Negative gravitational potential energy

[Isaac Dupree]

https://en.wikipedia.org/wiki/Gravitational_energy

According to e=mc^2 but taking Newtonian mechanics for the rest of
physics (ha, ha), two 1kg objects must be less than 3.71282(8) * 10^-28
m apart to have negative overall energy. That's 10^12 times smaller
than the charge radius of a proton.

gravitational potential energy: -G(1kg)(1kg)/r
mass energy: (1kg)c^2
G = 6.67384(80)*10^-11 m^3 kg^-1 s^-2
c = 299792458 m/s
c^2 = 89875517873681764 m^2/s^2
when is (1kg)c^2 + -G(1kg)(1kg)/r < 0 ?
let's solve the =0
(1kg)c^2 = G(1kg)(1kg)/r
r = (G(1kg)(1kg)) / ((1kg+1kg)c^2)
= 3.71282(8) * 10^-28 m

The higher the masses, the higher the distance (linearly). Unequal-mass
objects don't help us break the world: it's probably harder for them to
be close to each other.

Physics at that scale is obviously special.

If one of the objects is the Earth and the other is 1 kg, the required
distance is about a millimetre; so the Earth would have to be <= 1mm
radius, which I believe makes it a black hole.

Indeed, the 3.71282(8) * 10^-28 m computed above is exactly a quarter of
a 1 kg object's Schwartzchild radius[1]: so indeed there is a close
relation between negative gravitational potential energy and black holes.

Let's just not let anything in Lasercake become dense enough to become a
black hole, 'kay? (A macroscopic black hole anyway; atom-massed black
holes are fine and evaporate in ridiculously small amounts of time: even
a black hole the mass of a car evaporates in a nanosecond [2].)

On a not-actually-related note, there are games that let you play around
near the speed of light:
http://scienceblogs.com/startswithabang/2012/11/11/weekend-diversion-the-first-relativistic-video-game/

[1] https://en.wikipedia.org/wiki/Schwarzschild_radius
[2] https://en.wikipedia.org/wiki/Black_hole#Evaporation