Gravitational Fields

Fields, potentials, and superposition

The gravitational field of a mass distribution is the force per unit test mass. Outside a spherical distribution of mass $M$,

The field is a vector defined at every location, independent of the test mass used to measure it. Its SI units are $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}$, equal to $\mathrm{m}{\mathrm{s}}^{-2}$. A test mass accelerates according to $\vec{a}=\vec{g}$ when gravity is the only force. This equality of acceleration for different test masses is an empirical statement about gravitational and inertial mass.

Field lines point in the direction of force on a positive test mass. For gravity they point inward toward an isolated mass. Their density in a diagram represents field magnitude qualitatively; field lines are not physical tracks followed by particles. A moving particle's path depends on its initial velocity as well as the field direction.

Reference potentials and measured differences.

Only differences in gravitational potential enter work and energy calculations. Replacing $V(\vec{r})$ by $V(\vec{r})+C$ leaves $\vec{g}=-\nabla V$ unchanged, because the gradient of a constant is zero. The reference is therefore a convention, not a measurable local property. For an isolated mass the standard convention is $V(\infty )=0$, giving $V=-GM/r$. Near Earth's surface, laboratory work often uses a local reference plane and writes $\Delta V\approx g\Delta y$. The two descriptions agree over small height ranges, but they should not be mixed carelessly. A local zero at sea level and a zero at infinity differ by a large constant.

Potential differences are directly connected with clock rates and height measurements in precision geodesy, although Newtonian mechanics alone treats them as energy per unit mass. A gravimeter measures acceleration, not absolute potential. To construct a potential map from measured acceleration, one must integrate along paths and impose a reference value at one point or at a distant boundary. The resulting map has a fixed arbitrary offset. A stated datum, such as a chosen equipotential surface, makes values from different surveys comparable.

The sign convention gives a direct physical check. Moving upward away from a positive mass makes $V$ less negative, so $\Delta V>0$, $\Delta U=m\Delta V>0$, and a slowly lifted test mass requires positive external work. Falling reverses all three work statements. A released mass does not move toward lower altitude because altitude is intrinsically special; it moves toward lower potential. On a rotating planet the effective potential also includes the centrifugal term, so a surface of constant geometric radius need not be an equipotential surface.

Far-field structure and multipole corrections.

At distances much larger than the size of a localized mass distribution, its potential has a hierarchy of scales. The leading term depends only on the total mass $M$ and falls as $1/r$:

This point-mass approximation is not a claim that the body is spherical. It is a statement about scale. If the observation distance is $r$ and the source has radius or separation scale $a$, departures from the leading term are suppressed by powers of $a/r$. Higher-order terms describe how mass is displaced relative to the chosen origin and how strongly the distribution departs from spherical symmetry. With the origin at the centre of mass, the mass-dipole term vanishes. The first important correction is then commonly a quadrupole term, associated with an elongated, flattened, or otherwise nonspherical distribution.

The far-field approximation gives a clean estimate of when a simplified model is adequate. A satellite hundreds of Earth radii away primarily responds to total Earth mass. A low satellite is sensitive to Earth's equatorial bulge and local mass anomalies, so a pure $-GM/r$ potential accumulates phase error over many orbits. The same distinction appears in binary systems: at a distance much larger than the binary separation, the pair resembles one mass; closer in, the changing direction and magnitude of the correction become dynamically significant.

Symmetry should be used before any series expansion. A spherical distribution has no correction outside its radius: the shell theorem makes the point-mass result exact. An axisymmetric but flattened body has a preferred axis, so its potential depends on both $r$ and polar angle. Its equipotentials are slightly distorted from spheres. The leading far-field term is insensitive to internal shape, while the smaller correction carries the shape dependence.

Measurement, gradients, and scale analysis

Orbital motion samples the gravitational field continuously. For a circular orbit in a spherical potential, the condition $v^2/r=GM/r^2$ fixes the speed and the period. In a more realistic potential, small departures from spherical symmetry change the orientation of an orbit, the timing of repeated ground tracks, and the relative motion of two nearby spacecraft. Measured deliberately, these departures carry information about the planet's internal mass distribution.

