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Problem number 23 in chapter 1 of Purcell's "Electricity and Magnetism" Second edition page 37 shows that 90% of the energy stored in the electrostatic field of a spherical charge distribution of radius R is contained in a spherical region of 10R.

The energy by this description is diffuse and only partially resides within what we would call the physical bounds of the particle. Gravity has the same form as the electromagnetic force and I would expect would have this same character of distribution of energy (i.e. 90% of the energy of the gravitational mass contained within a radius of 1R is stored within a sphere of 10R).

Is this true of gravitational energy and has this little detail been considered in the galaxy rotation curve explanations? I would think that something this elementary would be well accounted for in galaxy rotation descriptions. Much of the electromagnetic and gravitational energy seems diffused into the area surrounding mass and I think that galaxy rotation curves require more diffuse gravitational energy distribution don't they?

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  • $\begingroup$ "The principle of superposition of charges..."? This is wrong for a start. And are you asking whether gravitational potential energy has been considered? Or are you asking whether the "mass equivalent" of the gravitational potential energy has been considered - which is equivalet to saying, have people used GR rather than Newtonian mechanics to look at the problem? The answer to these are yes or course; and yes, but it isn't necessary. $\endgroup$ – Rob Jeffries Dec 2 '17 at 15:47
  • $\begingroup$ @Rob Jeffries could you elaborate on ---"The principle of superposition of charges..."? This is wrong for a start.--- I'm not sure what's wrong with this. $\endgroup$ – DMac Dec 2 '17 at 22:33
  • $\begingroup$ What I'm asking is regarding the diffuse nature of the gravitational energy (g(m^2/r)). Normally the gravitational potentials are idealized to point sources so that numerical calculations are not intractable computationally. However analogizing to the electrostatic case where the energy (q^2/r) apparently resides in considerable measure in a location other than within the physical confines of the charge, does the gravitational energy have this same character and has that detail been included in the calculations. $\endgroup$ – DMac Dec 2 '17 at 23:06
  • $\begingroup$ Your statement about the nature of charges and electrostatic energy is incorrect. The idea that gravitational potentials are "normally" treated as point sources is incorrect. $\endgroup$ – Rob Jeffries Dec 3 '17 at 0:11
  • $\begingroup$ I'll accept your statement that gravitational potentials are "normally" treated as point sources is incorrect because my statement didn't specify the circumstances. As for the nature of charges and electrostatic energy, did you mean that --- 90% of the energy stored in the electrostatic field of a spherical charge distribution of radius R is contained in a spherical region of 10R--- is incorrect? or some other aspect? $\endgroup$ – DMac Dec 3 '17 at 2:55
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You appear to be asking whether gravitational potential energy acts as a significant contributor to the total mass-energy of a galaxy - a question that is only relevant within the context of General Relativity, where indeed other forms of energy besides the rest-mass energy contribute to the gravitational effects.

It turns out that GR is not required to interpret galaxy rotation curves because the influence of the other forms of mass-energy (or more formally, other contributors to the stress-energy tensor) are very small.

The contribution of gravitational potential energy can be assessed using the ratio $GM/rc^2$, which is approximately the ratio of gravitational potential energy to rest mass energy. For example, if $10^{12}$ solar masses exists within a 100 kpc radius (reasonable numbers for the Milky Way); then the ratio is $5\times 10^{-7}$. Of course, if there is no dark matter then this would be even smaller.

I should also add that this tiny correction, like the gravitational potential energy, is negative. That is, it reduces the gravitational influence of the mass/energy in a galaxy. The (positive) total kinetic energy of the Galactic components (also tiny) would also need to be added, but since bound systems have a total energy (kinetic plus gravitational potential) $<0$, there is still a net reduction in the gravitational mass.

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