Mass of potential gravity energy?

Question

Assume a system where there only two 1 kg solid iron balls floating in space. The two balls are touching each other, so the potential gravitational energy between them is 0. Now I move them 1000 m apart, now the potential gravitational energy between them is $\int_{2r}^{2r+1000} \frac{G}{x^2} dx$, which is approximately $1.07 \times 10^{-9}$ J (considering a radius derived from pure iron density). According to Einstein's $M = \dfrac{E}{c^2}$, the added mass to the system, which was originally 2 kg, is $1.19 \times 10^{-17}$ kg, which I am not sure it could be detected at all. But the fact is that we had a system, added energy to it, and its mass increased.

If instead of solid iron balls, we have sun-like stars (same mass and radius), and instead of 2, we have 90 billions, and the average distance between each two is $1.28 \times 10^{21}$ m (as in a spiral galaxy with 150 kly radius), and finally (and most importantly, as most of the potential gravitational energy is gained in close proximity) we assume the radius of two of the stars, if merged, is $\sqrt[3]{2}$ greater than the original radius (i.e. the volume is preserved), then we are separating 90 billions stars from each other, from a distance of 2 half-sphere barycenters (of the merged stars) to the average star distance in a spiral galaxy, (90 billions)² times.

Using that simple galaxy model, the potential gravity energy accounts for 99.9995 % of the total mass of the galaxy (my python code), spread diffusely through it.

So, logic says such mass exists, because the potential energy exists in the system, and judging from the numbers, it seems very relevant, but I have never heard anyone talking about it. Is such energy accounted for in the estimates of the total mass of visible matter in the galaxy? If not, was it ever considered as a possible explanation for the dark matter?

Answer 1

To answer the last question first: physicists are very well aware of this sort of argument; it is the sort of thing they were brought up on and now have for breakfast (or a light afternoon snack) every day.

The energies you have calculated are indeed relevant to astrophysics, but they are fully incorporated in all calculations, because they are part of the standard physics here. I am glad you have noticed these things though. It was by these sorts of energy arguments that people such as Chandrasekhar began to explore the possible formation of black holes by the collapse of stars. The main fault in your reasoning is to take the close-together situation as your "zero" of energy and then say the stars have extra mass-energy when they are far apart. It turns out that what really happens is that stars have their ordinary energy when they are far apart, and less energy than that when they are close together. I mean, when they are not moving at different velocity but simply displaced closer to one another.

Here's a nice example: if you lower a 1 kg (or 1 billion kg) object on a rope gradually down towards the horizon of a black hole, then when it reaches the horizon and is released, the mass of the black hole does not grow because all the mass-energy of the object has been gathered in at the top end of the rope! It is just as if the lowered item lost all its mass energy in this process.

So, to repeat, galaxies have their ordinary energy when the stars are far apart, not when the stars are close together. By "ordinary energy" of a star, I mean approximately the number of protons and neutrons it contains, multiplied by the standard mass of a proton, multiplied by $c^2$. Of course it is believed that most mass is not from baryons but from unknown but gravitating stuff called dark matter, as you mention. But it would be reasonable to suppose that the gravitational effects for dark matter are similar to those for ordinary matter.

Answer 2

I don't think it is the dark matter we are looking for. I am not sure that we can use $m=E/c^2$ as a source of mass caused by the potential energy. But even we assume that could be possible, the hypothesis has still problems.

1- How can you explain the dark matter energy density in the early universe that effects the CMBR and later its effect on the galaxy structures.

2- Is your model can explain the density profiles described by the Navarro-Frenk-White profile?

3- The dark matter halo is usually around the galaxies and even considered to be further than the radius of the galaxies. How can your model explain these effects?

Answer 3

You may have added energy to it but that doesn't necessarily increase the systems mass. That energy is equivalent to some amount of mass. If for example the two separated masses accelerated towards each other, the potential energy turns into kinetic energy. When they are just about to collide, the gravitational potential energy approaches 0, and all of the original potential energy is now kinetic. The collision will increase the temperature of the combined blob of the masses. Matter was not necessarily created or destroyed. Energy and matter two sides of the same coin, and can't be destroyed or created. Matter can turn into energy, and energy can turn into matter. Gravitational potential energy is not new matter. When the gravitational potential decreases, increasing kinetic energy, the creation of matter is possible when they collide, but this consumes some of the original energy.