A gravity-gradient instrument compares acceleration at separated proof masses. If two masses lie a short radial distance $d$ apart, their common fall toward the planet is mostly irrelevant to the instrument. The signal is the difference, approximately $(\partial g_r/\partial r)d$. For $g_r=-GM/r^2$, the radial derivative has magnitude $2GM/r^3$. A separation transverse to the radial line has a different sign and factor. Measuring several orientations therefore gives more information than a single scalar gravimeter reading.

The scale dependence explains both the power and the difficulty of satellite gravity missions. Low altitude strengthens sensitivity to short-wavelength mass features, because their signals decay rapidly with distance. It also increases atmospheric drag, terrain-related modelling requirements, and sensitivity to orbit errors. High altitude produces a cleaner broad-scale signal but smooths away local structure. A reported anomaly must consequently state altitude, sampling interval, instrument baseline, and the background model removed before the residual was interpreted.

The radial arrows in the figure show a simple tidal measurement. Both proof masses accelerate inward. The lower mass, being closer to the source, has a slightly larger inward acceleration, so their separation tends to grow. The instrument reports this relative acceleration, not the full free-fall acceleration that every part of the spacecraft shares.

Error budgets and scale analysis.

An approximation should carry an estimate of the scale it neglects. The near-surface result $\Delta V\approx g\Delta y$ omits curvature of the inverse-square potential. Its fractional correction over a vertical range $h$ is of order $h/R$, where $R$ is the distance from the source centre. For a laboratory height this is tiny; for an altitude change comparable with a planet's radius it is not. The point-mass approximation outside a nonspherical body has a separate fractional error controlled by powers of source size divided by observation distance. These estimates decide the model before numerical values are substituted.

Measurements have their own error scales. An uncertainty $\delta r$ in position produces an uncertainty of roughly $2\delta r/r$ in the magnitude $GM/r^2$ when $GM$ is known. If $GM$ itself is fitted from orbit data, its uncertainty and the position uncertainty must be propagated together. Differencing two nearby values to obtain a gradient can enlarge random noise even while it removes a common offset. Smoothing suppresses that noise but also suppresses genuine short-scale features. A gravity map without stated resolution cannot distinguish a small physical anomaly from a processing artifact.

Dimensional analysis is a rapid defence against algebraic mistakes. Potential must have units of $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}={\mathrm{m}}^2{\mathrm{s}}^{-2}$; field must have units of $\mathrm{m}{\mathrm{s}}^{-2}$; a gradient of field has units of ${\mathrm{s}}^{-2}$. The tidal estimate $GM\ell /r^3$ has units of acceleration because multiplication by $\ell$ restores the missing length. An expression such as $GM/r^3$ is therefore a gradient scale, not an acceleration scale, until multiplied by a separation.

Two model curves can agree closely at one scale and diverge at another. Check a model against the required accuracy over the full range of distances, not at a single convenient reference point.

Order-of-magnitude estimates also expose numerical precision limits. When two large, nearly equal potential values are subtracted to find a small difference, the relative uncertainty of that difference can be much larger than the relative uncertainty of either value. It is often better to calculate a potential difference from an analytic expression, or to use a reference point near the region of interest, than to subtract two rounded absolute values. Conversely, a reference shift leaves a poorly resolved gradient unresolved: it changes every potential value by the same constant and leaves all finite differences unchanged. Separating these two issues—choice of datum and numerical resolution—keeps error analysis physically meaningful.

Potential, work, and spherical distributions

Newtonian gravity obeys superposition. The field of several point masses is the vector sum

Components must be added, not magnitudes. At a midpoint between equal masses, the individual fields are nonzero but cancel. At points away from the symmetry line, their transverse components can add while longitudinal components cancel.

Gravitational potential is potential energy per unit mass. With its zero at infinity,

Potential is scalar, so several source potentials add algebraically. The field is the negative spatial gradient: it points in the direction of decreasing potential. Equipotential surfaces for a point mass are concentric spheres. Motion along one equipotential requires no gravitational work because the displacement is perpendicular to the field.

Work and near-surface approximation.

The exact gravitational potential-energy difference between radii $r_i$ and $r_f$ is

Near a planet surface, write $r=R+h$ with $h\ll R$. Expanding the reciprocal gives $V_g(R+h)\approx -GM/R+gh$, where $g=GM/R^2$. The constant first term has no physical effect on force, and the potential-energy difference becomes $mg\Delta h$. The approximation deteriorates over satellite distances or deep vertical shafts.

