Exploring Plank Length & Unrelated Concepts

[Jack]

What is so important about plank length i.e. why are seemingly unrelated things such as curled up dimensions and superstrings plank length?

 

[damgo]

Planck length is basically the distance scale at which quantum effects start warping spacetime. If you try to measure distances smaller than the Planck length, you have to use a high-enough energy probe that GR predicts it will spacetime enough to mess up whatever you're trying to measure. It's also the distance where you would notice quantum vacuum fluctuation seriously warping local spacetime, according to GR.

 

[marcus]

Originally posted by Jack
What is so important about plank length i.e. why are seemingly unrelated things such as curled up dimensions and superstrings plank length?

The Planck units are all of a piece so that this question can be rephrased and answered in various ways.

The Planck length is the wavelength associated with Planck mass or Planck energy or Planck frequency. So if that mass scale is important or the energy or frequency scale is important in the context of discussion then the length will come up too.

Here's a paraphrase of part of what I think Damgo said: Planck mass is 22 micrograms (the mass of a flea) and its wavelength (which says something about how reliably and precisely you can locate it) is Planck length. On the other hand a black hole with
Planck mass has schw. radius twice Planck length. This already doesn't make good sense. How can the black hole geometry work if the uncertainty about locating the singularity is comparable to the size of the hole itself? So QM and GR are not compatible at that scale. Its the scale at which 20th century theories break down and 21st century theories are being constructed.

But there were already clues as early as 1916 that this scale is basic to understanding nature. The main equation of GR has the Planck force in it. the unit of force that belongs to that system of units----pl. mass, pl. length, pl.time, pl. energy etc.----just the way the Newton belongs to the metric units. Planck force is c^4/G. You can calculate it out and it comes to 12E43 Newtons.
or roughly E40tonsforce. It is the central constant in the Einstein equation, the prevailing model of how gravity works.

R_μν - (1/2)g_μν R = (8piG/c^4)T_μν

The stress-energy tensor T has units of energy density or equivalently pressure. Dividing a pressure by a force gives the reciprocal of area----one over an area---also the unit of curvature.
This caltech source begins with a brief chapter reviewing standard cosmology and the chapter starts off with this equation, in case you'd like more discussion, theirs is comparatively clear and carefull.

http://nedwww.ipac.caltech.edu/level5/Sept02/Reid/frames.html

My point is that the equation says that a certain curvature expression (one over some area) is equal to 8pi times an energy density DIVIDED BY THE PLANCK FORCE of c^4/G.

This force is what connects the mass-energy density in a region with the spacetime curvature there.

So it is a fundamental scale of force in nature and this force pushing at the natural unit of speed (c) is a certain power which in turn by Planck's hbar constant is connected to the natural unit of frequency (1 over Planck time). The whole set of scales is so interconnected that when one unit is brought into play the other units are apt to start cropping up as well.

There are three articles by Franck Wilczek in Physics Today online that discuss the many ways in which this scale is basic to emerging theories of nature and the third article has links to the first two so I will give you URL for the third in the series, in case
you want to read something like an authoritative account---

http://www.aip.org/web2/aiphome/pt/vol-55/iss-8/p10.shtml

In a certain way Wilczek is just saying the same thing as Damgo just did, but maybe turned inside out to show the positive side of it----conventional theories don't work well at this scale so here is where we have to start looking to see how things fit together.
Just my impression.