What is the relationship between the Planck length and black holes?

[edguy99]

This article is posted on twitter with the byline: "You, me, and even aliens from Alpha Centauri could arrive at the same Planck units." but does not explain why.

http://www.symmetrymagazine.org/article/the-planck-scale

I believe I have read (but cannot find the reference now) the following explanation:

The energy of an electromagnetic wave is inversely proportional to its wave length. The Planck Length is so small that calculations show something vibrating within that length, would be so energetic it would become a black hole.

Does anyone know if this is true or false?

 

[phinds]

Have you tried simply Googling "what is Plank length"? It is well defined and

The Planck Length is so small that calculations show something vibrating within that length, would be so energetic it would become a black hole

sounds ridiculous to me.

https://en.wikipedia.org/wiki/Planck_length

 

[Nugatory]

This question comes up so often that it merits its own Insights article: https://www.physicsforums.com/threads/a-hand-wavy-discussion-of-the-planck-length-comments.831640/

I believe I have read (but cannot find the reference now) the following explanation:

The energy of an electromagnetic wave is inversely proportional to its wave length. The Planck Length is so small that calculations show something vibrating within that length, would be so energetic it would become a black hole.

"I believe I have read" doesn't even come close to meeting the minimum standards for citing a source here - and without more to go on, we have no way of knowing whether your source is wrong or you have misinterpreted or misremembered what you might or might not have read.

 

[edguy99]

Perhaps I should rephrase the question "Since the energy of an electromagnetic wave is inversely proportional to its wave length, at what wavelength would a photon become so energetic that schwarzschild radius would match the wavelength of the photon.?"

Is the answer to this question 1.616199(97)×10−35 meters?

 

[edguy99]

@Nugatory Thank you for the reference. "Basically, the Planck length is the length-scale at which quantum gravity becomes relevant. It is roughly the distance things have to be before you start to consider “hmm I wonder if there’s a chance this whole system randomly forms a black hole.” I did not really understand this until I convinced myself with the following derivation, which was the main inspiration for this article."

Found it! http://math.ucr.edu/home/baez/lengths.html#planck_length - thanks @john baez

 

[jimgraber]

""The Planck Length is so small that calculations show something vibrating within that length, would be so energetic it would become a black hole ""

"sounds ridiculous to me."
On the contrary,
actually this (first quotation) is a slightly poetic version, but it is essentially true.

For every length, you can determine two energies, one for a photon (of that wavelength) and one for a black hole (of that Schwarzschild radius). At the Planck length, those two energies will be equal. At longer lengths, the black hole will be more energetic than the photon. At shorter lengths, the photon is more energetic than the black hole.

One can check it out using these three formulas:

E=Mc^2,

E=hc /lambda (photon energy)

R =2G M/c^2 (Schwarzschild radius for black hole)Jim Graber

 

[phinds]

Oh, well. I should know better by now than to think that just because something sounds ridiculous to me, that that has any bearing on whether it's true or not when it comes to quantum mechanics

 

[edguy99]

Some pretty interesting discussion around the internet concerning Planck Length. Wikipedia does not even mention that fact that the Planck Length is where the wavelength of a photon matches the schwarzschild radius of the energy the photon contains. Seems strange not to mention something that important.

 

[Nugatory]

Wikipedia does not even mention that fact that the Planck Length is where the wavelength of a photon matches the schwarzschild radius of the energy the photon contains. Seems strange not to mention something that important.

It's not as important as it appears. The relationship between energy and Schwarzschild radius requires the assumption that quantum effects are negligible and the relationship between energy and wavelength requires the assumption that gravitational effects are negligible. Both assumptions fail under conditions where the two energies are anywhere near approximately equal, so neither calculation produces a meaningful result under those conditions. Thus, we have two numbers that don't really mean much of anything but happen to come out the same.

 

[edguy99]

I am open to other suggestion (and would like to hear them), but to me, "where the wavelength of a photon matches the schwarzschild radius of the energy the photon contains" is the easiest and most sensible explanation of what a Planck Length is. Love it :)