Potential differences directly determine energy changes. A negative potential records the chosen zero at infinite separation and the work required to remove a bound mass to infinity. The escape condition is total energy zero, not potential energy zero at launch.

Spherical mass distributions.

The shell theorem has two results. Outside a spherically symmetric shell or solid body, its field equals the field of a point mass equal to the total mass placed at the centre. Inside a thin spherical shell, the net field is zero. Contributions from near shell elements are stronger but cover less solid angle. The solid-angle factor balances the weaker contributions from more distant larger areas.

Inside a uniform solid sphere of radius $R$, only enclosed mass contributes:

The field grows linearly from zero at the centre to $GM/R^2$ at the surface. The potential remains finite and has a minimum at the centre. A tunnel through an ideal uniform planet therefore produces simple harmonic motion in the absence of air resistance and rotation, because force is proportional to displacement toward the centre.

Measurement and limitations.

Local $g$ is measured with pendulums, falling bodies, or gravimeters. It varies with altitude, latitude, planet rotation, local density structure, and terrain. The effective weight measured on a rotating planet includes centrifugal effects and therefore differs slightly from the Newtonian gravitational field. Geophysical surveys use small anomalies in measured $g$ to infer density contrasts below the surface.

Newtonian fields give an excellent description in weak gravity and ordinary velocities. General relativity modifies gravity in precision timing, strong fields, and cosmological settings. The field-potential framework remains the correct Newtonian language for the scales assumed in this chapter.

Derivation from conservative work.

Moving a test mass radially from infinity to $r$ gives gravitational work

Potential-energy change is the negative of conservative-force work, producing $U(r)=-GMm/r$. The integral does not diverge at infinity because the inverse-square force decreases sufficiently rapidly. The work required from an external agent for a slow outward displacement is positive and equals the increase in potential energy. A rapid displacement can have different external work because kinetic energy also changes; potential difference remains a property of endpoints.

The relation $\vec{g}=-\nabla V_g$ in spherical symmetry reduces to

The radial component is negative because outward $r$ is positive while gravity points inward. Writing only the positive magnitude $GM/r^2$ is acceptable when direction is separately specified, but it cannot be substituted into a signed vector equation without restoring the sign and unit vector.

Superposition examples.

Two equal point masses separated by distance $2a$ have zero field at their midpoint. Their potential there is not zero:

This contrast illustrates the scalar-vector distinction. Field cancellation means the first spatial derivative of potential is zero at that point; it does not mean the potential itself vanishes. A small test mass placed exactly at the midpoint has no initial acceleration, but the equilibrium can be unstable in the direction joining the masses and stable in transverse directions only under additional constraints. Local field value alone does not determine the global potential function.

For continuous mass distributions, replace the sum by

The potential integral is often easier because it has no vector direction inside the integrand. Differentiating a symmetric potential afterward recovers the field. This method is particularly efficient for rings, disks, and spherical shells.

Shell theorem scope and interior potential.

The shell theorem requires exact spherical symmetry. It does not apply to a hemisphere, an oblate planet, or a shell with nonuniform surface density. Outside any spherically symmetric body, the point-mass result applies even if density varies with radius. Inside a hollow shell, the field is zero but potential is a nonzero constant $-GM/R$, where $R$ is shell radius. A zero field therefore does not imply zero potential energy.

Inside a uniform solid sphere, integration or matching at the surface gives

At $r=R$, this equals $-GM/R$ and its derivative matches the exterior field. At the centre, $V(0)=-3GM/(2R)$, lower than the surface potential. The local force on a mass at position $\vec{r}$ is $-GMm\vec{r}/R^3$, which has the same form as a three-dimensional isotropic harmonic oscillator.

Escape, field energy, and numerical checks

Escape is a consequence of potential difference. Starting at radius $r$ with speed $v$, a mass has specific mechanical energy

The boundary between bound and unbound motion is $\epsilon =0$, giving $v_{\mathrm{esc}}=\sqrt{2GM/r}$. A negative potential does not somehow pull energy from the object; it records energy that must be supplied to separate the object to infinite distance. A rocket can reach a bound orbit without reaching escape speed, and an object can exceed escape speed yet strike a planet if its trajectory intersects the surface.

