Gravity Potential Theory and Earth's Gravitational Field

Core Idea

The gravitational potential U satisfies Laplace's equation ∇²U = 0 in mass-free regions and Poisson's equation ∇²U = −4πGρ in regions with density ρ. The gravity field g = −∇U and gravitational anomalies arise from lateral density variations in the crust and mantle. Forward modeling of gravity anomalies allows estimation of crustal thickness, density structure, and subsurface mass distribution; inverse methods recover density models from observed gravity data.

Explainer

Gravity potential theory extends the point-mass formula from classical mechanics to the full, continuous density distribution of the Earth. Instead of summing the gravitational pull of individual mass points, we define a scalar field U at every point in space such that the gravitational acceleration vector g = −∇U. This means you can recover the direction and magnitude of gravity everywhere by taking the spatial gradient of a single scalar quantity — a powerful simplification that draws directly on the potential theory framework you learned in mathematics.

In mass-free regions (above the surface, in air, or in low-density rock), U satisfies Laplace's equation ∇²U = 0. Where matter is present with density ρ, the equation becomes Poisson's equation ∇²U = −4πGρ. These two equations are not different physics — Poisson's equation reduces to Laplace's when ρ = 0. The analogy with electric potential is close: just as electrostatic potential satisfies Laplace's equation in charge-free space and Poisson's equation where charge exists, gravitational potential obeys the same mathematical structure (with mass density replacing charge density and G replacing 1/ε₀).

The practical power of this framework lies in gravity anomalies — departures from the expected gravity of a smooth, idealized reference Earth (the normal gravity field). If the crust beneath your gravimeter is unusually dense (like a buried iron ore deposit), the observed gravity will exceed the reference value: a positive anomaly. If the crust is unusually thin or contains a low-density salt dome, gravity will fall below reference: a negative anomaly. The shape and magnitude of the anomaly encode information about the depth, geometry, and density contrast of the causative body.

Forward modeling works from cause to effect: given an assumed density structure, compute the predicted gravity field by integrating Poisson's equation. This is unique and mathematically tractable. The inverse problem — recovering density structure from observed anomalies — is fundamentally non-unique: infinitely many density distributions can produce the same surface gravity field, because gravity measurements at the surface cannot distinguish a shallow weak density contrast from a deep strong one. Resolving this ambiguity requires additional constraints from seismic data, borehole samples, or geological reasoning. This non-uniqueness is not a limitation of our methods but a mathematical property of potential fields, and managing it is central to applied geophysics.