## Saturday, November 06, 2021

### [utewywvu] replacing the sun with two electrons

consider replacing the sun with two electrons placed very close to each other so that the potential energy of their electrostatic repulsion is equal to the (former) sun's mass-energy.  the distance between the electrons should be

(coulombconst*e^2) / (sunmass*c^2) = 1.2905183e-75 m

according to general relativity, the earth's orbit around the sun will be unaffected by the substitution.  energy bends spacetime exactly as well as matter.  the two electrons do need to remain magically nailed into place.

inspired by the Tardis replacing the sun in Dr. Who.

we have assumed electrons to be point particles.  for reference, Planck length = 1.6e-35 m.  our distance is much smaller, so weird stuff beyond known physics will definitely happen in this scenario.  maybe a naked singularity.

potential energy between two point charges is relative to some arbitrarily chosen zero.  in contrast, zero mass in Newtonian gravity is not arbitrarily chosen.  is it correct to have chosen zero energy to correspond to infinitely separated charges?

instead of 2 electrons, what if we had an electron and a positron separated by that tiny distance?  the potential energy is now of their attraction.  I'm pretty sure this will not gravitationally work as a substitute sun but cannot explain why.  what then happens if we then add nearly a sun's worth of mass-energy into the system and separate the two lovers to a macroscopic distance?  surely, adding that much mass-energy to a system should drastically and macroscopically affect the surrounding gravitational field / shape of spacetime.  but the shape of spacetime at the end is pretty boring (an electron and positron separated by a macroscopic distance), so the shape at the beginning must have been interesting.

consider an (uncharged) spherical shell of a given mass. consider changing its outside radius, keeping mass constant.  (to maintain the same mass, thickness or density needs to change.)  what happens to the shell's external gravitational field?  the shell maintains the same mass but changes its gravitational binding energy.  general relativity cares about total mass-energy.

hilariously tiny correction to original set up: the 2 electrons that replaced the sun exert gravity due to their mass.  therefore, we actually do not need to generate that (tiny) portion of the sun's mass-energy through electric potential.  corrected distance:

(coulombconst*e^2)/((sunmass - 2*electronmass) * c^2) = 1.2905183e-75 m

of course, the answer is essentially the same because sunmass/(2*electronmass) = 1.09179e+60 .

1/(1-x) = 1 + x + x^2 + ...

accounting for the electrons' masses allows them to be slightly further apart.  using a linear approximation of the above Taylor series, the electrons are further apart by 1.182e-135 m.  not making the linear approximation adds an additional 1.083e-195 m, mostly the quadratic term.

there are no metric prefixes for these tiny factors of 10.