- Proof of the corollary to
the shell theorem:
F_1_2 = F_1p_2 (by shell theorem) = -F_2_1p (by 3rd law which is explicitly true for Newtonian gravity) = -F_2p_1p (by shell theorem) = F_1p_2p (by 3rd law again), and so F_1_2 = F_1p_2p QED.

Of course, the interaction between the 2 masses can break their spherical symmetry and then neither shell theorem nor its corollary apply at least**NOT**exactly. - The general relativity
analogue of the
shell theorem is
Birkhoff's theorem.
Birkhoff's theorem tells the same story. Any spherically symmetric object's gravity is exactly the same as if all its mass were concentrated at a point at its center for objects outside of its radius.

For example, a 1 solar mass black hole and the Sun have the same gravitational field as long as you are ≥ 1 solar radius from their centers.

To conclude, the powerful gravity and exotic effects of a black hole only occur when you are relatively close to its event horizon or at least closer than the radius of an non-black-hole of the same mass as the black hole.

So black holes are

**NOT**monsters of gravity swallowing eveything in the observable universe. - By the by, progress is being made toward
a theory
of quantum gravity
In 2021,
gravity was measured to within ∼ 10 %
accuracy/precision
between gold
spheres of
mass 92 mg ≅ 10**(-4) kg
and the possibility exists of measuring
gravity for
masses
to ∼ 10**(-8) kg
(Christian Rothleitner 2021, Nature;
Westphal et al. 2021, Nature).
Now 10**(-8) kg (corresponding to a length scale for
typical density
solids of
100 μm)
is still much above the
atomic scale (∼ 0.1--1 nm)
(note atomic mass unit (AMU) = (1/12) C-12
= 1.660 539 066 60(50)*10**(-27) kg) and
even the
mesoscopic scale (∼ 100 nm -- 1 μm),
but it is in the low end the
macroscopic scale (>∼ 1 μm).
If quantum gravity effects
are larger than we currently think, then
they may show up for masses
to ∼ 10**(-8) kg.
Maybe there are sophisticated procedures to
measure gravity for even smaller
masses.
Time will tell.
Actually, it has been shown that gravity of the size of Earth surface gravity (fiducial value 9.8 N/kg) probably quantizes atomic states like other forces (see Evidence for Quantized Gravitational States of the Neutron, Frank Rioux, 2020; Quantum effects of gravity, Thomas J. Bowles, 2002, Nature; and for the research article itself, Nesvizhevsky et al. 2002, Nature). However, this result,

**IF VALID**, is just what we thoroughly expected and does**NOT**tell us more than confirming what we thought. So it's wonderful, if valid, but does**NOT**elucidate quantum gravity very much.