## LIGHT AS A BLACK HOLE: PLANCK LENGTH, Or How To Find What Matters In Physics

The following show what is important in physics, and what is not. Some may object to the following: how come I don’t use Quantum Field Theory (QFT) and General Relativity? Well first, nobody knows how to use QFT (except in reduced and admirable circumstances which brought zillions of Nobel Prizes).

However the two most basic axioms of QFT will be used. They are the basic Quantum of Energy axiom: E = hf, which is tied in with the Quantum Wave axiom. What does that represent? h is Planck constant, and “f” the frequency of the “particle”… Here a photon . Frequency? To all particles is associated a wave . Here that’s the plain old electromagnetic wave. So we have: rf = c… where “r” is just the photon wavelength, and c the speed of light. We will use “r’ to squeeze the photon inside itself.

NAIVE COMPUTATION  OF SIZE OF LIGHT INCAPABLE OF ESCAPING ITS OWN MASS-ENERGY: (1) Gravitation law. (2) Energy as the integration of Force over Distance. (3) Quantum Energy. (4) Definition of Speed of Light. (5) Mass = Energy. End Result: PLANCK LENGTH = square root of (Gh/ccc)

Another ingredient is the old F = G m M/ rr, the universal attraction law, older than Newton.

Here we will use it as self gravitation: m = M = “mass” of photon. Self gravitation is nothing surprising: that’s what holds all celestial bodies together, and not just stars and planets, but also galaxies and clusters of galaxies (so-called “virial” theorem). So here I just apply it to light.

Another possible controversy is whether a photon has “mass”. That was demonstrated by the French Henri Poincare (and stolen by Einstein). The root of the proof is as old as Buridan (and the big difference with Aristotle ridiculous physics). What Buridan said was that force = change of impetus. This is the so-called Second Law of Newton, F = ma… except in the slightly more general notion F = m dp/dt… where p is the “impetus” aka ‘momentum… 

Thus, according to Buridan, whatever exerts a force has an impetus, and the instantaneous force is the quotient of the infinitesimal change of impetus by the infinitesimal change of time. As electromagnetic radiation (“light”) exerts a pressure (Poynting vector, light sails), thus a force, it has impetus, thus mass… “inertial” mass. But then observations tell us inertial and gravitational masses are the same (“Principle of Equivalence”). So photons have gravitational mass m, given by E = mcc. Hence m = E/cc = hf/cc.

Now we squeeze the photon inside its own wave. So “r” will be both the scale at which the gravity is used and from which distance one had to make light work against itself to expand to infinity (so we expand from a particular r, R(index zero in the drawing). So “r” is a variable, gives rise to an infinitesimal, dr, and we have

F = F(r) = G mm/rr.

So the total energy to escape to infinity from R is the integral of   (Gmm/rr) dr = Gmm/R = G hhff/cccc R =

Ghhcc/ccccRRR

and that has to be less than the energy contained in the light to start with, namely: hf = hc/R.

So we get: G h/(c^3) (R^2) < 1, or: R = square root of (Gh/c^3).

Notice I used no fancy General Relativity, no Schwarzschild Radius…. And still I get the same result as if I had! It is as if General Relativity didn’t matter! How come?  Because, in truth it doesn’t indeed matter very much.
In a full relativistic treatment, the force-acceleration relationship is different from the classical one. Full relativity shows: Force = m ((GAMMA)^3) a… … However, this Relativistic Force is irrelevant here, as the Force used is the gravitational force… which is, in General Relativity, the same good old one used by Newton.

Also I didn’t use the Uncertainty Relationship(s)… as some treatment of the Planck Length do. But there again, this is because the Quantum Wave Principle contains it already (QWP –> UR).  hence we can see what principles are important, and which ones are less so. General Relativity is not a deep theory, just a slight modification of Classical Mechanics: time slows down near large masses, so forces are exerted longer. That force = curvature is an idea from Riemann, already nearly explicit in Buridan (who said planets were inertial on their circular orbits around the Sun).

Now the Planck Length tells us that gravity and the inner geometry of elementary particles are intimately tied in. Also there should be an absolute energy transfer limit…

Reconstructing basic physics from basic principles is a valuable exercise…

Patrice Ayme

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 The remarkable idea of E = hf came from Planck, and then was used spectacularly, but very simply, by Einstein to explain the photoelectric effect… to Planck’s anger (!), and was then expanded by De Broglie to all particles… As a Principle.

