Shell theorem

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body.

Isaac Newton proved the shell theorem[1] and stated that: A corollary is that inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of mass.

These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus.

In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law.

The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force.

Finally, arranging an infinite number of infinitely thin discs to make a sphere, this equation involving a disc will be used to find the gravitational field due to a sphere.

Suppose that this mass is evenly distributed in a ring centered at the origin and facing point

Applying Newton's Universal Law of Gravitation, the sum of the forces due to the mass elements in the shaded band is However, since there is partial cancellation due to the vector nature of the force in conjunction with the circular band's symmetry, the leftover component (in the direction pointing towards

, the shaded blue band appears as a thin annulus whose inner and outer radii converge to

Performing an implicit differentiation of the second of the "cosine law" expressions above yields and thus It follows that where the new integration variable

using the first of the "cosine law" expressions above, one finally gets that A primitive function to the integrand is and inserting the bounds

, i.e., Therefore, the total gravity is As found earlier, this suggests that the gravity of a solid spherical ball to an exterior object can be simplified as that of a point mass in the center of the ball with the same mass.

For a point inside the shell, the difference is that when θ is equal to zero, ϕ takes the value π radians and s the value R − r. When θ increases from 0 to π radians, ϕ decreases from the initial value π radians to zero and s increases from the initial value R − r to the value R + r. This can all be seen in the following figure Inserting these bounds into the primitive function one gets that, in this case saying that the net gravitational forces acting on the point mass from the mass elements of the shell, outside the measurement point, cancel out.

): The shell theorem is an immediate consequence of Gauss's law for gravity saying that where M is the mass of the part of the spherically symmetric mass distribution that is inside the sphere with radius r and is the surface integral of the gravitational field

The shell bound by two concentric, similar, and aligned ellipsoids (a homoeoid) exters no gravitational force on a point inside of it.

The use of infinitesimals and limiting processes in geometrical constructions are simple and elegant and avoid the need for any integrations.

In the following, it is considered in slightly greater detail than Newton provides.

Although the following are completely faithful to Newton's proofs, very minor changes have been made to attempt to make them clearer.

Through P draw two lines IL and HK such that the angle KPL is very small.

The lines KH and IL are rotated about the axis JM to form two cones that intersect the sphere in two closed curves.

1 the sphere is seen from a distance along the line PE and is assumed transparent so both curves can be seen.

Note: Newton simply describes the arcs IH and KL as 'minimally small' and the areas traced out by the lines IL and HK can be any shape, not necessarily elliptic, but they will always be similar.

Extend PI to intersect the sphere at L and draw SF to the point F that bisects IL.

Extend PH to intersect the sphere at K and draw SE to the point E that bisects HK, and extend SF to intersect HK at D. Drop a perpendicular IQ on to the line PS joining P to the center S. Let the radius of the sphere be a and the distance PS be D. Let arc IH be extended perpendicularly out of the plane of the diagram, by a small distance ζ.

Generate a ring with width ih and radius iq by making angle

Solving the quadratic for DF, in the limit as ES approaches FS, the smaller root,

Therefore, the force on a particle any distance D from the center of the hollow sphere is inversely proportional to

Spherical symmetry implies that the metric has time-independent Schwarzschild geometry, even if a central mass is undergoing gravitational collapse (Misner et al. 1973; see Birkhoff's theorem).

This reduces the metric to flat Minkowski space; thus external shells have no gravitational effect.

This result illuminates the gravitational collapse leading to a black hole and its effect on the motion of light-rays and particles outside and inside the event horizon (Hartle 2003, chapter 12).