives inside

$$\frac1{r^2}\frac{d}{dr}\left(r^2\frac{d\Phi}{dr}\right)=4\pi G\rho.$$

Regularity at $r=0$ removes the $1/r$ homogeneous term, so $d\Phi_{\rm in}/dr=GMr/R^3$. Outside, decay gives $\Phi_{\rm out}=-GM/r$. Continuity of $\Phi$ and its derivative at a surface without a singular shell yields

$$\Phi(r)= \begin{cases} -\dfrac{GM}{2R^3}(3R^2-r^2),&r\le R,\\[4pt] -\dfrac{GM}{r},&r\ge R. \end{cases}$$

The interior field $\mathbf g=-GM\mathbf r/R^3$ is regular and matches the exterior field at $R$. Every term in $\Phi$ has dimension $L^2T^{-2}$.

There are three independent checks. Applying the radial Laplacian inside returns $3GM/R^3=4\pi G\rho$. Outside it returns zero for $r\gt0$. Finally, the derivative at the surface is $GM/R^2$ from both sides, so there is no delta-function surface mass. Had $d\Phi/dr$ jumped, a pillbox flux integral would identify the corresponding idealized surface density. These checks test the equation, the sign, and the matching condition separately.

Solving Laplace’s equation outside a source does not determine its multipoles without source matching or boundary data. Arbitrary Neumann data may violate flux compatibility; delta sources must be interpreted distributionally. Time-dependent Poisson theory is still instantaneous.

Tidal Gravity and Relative Acceleration extracts the local field variation that nearby test bodies can measure.

References

• Reall, Harvey S. Part IB Electromagnetism. University of Cambridge lecture notes, 2022. §2.6, pp. 29–30, PDF.
• Skinner, David. Mathematical Methods. University of Cambridge Part IB lecture notes, Michaelmas 2015. §10.3.3, pp. 136–137, PDF.
• Tong, David. Vector Calculus. University of Cambridge lecture notes, n.d. §§5.1–5.2.3, pp. 92–106, PDF.