In the previous article, “Rearrange the Inside of a Box and the Gravity Outside Does Not Change at All”, what I found most interesting was that it holds not only for spheres but for all kinds of shapes, such as a cube. Wrap a thing in a closed skin, and whatever happens inside, only a fixed set of numbers can be read from outside. I think things are made up of modules.
Then, for a module whose shape is not a sphere, how small can the contents be gathered? For a sphere, gravity seen from outside is the same as if all the mass were placed at the single point in the middle (Newton’s shell theorem). This page checks this by shrinking an ellipsoid within its confocal family (explained below). The end point was a flat disc.
The calculations in this article were made for this page or come from papers that I published myself as preprints; they have not been peer reviewed.
Key points
- Seen from outside, the gravity of a uniformly filled ellipsoid is the same as that of a flat elliptic disc placed inside it. For an ellipsoid with semi-axes 3, 2, 1, the core is a disc with semi-axes √8 and √3, and the relative difference in the field outside was of order 10⁻¹⁴.
- Ellipsoids obtained by subtracting the same number from the squared semi-axes (confocal ellipsoids) cannot be told apart from outside if they have the same mass. The numbers per angular pattern that can be read from outside are also the same.
- The core reached by squeezing as far as possible while staying confocal is a point for a sphere, a segment for a prolate spheroid and a flat disc for an ellipsoid. The dimensions of the core are 0, 1 and 2, and the shell theorem was the smallest case, in which the core shrinks to a single point.
- It is not only inside a spherical skin that the force is 0. For an ellipsoidal skin too, if the thickness is made proportional to the distance from the centre to the tangent plane, the force inside is 0. With the same thickness everywhere, it is not 0. On a sphere, the two ways of painting give the same skin.
- The confocal core and the flat inside of the homoeoid hold when the field obeys the massless Laplace equation (when the force weakens as the square of the distance). They break down for a force that weakens as the cube, and for the Yukawa field, which has mass even though its equation is of second order. Only the sphere still gives, from outside, a field of the same form as a point even for the Yukawa field.
- A sphere is a point
- Confocal
- The core
- Force inside
- Why 0
- Sharing out 4π
- If not the inverse square
- The Earth
- Modules
- Separator
- Questions
- References
From outside, a sphere is a single point
Outside a uniform ball of mass 1, gravity is the same as for a point of mass 1 placed at the middle, and it weakens as the square of the distance. This is Newton’s shell theorem. Paper 1114 found by calculation that four things, a point, a skin, a 1/r² ball and a uniform ball, give the same field outside and split apart inside. For a sphere, shrinking the contents to a single point does not change the outside.
For shapes other than a sphere, it does not work that way. As Paper 1107 showed, what can be read from outside is one number per angular pattern. Besides the total mass (pattern 0), the numbers of an elliptical stretch (pattern 2) and of finer patterns also go out. A single point cannot give out all of them. How far, then, can the contents be shrunk?
Shrinking while staying confocal does not change the outside
Square the three semi-axes of an ellipsoid and subtract the same number λ from each. The ellipsoid made this way shares its foci with the original one, so it is called a confocal ellipsoid. Subtracting 0.5 from the ellipsoid with semi-axes 3, 2, 1 gives semi-axes 2.9155, 1.8708, 0.7071, a much flatter shape. Even so, when it was filled uniformly with the same mass, the relative difference of the fields at 3 outer points (the size of the difference ÷ the size of the force) ranged from 6.81×10⁻¹⁵ to 4.51×10⁻¹⁴, the size of the numerical error. This is a theorem that Maclaurin (1742) showed for spheroids and that, for general ellipsoids, is known under the names of Laplace and Ivory (1809). For comparison, when a non-confocal ellipsoid (semi-axes 2.9155, 1.9644, 0.6734) was given the same mass, the difference ranged from 7.24×10⁻³ to 1.91×10⁻².
Choose a shape and shrink it with the slider. Compare “shrink confocally” with “shrink only the short semi-axis”. The direction of the outer point can be moved too.
