Interesting question.

Your question is a statement of fact really. But the tendency towards seems to go contrary to your comment.

If you take the case of a free moving electron for example, that has only translational carrying energy, if it is attracted to an ionized hydrogen nucleus (a proton), it will accelerate until it it is stopped in energy equilibrium at a precise distance from the nucleus which averages out at a distance of 5.291772083E-11 meter from the nucleus. At that moment, all of its translational energy will be released as a photon.

But at  this distance from the nucleus, it is permanently induced with 27.2 eV of unreleasable carrying energy as a function of the inverse square Coulomb law of distance which is in action between the electron and the proton.

If the electron axially wiggles about this mean orbital distance, this amount of unreleasable energy will vary accourdingly, diminishing if the electron moves away and increasing if the electron comes closer. If the electron is chased away from the rest orbital and escapes the proton, this energy will diminish accordingly.

There seems to be no reason for gravity to be acting differently, since it also obeys the very same law of inverse square of the distance.

1.What kind of energy equilibrium could be in case of a black hole?

2. Parallels with electromagnetic interactions are not correct because there is no gravitational repulsion, but  the universe expands with increasing speed.

Dear N. K. Zhevago

I think you overestimate the importance of electrostatic repulsion.

It is well understood that repulsive electrostatic force between same sign particles has an infinite range, just like electrostatic attraction between opposite signs particles. But there is an assumption that electrostatic repulsion can have an important effect at large distances just like electrostatic attraction.

Such large effects are definitely measurable at distances such as between both up quarks in a proton or both down quarks in a neutron, and even at distances such as between nuclei and electronic escorts in atoms.

But the fact is that since electrostatic repulsion decreases in intensity as a function of the inverse square of the increasing distance between any pair of like sign particles, its effect quickly becomes infinitesimal as the distance increases between any pair of these particles; to the point of becoming barely detectable, if at all, even at millimeter range distances between any single pair of same sign elementary particles.

It is well known that large numbers of atoms and molecules need to be ionized at our scale for this repulsion to even become measurable with lab instruments. The largest concentrations of like sign ionized atoms and molecules known to us naturally occur only momentarily in gaseous media such as the Earth's atmosphere, and they quickly dissipate, sometimes brutally through lightning, but more often by simply dissipating due to the very repulsion between these same sign ions which permanently tend by nature to move as far away from each other as fast as they can, until they eventually end up neutral as weakly bound electrons in the surrounding environment are captured by the positive ions.

Consequently, electrostatic repulsion is an important factor at elementary particles close quarters range, a barely noticeable factor at our range except when momentary huge concentrations of ions occur and an absolutely insignificant factor at astronomical range.

The best proof of the insignificance of electrostatic repulsion at our macroscopic level can be verified by anybody just considering that all atoms making up all molecules in all bodies about us including our own bodies systematically present to the outside world their mutually repelling electronic escorts.

This does not prevent us from getting close to these bodies without feeling any repulsion whatsoever until "touching" contact is established. In fact, this "touching" contact that prevents interpenetration is the most intense manifestation of the electrostatic repulsion between like charged electronic escorts that can be perceived at our level and it is easily verifiable that it occurs at submicroscopic distances, even with the massive amounts of electronic escorts that are involved in such "touching" encounters.

So even if gravitation and electrostatic attraction, both acting as a function of the inverse square of the distance between bodies, were the same force, electrostatic repulsion would play no role at all even at our macroscopic level, let alone astronomical distances.

Neither in gravity nor elsewhere does such a tendency exist: clearly Jupiter would lose a great deal of energy if it fell into the Sun, yet it does not. Examples could be multiplied almost indefinitely.


Energy is conserved, yet this conservation in no way determines the motion. When friction is present, then the general tendency for energy to decrease may partly create such an appearance

To be in gravitation energy equilibrium the Universe should collapse into the singularity in the end, but it is well known now that it expands with increasing speed. Consequently, there are always other kinds of forces to be taken into account when energy conservation is the starting point for our conclusions.

There is a positive energy theorem for general relativity, which states that energy of an asymptotically flat space-time is non-negative, and it vanishes for the Minkowski space time. Therefore you can safely say that  empty regions should have less energy than the regions which contain mass. Therefore empty regions are energetically preferred than regions containing massive bodies.

Ref. E. Witten, "A new proof of the positive energy theorem", Commun. Math. Phys. 80, 381 (1981).

If you want to deduce Newton's inverse square force law starting from GR, you have to first consider a solution of Einstein's field equations outside a spherically symmetric mass distribution. This solution was first derived by K. Schwarzschild in 1916; and thus named as the Schwarzschild metric. Then taking a weak field limit, one gets inverse square force law.

http://www.mth.uct.ac.za/omei/gr/chap7/node3.html#SECTION00030000000000000000

The gravitation potential reflects the influence that massive particles have on the space in which we live. Thus, the most convenient interpretation of this situation is that the space in which we live can be represented by a field (NOT THE EM FIELD !!!) and that the gravitation potential is no more and no less than a smoothed version of this field. The "space field" is contracted by the influence of point-like particles that obstruct the continuity of this field. A coherent swarm of such obstructions causes a smooth contraction in the smoothed version of the field. The number of elements in the swarm and the spread of the swarm are characteristics of the impact that the swarm has on the deformation of the field. A Gaussian distribution of the  obstructions would result in a smoothed shape that is described by ERF(r)/r. At some larger distance r this behaves as 1/r. The spread of the Gaussian distribution and the number of contributing obstructions determine the impact of the swarm. 

http://www.e-physics.eu/TheHilbertBookTestModel.pdf

"If you want to deduce Newton's inverse square force law starting from GR, you have to first consider a solution of Einstein's field equations outside a spherically symmetric mass distribution. This solution was first derived by K. Schwarzschild in 1916;"

I'm afraid not. Here we can talk about gravitational potentials, to speak about any force we have to add the mass of the test body which was eliminated with the postulate of the equivalence principle. The geodesic motion is obtained by first get rid of any mass, so force cannot be reinstated unless we do it in a artificial way.