# Re: Planck Length

```> On 21 Jan 2019, at 00:17, Lawrence Crowell <goldenfieldquaterni...@gmail.com>
> wrote:
>
> 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```
```
In a physical reality.? But once we assume mechanism, we cannot do that
assumptions. Halting and non halting computations is a very solid notion which
does not depend on the physical reality, nor of any choice of the universal
complete theory that we presuppose. We still have to assume one Turing
universal system, but both theology and physics are independent of which
universal system we start with. I use usually either arithmetic, or the
combinators or a universal diophantine polynomial.
With mechanism, the physical laws are not fundamental, but are explained
“Turing-thropically”, using the logics of self-reference of Gödel, Löb,
Solovay.
To test empirically the digital mechanist hypothesis (in the cognitive science)
we have to compare the physics deducible by introspection by Turing machine,
with the physics observed. Thanks to QM, it fits up to now. But we are light
years aways from justifying string theory, or even classical physics. The goal
is not to change physics, but to get the metaphysics right (with respect to
that mechanist assumption and the mind-body problem). The notion of computation
is the most solid epistemological notion, as with Church’s thesis, it admit a
purely mathematical, even purely arithmetic, definition. Analysis and physics
are ways the numbers see themselves when taking their first person
indetermination in arithmetic into account.

> 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.

I am not sure I understand this.

> 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,

The problem of the existence of infinite computation in the physical universe
is an open problem in arithmetic. Arithmetic contains all non halting
computations, but it is unclear if the physical universe has to be finite or
not. The first person indeterminacy suggests a priori many infinities,
including continua, but the highly counter-intuitive nature of self-reference
suggests to be cautious in drawing to rapidly some conclusion. With mechanism,
a part of our past is determined by our (many) futures.

> and with gravitation or quantum gravity the moduli space is nonHausdorff

That could be interesting. The topological semantics of the theology (G and G*)
are nonHausdorff too.
Could be a coincidence, of course, as physics should be in the intensional
variants of G*.

> 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.

OK, I think. That would make Mechanism wrong. That is testable, but the
evidences favours mechanism.

>
>>
>> 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.

I cannot really judge this. I can agree that this is a bit the ugly part of
that theory (I mean the compactififed dimension), but that is not an argument,
and taste can differ ...

>
>
>
>
>> 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.

That is interesting. Note that with mechanism, we know "for sure” that the
ultimate reality (independent of us the Löbian universal machine) has to be non
dimensional (as arithmetic and elementary computer science is).

Bruno

>
> LC
>
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