- current semi-axes
- force at the outer point (G = 1, mass 1)
- difference from the original shape
- core of this shape
I also checked this with the numbers readable from outside. An ellipsoid keeps its shape when mirrored in each of the three coordinate planes, so the numbers of odd patterns, among others, are 0. Of the numbers that remain, writing the squared semi-axes as P, Q and R, the two numbers of pattern 2 are (2R−P−Q)/5 and (P−Q)/5, and the three numbers of pattern 4 are also fixed by the differences of P, Q and R alone (the symbolic check goes up to pattern 4). Subtracting the same λ from all three does not change the differences. The numbers readable from outside are the same, and that is why the two cannot be told apart from outside. How the inside is filled does differ: for example, the average of z² over the contents, R/5, changes under the confocal shift.
Squeezed as far as possible, it becomes the core
Raise λ up to the square of the short semi-axis, and the short semi-axis becomes 0: the ellipsoid becomes a flat disc. For semi-axes 3, 2, 1, the core is an elliptic disc with semi-axes √8 and √3 (2.828427 and 1.732051). Its density is greatest in the middle and falls to 0 at the rim, in proportion to √(1−q²) (q is the distance from the centre, with the rim of the disc set to 1). This has the same form as the thickness of the original ellipsoid pressed flat from straight above.
Change the shape, and the core changes too.
| shape (semi-axes) | core | dimension of the core |
|---|---|---|
| sphere (1, 1, 1) | the point at the centre | 0 |
| prolate spheroid (1, 1, 2) | the segment of length 2√3 between the two foci | 1 |
| oblate spheroid (2, 2, 1) | a round disc of radius √3 | 2 |
| ellipsoid (3, 2, 1) | an elliptic disc with semi-axes √8 and √3 | 2 |
In all four cases the core gave the same field as the original shape at 3 outer points (the relative difference was of order 10⁻¹⁴; for each shape the points were chosen outside the original shape). The size of the core is fixed by the differences of the squared semi-axes. For a sphere the differences are all 0, so the core shrinks to a single point. The shell theorem was the smallest case of this “shrinking down to the core”.
The same outside does not mean the same inside. At the point (0, 0, 0.5) inside the ellipsoid, a little away from the disc, the difference between the force of the solid ellipsoid and the force of the core disc was 1.911392 times the force of the solid.
The ellipsoidal skin with no force inside
Inside a spherical skin, the force is 0 wherever you are. What about an ellipsoidal skin? We compare two ways of painting the skin. In one, the thickness is made proportional to the distance p from the centre to the tangent plane; this is the shell between the original ellipsoid and a similar ellipsoid made slightly larger. It is called a homoeoid. In the other, the skin is painted with the same thickness everywhere.
For semi-axes 3, 2, 1, p is 3 at the ends of the long axis and 1 at the ends of the short axis. With the mass of the skin set to 1 and its pull added up at 3 inner points, the force was of order 10⁻¹⁵ (the size of the numerical error) for the homoeoid, and from 1.28×10⁻² to 3.79×10⁻² for the skin with the same thickness everywhere. On a sphere p is the same everywhere, so the two ways of painting give the same skin. That is why, looking only at spheres, one cannot notice this difference.
Choose how the skin is painted, and use the slider to change the shape from a sphere to an ellipsoid. The position of the inner point can be moved too.
- semi-axes
- range of the distance to the tangent plane
- force at the inner point (G = 1, skin mass 1)
Why the force inside is 0
Inside a uniform ellipsoid, the gravitational potential is a quadratic in the coordinates (three terms in x², y² and z², plus a constant). For an ellipsoid with all its semi-axes enlarged in the same proportion, the quadratic terms are the same as before and only the constant changes. So in the cavity enclosed by the shell between the two (inside the inner ellipsoid), the potential is the same everywhere, and the force is 0. The difference between the ellipsoid with semi-axes 1.1 times as large and the original was −4.02684457640873 at each of 3 inner points. And that the only uniform bodies whose potential inside is a quadratic are ellipsoids is cited as a result of Dive (1931) and others (the original has not been read for this page).