The gravitational field itself stores energy in a broader field-theory treatment, but Newtonian introductory mechanics defines potential energy for the interacting mass configuration. This convention is sufficient for work and orbit calculations provided one system boundary and one zero reference are retained consistently.

Vector superposition in a gravitational field

Superposition is most transparent when each source contribution is written as a vector before any symmetry argument is used. For point masses $m_i$ at positions ${\vec{r}}_i$, the field at observation point $\vec{r}$ is $\vec{g}(\vec{r})=-G{\sum }_i{m}_i(\vec{r}-{\vec{r}}_i)/|\vec{r}-{\vec{r}}_i{|}^3$. The denominator contains the cube of separation because the numerator contains one factor of separation direction. This form supports component calculations: it keeps the attractive direction and inverse-square magnitude in one expression.

Two equal masses illustrate why magnitudes must not be added. Put the masses at $(\pm a,0)$ and evaluate the field at $(0,y)$. The two separation lengths are equal. Their horizontal field components have equal magnitude and opposite signs, so they cancel. Their vertical components both point downward toward the masses and add. The net field therefore lies on the symmetry axis, with magnitude $2GMy/(a^2+y^2{)}^{3/2}$. At the midpoint $y=0$, every component cancels and the field is zero, although the potential remains negative.

Superposition applies to potential more simply because potential is scalar. At the same point, $V=-GM/\sqrt{a^2+y^2}-GM/\sqrt{a^2+y^2}$. No components are needed. The field can then be found by differentiating this potential with respect to position. This route is often faster for a continuous symmetric source: integrate the scalar contribution first, then take the gradient after symmetry has reduced the variables.

Symmetry arguments must include both source arrangement and observation point. Two identical masses have a zero field only at their midpoint, not everywhere on the perpendicular bisector. A ring has zero field at its centre but a nonzero axial field away from the centre. The appropriate statement is that a symmetry operation leaves the source distribution unchanged while changing the sign of a possible field component; that component must then vanish. Components not reversed by the symmetry can remain nonzero.

Potential gradients and field measurements

Potential difference is the work per unit mass required to move slowly against gravity. Between two points A and B, $V_B-V_A=-{\int }_A^B\vec{g}\cdot \mathbf{d}\vec{\ell }$. The line integral depends only on endpoints because Newtonian gravity is conservative. Along a radial path outside a spherical mass, integration of the inverse-square field gives $V=-GM/r$ after choosing zero potential at infinity. Along a tangential displacement on a sphere of fixed radius, the dot product is zero, so potential does not change.

The field is the spatial rate of potential decrease. In one radial coordinate, $g_r=-\frac{\mathbf{d}V}{\mathbf{d}r}$. A steep potential curve corresponds to a large field magnitude; a flat curve corresponds to a weak field. The negative sign is essential. As radius increases, the potential of an isolated mass rises toward zero, so its derivative is positive while the gravitational field points inward and has negative radial component. A joint potential-and-field plot displays this sign relation without relying on a memorized arrow convention.

Potential measurements can be inferred from work or energy differences even when a direct force measurement is difficult. A falling object exchanges gravitational potential energy for kinetic energy, while a slowly raised calibration mass measures potential difference through external work. The reference value of potential is arbitrary, but potential differences and field gradients are not. Shifting every potential value by one constant changes neither field nor motion.

Equipotential surfaces support a geometric field measurement. The field is normal to each equipotential because a tangential component would do work during an equipotential displacement and change potential. Closely spaced equipotentials indicate a larger field magnitude for equal potential increments. This convention is qualitative unless the contour interval is stated; a diagram with uneven contour increments cannot use spacing alone as a magnitude scale.

Spherical shells and interior behavior

The shell theorem separates field behavior from potential behavior. Inside a thin spherical shell, the gravitational field is zero at every interior point. The potential is not zero; it is the constant $-GM/R$ when the shell has mass $M$ and radius $R$. Moving a test mass anywhere inside therefore requires no gravitational work, but taking it from the shell interior to infinity still requires positive external work. Zero field means zero local gradient of potential, not zero potential itself.

