On Mon, Jan 7, 2019 at 8:03 AM <agrayson2...@gmail.com> wrote:

*> How does one calculate Planck length using the fundamental constants G,
> h, and c, and having calculated it, how does one show that measuring a
> length that small with photons of the same approximate wave length, would
> result in a black hole? TIA, AG*


In any wave the speed of the wave is wavelength times frequency and
according to
Planck E= h*frequency  so E= C*h/wavelength.  Thus the smaller the
wavelength the greater the energy. According to Einstein energy is just
another form of mass (E = MC^2) so at some point the wavelength is so small
and the light photon is so energetic (aka massive) that the escape velocity
is greater than the speed of light and the object becomes a Black Hole.

Or you can look at it another way, we know from Heisenberg that to
determine the position of a particle more precisely with light you have to
use a smaller wavelength, and there is something called the  "Compton
wavelength" (Lc) ; to pin down the position of a particle of mass m to
within one Compton wavelength would require light of enough energy to
create another particle of that mass. The formula for the Compton
Wavelength is Lc= h/(2PI*M*c).

Schwarzschild told us that the radius of a Black Hole (Rs), that is to say
where the escape velocity is the speed of light  is:  Rs= GM/c^2. At some
mass Lc will equal Rs and that mass is the Planck mass, and that Black Hole
will have the radius of the Planck Length, 1.6*10^-35 meters.

Then if you do a little algebra:
GM/c^2 = h/(2PI*M*c)
GM= hc/2PI*M
GM^2 = hc/2*PI
M^2 = hc/2*PI*G
M = (hc/2*PI*G)^1/2    and that is the formula for the Planck Mass , it's .02
milligrams.

And the Planck Length turns out to be (G*h/2*PI*c^3)^1/2 and the Planck time
is the time it takes light to travel the Planck length.

The Planck Temperature Tp is sort of the counterpoint to Absolute Zero, Tp
is as hot as things can get because the black-body radiation given off by
things when they are at temperature Tp have a wavelength equal to the
Planck Length, the distance light can move in the Planck Time of 10^-44
seconds. The formula for the Planck temperature is Tp = Mp*c^2/k where Mp
is the Planck Mass and K is Boltzmann's constant and it works out to be
1.4*10^32 degrees Kelvin.  Beyond that point both Quantum Mechanics and
General Relativity break down and nobody understands what if anything is
going on.

The surface temperature of the sun is at 5.7 *10^3  degrees Kelvin so if it
were 2.46*10^28 times hotter it would be at the Planck Temperature, and
because radiant energy is proportional to T^4 the sun would be 3.67*10^113
times brighter. At that temperature to equal the sun's brightness the
surface area would have to be reduced by a factor of 3.67*10^113, the
surface area of a sphere is proportional to the radius squared, so you'd
have to reduce the sun's radius by (3.67*10^113)^1/2, and that is  6.05*10^56.
The sun's radius is 6.95*10^8   meters and  6.95*10^8/ 6.05*10^56  is
1.15^10^-48 meters.

That means a sphere at the Planck Temperature with a radius 10 thousand
billion times SMALLER than the Planck Length would be as bright as the sun,
but as far as we know nothing can be that small. If the radius was 10^13
times longer it would be as small as things can get and the object would be
(10^13)^2 = 10^26 times as bright as the sun. I'm just speculating but
perhaps that's the luminosity of the Big Bang; I say that because that's
how bright things would be if the smallest thing we think can exist was as
hot as we think things can get.

John K Clark

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