Paper 156 split the reasons why the inside of a spherical shell is flat into three: being harmonic, spherical symmetry, and the value not becoming infinite at the centre. The homoeoid is not spherically symmetric, yet its inside is flat. What takes the place of spherical symmetry here is that the potential inside is a quadratic, together with the special way the thickness is painted.
Sharing out 4π among three directions
Differentiate that quadratic twice in each of the three directions and add the three values, and you get 4π: the 4π of the equation of gravity, ∇²Φ = 4πρ. According to the shape, 4π is shared out among the three directions. For a sphere each gets 4π/3, the number that Paper 1084 called the “longitudinal share”. Paper 1084 also computed that for spheroids this share moves with the shape. For semi-axes 3, 2, 1, the shorter the axis, the more its direction gets: 1.964, 3.357 and 7.245 (adding up to 4π).
“Add Up the Dots on Graph Paper, and the Answer Depends on the Shape” (Paper 1097) computed that the sum of the fields of dipoles arranged on a lattice is 0 for a sphere, 4π/3 for a thin needle and −8π/3 for a thin plate. Paper 1098 computed, for spheroids, that these three come only from a shape term that can be written with the depolarization factor. These three are 4π/3 minus the share given to that direction. For a sphere, 4π/3 − 4π/3 = 0. Along the long direction of a thin needle almost nothing is given, so the value is 4π/3; across the thickness of a thin plate almost all of 4π is given, so it is 4π/3 − 4π = −8π/3. This is because the field inside a uniformly polarized ellipsoid is fixed by the same quadratic. The zero force inside the homoeoid and these three values come from the same quadratic.
If it is not the inverse square, it breaks down
I did the same calculation with a force that weakens as the cube of the distance. The difference between the two confocal ellipsoids at the 3 outer points was from 2.39×10⁻² to 2.82×10⁻², and the force inside the homoeoid was from 1.92×10⁻² to 1.45×10⁻¹, not 0. With the massive Yukawa field, it breaks down as well, even though the field equation is of second order. The difference between the two confocal ellipsoids at the 3 outer points was from 3.57×10⁻² to 5.20×10⁻², the force inside the homoeoid was from 1.35×10⁻² to 4.27×10⁻², and even inside a spherical skin of radius 1 it was 6.80×10⁻². Paper 1107 separated one skin sufficing for the numbers readable from outside (second order) and 4π remaining on the skin (the field having zero mass) as distinct roots. For both the core and the flat inside, being of second order is not enough: they rest on the field obeying the massless Laplace equation.
A uniform ball, however, still gives from outside a field of the same form as a point, even for the Yukawa field. The ratio of the forces of a uniform ball of radius 1 and of the point at its centre was 3/e = 1.103638 at all 3 outer points; only the mass changed. Shrinking a sphere to a point has one more support: symmetry. Shrinking an ellipsoid to a disc needs the massless Laplace equation.
If the Earth were a uniform ellipsoid
The ellipsoid that represents the shape of the Earth (WGS84) has an equatorial radius of 6378137 m and a polar radius of 6356752.314245 m. If the Earth were, hypothetically, a uniform ellipsoid, its core would be a round disc of radius 521.854 km. That is 0.0818192 times the equatorial radius, and this factor is the eccentricity of the ellipsoid itself. The real Earth is heavier towards the middle, and its layers differ in flatness, so its gravity outside is not the same as that of this disc. This number only shows how large the core of a uniform body of this shape is.
On the intuition of modules
Back to the first idea. When a thing is wrapped in a closed skin, what can be read from outside was one number per angular pattern. How many there are is fixed by the order of the field equation and does not depend on the shape (Paper 1107). What this page found is that, within the confocal family, how small the “core” that gives out those numbers can be made is fixed by the shape. A sphere can be shrunk down to a point, a prolate spheroid down to a segment, and an ellipsoid down to a flat disc. Whether an arrangement that is not confocal could make it smaller still has not been checked.
Seen from outside, a solid ellipsoid and a flat disc cannot be told apart. But this does not say that a flat disc gives rise to the contents of the solid. It says only that they cannot be told apart by gravity seen from outside. Inside, the forces of the solid and the disc clearly differ.