Outside a shell, spherical symmetry makes the field identical to that of a point mass at the centre. The result does not say that individual shell elements exert zero force. Near elements exert stronger forces, but far elements subtend larger areas and supply compensating components. The exact cancellation requires a closed shell with uniform spherical symmetry. A cap, a hollow hemisphere, or a nonuniform shell has a nonzero interior field and must be integrated without invoking the theorem.

Inside a uniform solid sphere, the situation differs because mass is enclosed at every radius. The shell theorem applied to nested shells leaves only the mass inside the observation radius contributing to field. Since enclosed mass grows as $r^3$, the field magnitude grows as $r$. The potential is a smooth quadratic function of radius, with its lowest value at the centre. This linear restoring field is why an ideal straight tunnel through a nonrotating uniform planet produces simple harmonic motion rather than the inverse-square motion found outside.

Shell behavior is also a caution about local-density intuition. Material outside a radius can contribute to potential even when it contributes no net field there. A geophysical gravity measurement is sensitive to departures from spherical symmetry, but potential and field interpretations require the chosen reference and spatial model. Treating all mass beyond an observation radius as irrelevant is correct for the shell-theorem field in a spherical model, not for potential energy or for an irregular real planet.

Near-surface approximation and its limits.

Near a spherical body's surface, write the radial coordinate as $r=R+h$ with $|h|\ll R$. The exact field magnitude is $g(h)=GM/(R+h{)}^2=g_0(1+h/R{)}^{-2}$, where $g_0=GM/R^2$. Expanding the reciprocal gives $g(h)\simeq g_0(1-2h/R)$ to first order. The common constant-$g$ model drops the correction entirely and is accurate only when the fractional height $h/R$ is small compared with the required precision.

Potential energy gives a related test. The exact rise from surface to height $h$ is $\Delta U=GMm[1/R-1/(R+h)]$. The approximation $mgh$ is the first term of this expression. Its fractional error is of order $h/R$. For a one-kilometre elevation change at Earth radius, the correction is about $1.6\times 10^{-4}$ and is usually smaller than ordinary measurement uncertainty. For a low-orbit altitude of $400\mathrm{km}$, the field magnitude is about $8.7\mathrm{m}{\mathrm{s}}^{-2}$, more than ten percent below the surface value, so constant $g$ is no longer suitable.

The direction of the approximation matters in long vertical systems. A deep mine, high-altitude balloon, or tall atmospheric column samples varying radius and, for Earth, varying density and rotation. At shallow depth inside a uniform-density ideal sphere, field magnitude decreases rather than increases because less mass is enclosed. Real Earth density rises with depth over some ranges, so a simple uniform-sphere formula is itself an approximation. Local gravimetry uses a model that separates altitude correction, latitude correction, terrain, and density anomalies.

Approximation checks should compare scales before equations are simplified. State the reference radius, maximum height or depth, and tolerated fractional error. If the correction is comparable with sensor uncertainty, retaining the exact formula may add complexity without improving a measured prediction. If the correction is larger, the constant-$g$ model can give systematically wrong work, pressure, and orbital estimates even when the algebra is otherwise correct.

Dimensional analysis checks every field expression. $GM/r^2$ has units $\mathrm{m}{\mathrm{s}}^{-2}$; $GM/r$ has units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, or joules per kilogram. The field is the derivative of potential with respect to distance, so it has one additional inverse length. A proposed potential proportional to $1/r^2$ would yield a $1/r^3$ force and cannot represent Newtonian point-mass gravity.

Effective gravity and orbital motion

On a rotating planet, a scale measures the contact force needed to keep an object at fixed latitude. In the rotating frame, centrifugal acceleration reduces the apparent downward field by ${\omega }^{2}R{\cos }^2\lambda$ in the simplest spherical model, where $\lambda$ is latitude. Planetary oblateness also changes the true gravitational field. Consequently, standard $g$ is a conventional reference value, not an exact field magnitude at every geographic location.

The distinction between field and effective weight is important in orbit. An astronaut in circular free fall still experiences a substantial gravitational field. The support force is nearly zero because spacecraft and astronaut accelerate together. Local tidal field differences, however, remain and become important for large spacecraft or very low orbits.

