# Re: Planck Length

```On Sunday, January 20, 2019 at 9:16:01 AM UTC-6, Bruno Marchal wrote:
>
>
> On 19 Jan 2019, at 01:42, Lawrence Crowell <goldenfield...@gmail.com
> <javascript:>> wrote:
>
> On Thursday, January 17, 2019 at 6:31:06 AM UTC-6, Bruno Marchal wrote:
>>
>>
>> On 17 Jan 2019, at 09:22, agrays...@gmail.com wrote:
>>
>>
>>
>> On Monday, January 7, 2019 at 9:25:16 PM UTC, John Clark wrote:
>>>
>>> On Mon, Jan 7, 2019 at 8:03 AM <agrays...@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
>>>
>>
>>
>> *Later I'll post some questions I have about your derivation of the
>> Planck length, but for now here's a philosophical question; Is there any
>> difference between the claim that space is discrete, from the claim or
>> conjecture that we cannot in principle measure a length shorter than the
>> Planck length? *
>> *TIA, AG *
>>
>>
>> That is a very good question. I have no answer. I don’t think physicists
>> have an answer either, and I do think that this requires the solution of
>> the “quantum gravity” or the “quantum space-time” problem.
>> With loop-gravity theory, I would say that the continuum is eventually
>> replaced by something discrete, but not so with string theory; for example.
>> With Mechanism, there are argument that something must stay “continuous”,
>> but it might be only the distribution of probability (the real-complex
>> amplitude).
>>
>> Bruno
>>
>
> The Planck length is just the smallest length beyond which you can isolate
> a quantum bit. Remember, it is the length at which the Compton wavelength
> of a black hole equals its Schwarzschild radius. It is a bit similar to the
> Nyquist frequency in engineering. In order to measure the frequency of a
> rotating system you must take pictures that are at least double that
> frequency. Similarly to measure the frequency of an EM wave you need to
> have a wave with Fourier modes that are 2 or more times the frequency you
> want to measure. The black hole is in a sense a fundamental cut-off in the
> time scale, or in a reciprocal manner the energy, one can sample space to
> find qubits.
>
>
> That makes some sense. It corroborates what Brent said. To “see” beyond
> the Planck resolution, we need so much energy that we would create a black
> hole, and ost any available information. This does not mean that a shorter
> length is no possible in principle, just that we cannot make any practical
> sense of it.
>
>
>
I think we talked a bit on this list about hyper-Turing machines. These are
conditions set up by various spacetimes where a Cauchy horizon makes an
infinite computation accessible to a local observer. A nonhalting
computation can have its output read by such an observer. These spacetimes
are Hobert-Malament spaces.The Planck scale may then be a way quantum
gravity imposes a fundamental limit on what an observer can measure.```
```
If one is to think of computation according to halting one needs to think
according to nilpotent operators. For a group G with elements g these act
on vectors v so that gv = v'. These vectors can be states in a Hilbert
space or fermionic spinors. The group elements are generated by algebraic
operators A so that g = e^{iA}. Now if we have the nilpotent situation
where Av = 0 without A or v being zero then gv ≈ (1 + iA)v = v.

A time ordered product of fields, often used in path integral, is a
sequence of operators similar to g and we may then have that g_1g_2g_3 …
g_n as a way that a system interacts. We might then have some condition
that at g_m for m < n the set of group operations all return the same
value, so the group has a nilpotent condition on its operators. This would
then bear some analogue to the idea of a halted computation.

The question of whether there are nonhalting conditions is then most likely
relevant to spacetime physics of quantum fields. If we have a black hole of
mass M it then has temperature T = 1/8πGM. Suppose this sits in a spacetime
with a background of the same temperature. We might be tempted to say there
is equilibrium, which is a sort of halted development. However, it the
black hole emits a photon by Hawking radiation of mass-energy δm so M → M -
δm it is evident its temperature increases. Conversely if it absorbs a
photon from the thermal background then  M → M + δm and its temperature
decreases. This will then put the black hole in a state where it is now
more likely to quantum evaporate or to grow unbounded by absorbing
background photons.

This might then be a situation of nonhalting, and with gravitation or
quantum gravity the moduli space is nonHausdorff with orbits of gauge
equivalent potentials or moduli that are not bounded. We might then
consider quantum gravitation as an arena where the quantum computation of
its states are nonhalting, or might they be entirely uncomputable. The
inability to isolate a qubit in a region smaller may simply mean that no
local observer can read the output of an ideal hyper-Turing machine from an
HM spacetime.

> The levels of confusion over this are enormous. It does not tell us that
> spacetime is somehow sliced and diced into briquets or pieces.
>
>
> I agree. Besides, this might depend heavily on the solution of the quantum
> gravity problem. Loop gravity, as far as I understand it, does seem to
> impose some granularity on space-time. Superstring do not, apparently.
>
>
>
String theory does some other things that may not be right as well. The
compactification of spaces with dimensions in addition to 3-space plus time
has certain implications, which do not seem to be born out.

>
>
> It does not tell us that quantum energy of some fields can't be far larger
> than the Planck energy, or equivalently the wavelength much smaller.
>
>
> OK.
>
>
> This would be analogous to a resonance state, and there is no reason there
> can't be such a thing in quantum gravity. The Planck scale would suggest
> this sort of state may decay into a sub-Planckian energy.  Further, it is
> plausible that quantum gravity beyond what appears as a linearized weak
> field approximation similar to the QED of photon bunched pairs may only
> exist at most an order of magnitude larger than the Planck scale anyway. A
> holographic screen is then a sort of beam splitter at the quantum-classical
> divide.
>
>
> This is a bit less clear to me, due to my incompetence to be sure. If you
> have some reference or link, but it is not urgent. I have not yet find to
> study the Holographic principle of Susskind, bu I have followed informal
> exposition given by him on some videos. Difficult subject, probably more so
> for mathematical logician.
>
> Bruno
>
>
This last part involves some deep physics on how the holographic screen is
in entangled states with Hawking radiation.

LC

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