The separator: how the thickness of the skin is painted, and whether two ellipsoids are confocal
A separator says what has to be moved for one thing to become another. Here there are two separators. One is how the thickness of the skin is painted: proportional to the distance to the tangent plane, the force inside is 0; with the same thickness everywhere, it is not. On a sphere the two give the same skin, so this separator does not act on a sphere. The other is whether two ellipsoids are confocal: if they are, at equal mass they cannot be told apart from outside; if not, they differ.
Questions
Does the flat disc make up the solid?
No. They only cannot be told apart by gravity seen from outside. The forces inside differ: at the point (0, 0, 0.5) inside the ellipsoid, the difference was 1.911392 times the force of the solid.
Does every shape have a core?
This page checked only ellipsoids. The cores of shapes that are not ellipsoids are not treated. For a general shape, too, the contents can be moved onto the skin without changing the field outside (sweeping out; Paper 1107).
Is this the same story as the “Maclaurin spheroid” of a rotating fluid?
No. The Maclaurin spheroid is about the shape in which a rotating fluid is in balance. The theorem on this page is about the gravity outside confocal ellipsoids, shown by the same Maclaurin, and it is a different thing.
Is it the same for electricity?
Yes. Whenever the force is inversely proportional to the square of the distance (the field obeys the massless Laplace equation), it is the same for electricity. In electricity, the way 4π is shared out is called the depolarization factors (Paper 1084, Paper 1097).
Papers and references
All the papers are published under CC BY 4.0.
- Paper 1116 “A Uniform Ellipsoid Pulls from Outside Like a Flat Focal Disc, and a Sphere Pulling Like a Point Is the Case in Which That Disc Shrinks to One Point”: the basis of the calculations on this page.
- Paper 1107 “When the Field Equation Is of Second Order, the Interior Can Be Read from Outside a Closed Skin Only as One Number per Angular Pattern”: what can be read from outside is one number per angular pattern.
- Paper 1114 “m = 4π Is the Mass of a Skin of Radius 1 at Density 1, and the 1/r² Ball Is That Skin Puffed Out Evenly”: for a sphere, a point, a skin and a ball are the same from outside.
- Paper 156 “The Inside of a Shell Is Not Flat Because of the Mean Value Theorem”: three reasons why the inside of a spherical shell is flat.
- Paper 1084 “4π Splits into a Longitudinal 4π/3 and a Transverse 8π/3, and the 21 cm Line of Hydrogen Measures the Transverse Part”: the share of 4π/3.
- Paper 1097 “A Lattice Sum of Dipole Fields Is Decided by the Shape of the Summation: 0 for a Sphere, 4π/3 for a Thin Needle, −8π/3 for a Thin Plate”: three values that change with the shape.
- Paper 1098 “In a Lattice Sum Whose Exponent Equals the Dimension, Shape Enters as the Mean Logarithm of the Distance to the Edge”: the three values come only from the shape term.
- Paper 51 “The Yamagishi Gravitational Multipole Aggregation Theory”: the permission to aggregate into a point reaches only as far as the shell theorem.
- Paper 787 “Chimborazo’s Summit, 2585.39 m Lower Than Everest’s, Is 2079.524 m Farther From Earth’s Centre”: the numbers of the Earth’s ellipsoid.
- Paper 300 “The Three Hundredth Census: Whether Two Things Share a Root Is Decidable”: separating equal numbers by what moves them.
- Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
- Maclaurin, C. (1742). A Treatise of Fluxions.
- Ivory, J. (1809). On the attractions of homogeneous ellipsoids. Philosophical Transactions of the Royal Society of London.
- Dive, P. (1931). Attraction des ellipsoïdes homogènes et réciproques d'un théorème de Newton. Bulletin de la Société Mathématique de France.
- Osborn, J. A. (1945). Demagnetizing Factors of the General Ellipsoid. Physical Review.
- Chandrasekhar, S. (1969). Ellipsoidal Figures of Equilibrium. Yale University Press.