Relation to orbital motion.

Field and potential give complementary orbital descriptions. The field equation sets the instantaneous acceleration in Newton's second law. The potential allows a global energy relation between two radii. In a circular orbit the field sets $v^2/r=GM/r^2$, while the potential gives the negative total energy. Neither description is more fundamental in this conservative central-force problem; each compresses a different part of the calculation.

For noncircular motion, angular momentum gives the missing directional constraint. The effective potential combines the gravitational potential with $L^2/(2m{r}^2)$, converting the radial part of the orbit problem into one-dimensional energy motion. Its centrifugal term is not a new physical outward force in an inertial frame; it is the energy associated with transverse velocity when angular momentum is held fixed.

Model errors to avoid.

The formula $g=GM/r^2$ gives a magnitude outside a spherical body, not a universal constant. Using $g=9.8\mathrm{m}{\mathrm{s}}^{-2}$ at every altitude overestimates gravity far from Earth. Conversely, applying the point-mass formula inside a uniform planet is incorrect because exterior shells cancel there. Potential and field must share the same reference: adding a constant to potential changes no field, but mixing two different zero conventions corrupts energy differences.

Field lines cannot cross at a point where the field is nonzero, since that would assign two directions to one vector. Equipotential surfaces cannot cross for the same reason: a scalar potential has one value at one location. These graphical checks are simple but catch many incorrect superposition sketches.

The same logic applies to numerical simulation. Grid values of potential should vary smoothly away from point sources, and finite-difference estimates of the field should point toward decreasing potential. Near a singular point mass, a finite grid cannot represent the divergence exactly; calculations either exclude the source location or replace it with a finite-size mass distribution. The regularization is a computational choice and must not be confused with a change to the physical inverse-square law outside the source.

The field, potential, and numerical gradient must remain mutually consistent.

Multiple masses and configuration energy

With a collection of isolated spherical masses, potential is usually the most economical starting point. Under the conventional reference $V(\infty )=0$,

Each contribution is a scalar and is negative. No direction has to be assigned while the sum is formed. The gravitational field follows afterwards from $\vec{g}=-\nabla V$. This order is valuable near a point at which the vector contributions nearly cancel. Two equal masses give zero field at the midpoint, yet the potential there is not zero; it is the negative sum of two finite contributions. A stationary test mass placed there is in an unstable equilibrium along the line joining the masses and in a stable direction perpendicular to that line only if other constraints are imposed. The local curvature of $V$, rather than the value of $V$ alone, decides the behaviour of a small displacement. Local curvature determines the nearby response.

Equipotential curves offer a compact picture of this sum. Where neighbouring curves crowd together, the magnitude of $-\nabla V$ is large. Their spacing is not itself a force scale unless the potential interval between adjacent curves is stated. In a symmetric two-mass diagram the curves pinch between the sources, whereas very far away their shape approaches the circular or spherical pattern of one mass equal to $M_1+M_2$.

The work needed to move a test mass slowly from $A$ to $B$ is $W_{\mathrm{ext}}=m[V(B)-V(A)]$. This result is independent of the route, provided the sources remain fixed. A route with long curved segments may involve more distance but not more net work. Along an equipotential segment the displacement is perpendicular to $\vec{g}$, so the gravitational work is zero there. The external agent supplies positive work when a mass is raised to a less negative potential; gravity supplies the same amount of positive work during the reverse motion.

Gravitational gradients, tides, and numerical maps.

A gravitational field is not fully described by its value at one point. Its spatial variation is the gravitational gradient. Across a body of size $\ell$ at distance $r$ from a nearly spherical source, the variation is of order $GM\ell /r^3$. The factor is important: a planet and a nearby spacecraft may share almost the same acceleration and still experience measurable relative acceleration. In the radial direction the nearer side is pulled more strongly and the farther side less strongly; in transverse directions the differential pull is inward toward the central line. These differences, rather than the common acceleration of the body's centre, are the tidal effects relevant to ocean tides, stretched satellite structures, and Roche-limit estimates.

