Шар тәрізді денелердің гравитациялық тартылысы туралы теорема

Statement on the gravitational attraction of spherical bodies
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy. Isaac Newton proved the shell theorem and stated that:
A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center. If the body is a spherically symmetric shell (i. e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell. 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. This can be seen as follows: take a point within such a sphere, at a distance from the center of the sphere. Then you can ignore all of the shells of greater radius, according to the shell theorem (2). But the point can be considered to be external to the remaining sphere of radius r, and according to (1) all of the mass of this sphere can be considered to be concentrated at its centre. The remaining mass is proportional to (because it is based on volume). The gravitational force exerted on a body at radius r will be proportional to (the inverse square law), so the overall gravitational effect is proportional to , so is linear in
These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Gauss's law for gravity offers an alternative way to state the theorem.) 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.

Propositions 70 and 71 consider the force acting on a particle from a hollow sphere with an infinitesimally thin surface, whose mass density is constant over the surface. The force on the particle from a small area of the surface of the sphere is proportional to the mass of the area and inversely as the square of its distance from the particle. The first proposition considers the case when the particle is inside the sphere, the second when it is outside. The use of infinitesimals and limiting processes in geometrical constructions are simple and elegant and avoid the need for any integrations. They well illustrate Newton's method of proving many of the propositions in the Principia. His proof of Propositions 70 is trivial. In the following, it is considered in slightly greater detail than Newton provides. The proof of Proposition 71 is more historically significant. It forms the first part of his proof that the gravitational force of a solid sphere acting on a particle outside it is inversely proportional to the square of its distance from the center of the sphere, provided the density at any point inside the sphere is a function only of its distance from the center of the sphere. Although the following are completely faithful to Newton's proofs, very minor changes have been made to attempt to make them clearer.

Fig. 2 is a cross section of the hollow sphere through the center, S and an arbitrary point, P, inside the sphere. Through P draw two lines IL and HK such that the angle KPL is very small. JM is the line through P that bisects that angle. From the inscribed angle theorem, the triangles IPH and KPL are similar. The lines KH and IL are rotated about the axis JM to form two cones that intersect the sphere in two closed curves. In Fig. 1 the sphere is seen from a distance along the line PE and is assumed transparent so both curves can be seen. The surface of the sphere that the cones intersect can be considered to be flat, and
Since the intersection of a cone with a plane is an ellipse, in this case the intersections form two ellipses with major axes IH and KL, where
By a similar argument, the minor axes are in the same ratio. This is clear if the sphere is viewed from above. Therefore the two ellipses are similar, so their areas are as the squares of their major axes. As the mass of any section of the surface is proportional to the area of that section, for the two elliptical areas the ratios of their masses
Since the force of attraction on P in the direction JM from either of the elliptic areas, is direct as the mass of the area and inversely as the square of its distance from P, it is independent of the distance of P from the sphere. Hence, the forces on P from the two infinitesimal elliptical areas are equal and opposite and there is no net force in the direction JM. As the position of P and the direction of JM are both arbitrary, it follows that any particle inside a hollow sphere experiences no net force from the mass of the sphere. 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.

An analogue for shell theorem exists in general relativity (GR). 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). The metric thus has form

(using geometrized units, where ). For (where is the radius of some mass shell), mass acts as a delta function at the origin. For , shells of mass may exist externally, but for the metric to be non singular at the origin, must be zero in the metric. 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).