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 what I call a QUANTUM wave. Traditional names are “matter waves”, or “De Broglie waves”… because De Broglie invented the idea, beyond light. Here of course the Quantum wave is just the plain old electromagnetic wave.

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 The difference between F = ma and F = m dp/dt is inexistent in classical physics… but NOT in Relativistic Physics… In relativistic physics, the momentum p, the impetus, is not just (mv), momentum multiplied by speed, as Buridan had it, but (GAMMA) mv. Where GAMMA is the usual inverse square root of (1-vv/cc).

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P/S: Virtual particles, that is, roughly the corrections QFT does to Field Theory, rests on time energy uncertainty (DeltaE)(Delta t) > h

### 14 Responses to “LIGHT AS A BLACK HOLE: PLANCK LENGTH, Or How To Find What Matters In Physics”

1. pshakkottai Says:

Amazing!

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2. Gmax Says:

Deeper than deep. So you tear the photon to infinity, and you saying much modern physics ain’t that deep?

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• Patrice Ayme Says:

Bib bang cosmology also tear photons to infinity…

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3. ianmillerblog Says:

Now, is that face a clue as to what is going on here?

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4. pshakkottai Says:

Is that face Nature’s mask of mystery?

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5. Dan Elbert Says:

Hi Patrice
I don’t follow the physical interpretation of your calculation. Do you mean that a sufficiently energetic photon will not propagate? Or that it cannot interact with matter?

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• Patrice Ayme Says:

Yes, well… It means there is a problem. I infuriated a number of household names, long long ago, by applying the same argument to the putative graviton… here I used it to say GR is not that important (as the GR computation gives exactly the same result… whereas in special relativity the force law is modified, here, doesn’t matter…)

So, sometimes I learned not to jump to conclusions too much… It probably means something deep in connection with my own SQPR…

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• Dan Elbert Says:

You mean that a graviton should nit be able to escape a black hole and therefore the black hole should not be able to interact gravitationally with the rest of the universe?

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6. pshakkottai Says:

When the Plank length and Schwarzfield radius are equal when m is the Plank mass. This is discussed in
http://math.ucr.edu/home/baez/planck/node2.html
and http://math.ucr.edu/home/baez/planck/node3.html
in connection with what an approach to quantum gravity should be like.
This is from
Higher-Dimensional Algebra
and Planck-Scale Physics

John C. Baez

Department of Mathematics, University of California
Riverside, California 92521, USA
January 28, 1999

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• Patrice Ayme Says:

Initially, Planck length was derived from the dimensional argument: make a quantity with one space dimension, using fundamental constants. It’s interesting that there is actually a physical argument for it. And not anymore stupid than the 3K cosmic background radiation, which also depends upon stretching photons…

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• pshakkottai Says:

Agreed! But do you think quantum mechanics and gravity will combine only at impossibly small scales?

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• Patrice Ayme Says:

Excellent question! Let me go on a rampage about various logical divergences in physics.
The fundamental problem of the modern gravitation, aka General Relativity, is that gravity is viewed as general inertia. The idea, weirdly is very old: Buridan, circa 1350 CE!
Buridan indeed explained the planets’ orbits around the Sun as circular inertia…
Fast forward to 1863, when Riemann created higher dimensional manifolds, complete with curvature… And the statement that force = curvature…
GR just embodies the Buridan-Riemann program, with [(Ricci) Curvature = Mass-Energy]…

All very nice and beautiful. But then, what about the other force, electromagnetism? Turns out it’s one dimensional… So Kaluza-Klein got a 5 dimensional theory… However then two other forces were found: the strong, and the weak. The strong force seemed to have a potential like electromagnetism and gravity: 1/r, but multiplied by a decreasing exponential: exp(-mr). So we get: (exp(-mr))/r… Then, turned out the strong force was even worse than that, with asymptotic freedom, quarks, gluons…
It’s not clear how to geometrize this.
Then a generalization of KK was tried, a sort of String Theory, with Calabi-Yau compact manifolds replacing the circle of KK…
From my point of view, that’s a wild goose chase… my relationship with Yau went down the drain…

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7. Gmax Says:

BTW, what is that absolute energy transferry limit? You seem to like the absolute. Absolute highest number, absolute wealth limit withe Romans

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