_________________________________________
Addendum: There's an unexpected surprise here. Things
didn't go as planned. You might want to read to the end.
__________________________________________
Why is it that no competent physicist has ever clamed
that HUP ought to prevent the formation of a black hole ?
After all, the existence of a ground state, even in the
face of zero orbital angular momentum, is a quintessential
result of QM. Well .... at least for a potential that falls
of no faster than - 1 / r.
Finite angular momentum prevents the collapse of classical
gravitating systems. As part of the effective potential, the
energy of orbital angular momentum, L, has the form
E_L = + L^2 / 2 * m * r^2
where m is the mass of the orbiting particle. If, in place of
the gravitational potentia, a potental of form
U(r) = - a^2 / r^2
existed, and if a^2 > L^2 / 2 * m,
the system would collapse to a central point.
Similarilly, In QM, there will be a collapse to the center when
a^2 > [ L^2 + hbar^2 / 4 ] / 2 * m
For this condition, there is no QM ground state.
In the external Schwarzschild metric, In addition to the terms
for gravitation (- 1 / r) and for angular momentum (+ 1 / r^2)
the effective potential contains a term proportional to - 1 / r^3:
E_effective = [Newton] - ( L^2 / 2 * m ) ( R / r^3 )
Where R is the Schwarzschild radius. Thus, in QM, for
( R / r ) L^2 > L^2 + hbar^2 / 4
There is no QM ground state. The particle falls to the center.
Heck !!!! This means r < R. It means that there's a fall to
the center (no QM ground state) when the particle is inside
the event horizon ???? This isn't going as planned. Guess
that Old Man isn't a competent physicist. So, ....
Why doesn't the HUP prevent the formation of a black hole ?
[Old Man]
.
|