Potential sampled on a rectangular grid with spacing $\Delta x$, a centred estimate is

with an analogous expression for $g_y$. Smaller spacing captures sharper variation but amplifies measurement noise when neighbouring potential values are subtracted. Near a point source, the grid must either omit the singular location or use a resolved finite body. A map is trustworthy only after the grid spacing, reference level, interpolation rule, and source model have been stated.

Gravity surveys follow this logic. A measured anomaly is compared with a smooth reference model; local differences may indicate denser rock, a cavity, or unmodelled topography. The inference is not unique. Different distributions of mass can produce very similar values above the surface, so independent geology and uncertainty estimates are part of a responsible interpretation. Numerical maps describe the measured potential or acceleration; they do not by themselves identify a unique underground structure.

Two numerical checks apply. Away from sources, the potential of a model that contains no mass in a grid cell should satisfy the discrete version of Laplace's equation: its value is close to the average of neighbouring values. At a cell containing a prescribed density, the corresponding departure from that average has the sign required by Poisson's equation, ${\nabla }^{2}V=4\pi G\rho$. These tests distinguish a physically meaningful map from a visually smooth map with no physical consistency. Setting $V=0$ on the edge of a small computational box makes the sources appear too close to an artificial zero-potential surface. Enlarging the box or imposing an exterior multipole approximation reduces that error.

The same map can be checked by a closed-route work calculation. Add $\vec{g}\cdot \Delta \vec{r}$ around a small rectangular loop. For a static Newtonian gravity model the result should approach zero as the grid is refined. A persistent nonzero loop sum signals numerical differentiation error, rounding error, or data that have been combined from incompatible reference systems. The check does not prove that the inferred mass model is unique; it verifies only that the reported potential and acceleration are mutually consistent.

Conductors and the limits of the gravity analogy.

Electrostatics supplies a mathematical comparison and a common source of false conclusions. Both inverse-square laws admit scalar potentials and superposition. Outside a spherical charge distribution or a spherical mass distribution, symmetry makes the result identical in form to that of a point source at the centre. A hollow spherical gravitational shell has zero net field everywhere inside, just as an ideal conducting cavity can have zero electric field under restricted electrostatic conditions.

The mechanisms are not the same. Charge comes in positive and negative signs and can move through a conductor. Mobile charge redistributes until the conductor is an equipotential, and surface charge can change the external electric field. Ordinary mass has one observed sign and does not rearrange in response to a gravitational pull in an analogous way. A massive shell produces zero interior field because the attractions from its fixed elements cancel by symmetry, not because mass has migrated to make its interior safe from gravity. There is no gravitational counterpart of electrostatic shielding made from ordinary matter.

The signs of the potentials also differ in their usual conventions. A positive electric source has positive potential and repels a positive test charge. A positive mass has negative gravitational potential when $V(\infty )=0$ and attracts a positive test mass. The shared equation structure supports Laplace-type boundary problems and equipotential maps, but physical language about conductors, induced charge, and screening must not be imported into gravity without a separate mechanism.

Work paths and configuration energy.

The path-independence statement can be tested directly. Break a route from $A$ to $B$ into short displacement vectors $\Delta {\vec{r}}_k$ and add $m{\vec{g}}_k\cdot \Delta {\vec{r}}_k$. As the steps become small, this sum tends to the same value for every route: $-m[V(B)-V(A)]$. A numerical calculation that gives different values for two routes through a fixed gravity model has usually used inconsistent signs, steps that are too large near a source, or a field that was not obtained from a single potential.

The distinction between potential and potential energy matters here. $V$ belongs to the source arrangement and has units of joules per kilogram. The potential energy of a specified test mass is $U=mV$. For several gravitating bodies, the total configuration energy is assembled one pair at a time,

The restriction $i<j$ is essential: summing over every ordered pair counts each interaction twice. This formula also records the reference condition implicitly. The energy is zero when every separation is infinite, and it is negative for a bound collection of ordinary positive masses. Bringing masses together from infinity releases energy; separating them requires external work.

With a planet treated as a fixed source, a radial transfer is often simplest to calculate. A curved route is equally legitimate. In the sketch, the curved part from $A$ to the right changes potential only when it crosses dashed circles; a motion around one dashed circle changes neither $V$ nor $U$. Equipotential maps therefore support surveying, orbital analysis, and numerical checks by converting a vector-work problem into differences between labelled scalar levels.