Moving objects in retarded gravitational potentials of an expanding spherical shell/Classical approach

Classical approach

This Wikibook assumes a huge amount of dark matter that is in a huge shell behind the visible part of the universe. This matter is rather cold, and it can be assumed that it is in a in thermal equilibrium. Therefore, the electromagnetic radiation emitted by this shell can be represented by the Planck radiation law. The emission of such a cold source of electromagnetic radiation is not visible and additionally the radiation has experienced a strong redshift due to the relative velocity of the fast expanding emitting shell as well as due to the gravitation of the mass of this shell.

The gravitational force $F$ between two masses $m$ and $M$ in a distance of $s$ is given by Newton's law of universal gravitation with the gravitational constant $G$:

$F=G\,{\frac {m\cdot M}{s^{2}}}$

In a thought experiment it is possible to investigate the behaviour of a mass $m$ that is located within a spherical shell with the mass $M$. The approach described in 1687 by Isaak Newton (1643–1727 in Gregorian dates) is called Newton's shell theorem. According to this theorem a mass $m$ within a homogeneous and spherically symmetric shell with a mass $M$ experiences no net gravitational force. Additionally, this is regardless of the location of the mass $m$ within the shell as well as of its velocity.

This behaviour can be easily explained with the following consideration: let a mass $m$ be within a homogeneous and spherically symmetric shell with the radius $r$ and the areal density $\rho _{a}$. Then the area $A$ and the mass $M$ of the shell can be expressed as:

$A=4\pi \cdot r^{2}$

$M=\int \limits _{A}\mathrm {d} M=\int \limits _{A}\rho _{a}\cdot \mathrm {d} A=\rho _{a}\cdot 4\pi \cdot r^{2}$

About the equivalence of the gravitational force to a mass $m$ in a homogeneous and spherically symmetric shell with the two area elements $\mathrm {d} A_{1}$ and $\mathrm {d} A_{2}$.

If we take an axis symmetrical double cone where the tips of the two cones are located at the position of the mass $m$ within a homogeneous and spherically symmetric shell, we get the situation shown in the adjacent figure.

For the two infinitesimally small area elements $\mathrm {d} A_{1}$ and $\mathrm {d} A_{2}$ in the distances $s_{1}$ and $s_{2}$ of a mass $m$ with the same infinitesimally small angle elements $\mathrm {d} \alpha$:

$\mathrm {d} A_{1}={\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} A_{2}={\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}$

For the appropriate masses $\mathrm {d} M_{1}$ and $\mathrm {d} M_{2}$ of the two area elements:

$\mathrm {d} M_{1}=\rho _{a}\cdot \mathrm {d} A_{1}=\rho _{a}\cdot {\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} M_{2}=\rho _{a}\cdot \mathrm {d} A_{2}=\rho _{a}\cdot {\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}$

With Newton's law of universal gravitation, we can express the two infinitesimally small gravitational forces $\mathrm {d} F_{1}$ and $\mathrm {d} F_{2}$ due to the masses of the two area elements:

$\mathrm {d} F_{1}=G\,{\frac {m\cdot \mathrm {d} M_{1}}{s_{1}^{2}}}=G\,{\frac {m\cdot \rho _{a}\cdot {\frac {\pi }{4}}\,s_{1}^{2}\cdot \mathrm {d} \alpha ^{2}}{s_{1}^{2}}}={\frac {\pi }{4}}\,G\cdot m\cdot \rho _{a}\cdot \mathrm {d} \alpha ^{2}$

$\mathrm {d} F_{2}=G\,{\frac {m\cdot \mathrm {d} M_{2}}{s_{2}^{2}}}=G\,{\frac {m\cdot \rho _{a}\cdot {\frac {\pi }{4}}\,s_{2}^{2}\cdot \mathrm {d} \alpha ^{2}}{s_{2}^{2}}}={\frac {\pi }{4}}\,G\cdot m\cdot \rho _{a}\cdot \mathrm {d} \alpha ^{2}$

It is obvious that both forces have the same value, and since they must be exactly directed to opposite directions, they add to zero. In other words: a mass within a spherical shell does not experience any net gravitational force at all.

It is worth noting that this classical approach assumes the infinite propagation speed of gravitational waves, which is applicable for static but not for dynamic situations as well as huge systems.