# Mass of potential gravity energy?

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?

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.

• But that mass can not be part of the local mass of the object. I mean, the Sun's mass we use to calculate Earth's orbit can not account for the mass of the added potential energy between the Sun and some distant star, because that energy (and corresponding mass) is not local to the sphere we call Sun, it is spread along both stars. It is different for a planet orbiting both stars on a binary system: the sum of individual stars' rest masses should be less than the stable binary system mass, pretty much like atom nuclei mass is different from the sum of individual protons and neutrons rest mass. Commented Jan 29, 2019 at 21:20
• I guess the best I can do for you in a short answer is to say that Newtonian physics is v. precise in the weak field limit of gravity, and if you want to know a more complete story then you need to understand general relativity where the energy of gravity is quite a subtle concept. As a starting point you can notice that the GR field equation has the ordinary stress-energy tensor (with no gravity contribution) as the source. Commented Jan 29, 2019 at 21:56

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?

• I only have a partial explanation to 3: it is the potential energy between galaxies. In milky way case, between the main galaxy and the satellites galaxies. But even if the model is far off in many orders of magnitude, if m=E/c2 can be applied to potential energy, we are talking about a mass of at least a few times of the visible mass itself (for a galaxy), which wouldn't go unnoticed (probably). Commented Jan 29, 2019 at 19:00
• Looking on 4, it seems the observation is not confirmed by later study: iopscience.iop.org/article/10.3847/2041-8213/aac216/meta Commented Jan 29, 2019 at 19:11
• Oh I didnt know that. That seems interesting. Maybe I should delete 4. Commented Jan 29, 2019 at 19:56

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.

• This view is a different from what I learned: you are saying there is matter, and there is energy, and one can be converted to the other. What I understand is that only energy have mass, and nothing else. Matter is just what we convention to call the energy that is trapped in the build-up of an atom (mostly at its nucleus). But just like the total mass of a nucleus is more than the sum of its protons and neutrons, the total mass of a stellar system is more than the sum of its matter, and includes all kinds of energies (kinectic, potential and electromagnetic radiation). Commented Jan 30, 2019 at 19:17
• I see energy and matter as the same thing-see pair production and annihilation. The mass of an atomic nucleus is less than the sum of the mass of its nucleons. mass defect. But you do have a point. There was a similar question that I found here. What do you mean the total mass of a stellar system is more than the sum of its matter? 99.9% of the mass is in the sun, and 99% of the remaining 0.1% is in the gas giants. Commented Jan 30, 2019 at 21:50
• Yes. But inside a stellar system this difference would be irrelevant, specially in single star systems. But the more stars/massive objects, more relevant would be this extra mass, increasing exponentially (proportional to the square of the number of objects). In a galaxy, this extra mass would be most of the total mass (at least, it seems so, considering only special relativity and Newton's gravity). Commented Feb 18, 2019 at 17:21
• Mass in special relativity is weird and you will have to define it better before you can get a concrete answer. Depending on the reference frame, relativistic mass is not always the same as rest or invariant mass. You tagged special relativity but there are also effects from general relativity on a systems mass to consider as well. I think you're onto something though. Check out this article. I did notice one error though, 1.07*10^-9 / (8.98*10^16) = 1.19^-26 grams or 1.19*10^23 kg, not 1.19*10^-17 kg. Commented Feb 19, 2019 at 16:06