From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Length scales in physics 4 - the Planck length
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Tue, 18 Aug 92 22:26:08 GMT


Now for one final length scale - still smaller.  This is the length
scale at which quantum gravity should become important - the Planck
length l.   On the scale of the Planck length, it's possible that the
structure of spacetime becomes quite different from the
four-dimensional manifold we know and love. Spacetime itself becomes a
foam (according to Wheeler) or a bucket of dust (according to Wheeler)
or a bubbling sea of virtual black holes (according to Hawking) or a
weave of knots or tangles (according to Ashtekar, Rovelli, and
Smolin).  In short, it's weird, but beyond that nobody really knows. 
To be more precise, the Planck length is the length scale at which
quantum mechanics, gravity and relativity all interact very strongly.
Thus it depends on hbar, c, and Newton's gravitational constant G. 
These have dimensions  

hbar = ML^2/T
c = L/T
G = force distance^2/mass^2 = (ML/T^2) L^2/M^2 = L^3/MT^2

so to get a length we have to use

l = (hbar G/c^3)^{1/2},

This makes some sense because the bigger hbar is, the more "quantum"
the universe is, so the bigger the length is at which quantum gravity
matters.  Also, the bigger G is, the stronger gravity is, so the
bigger the length is at which quantum gravity matters.  The bigger c
is, the less "relativistic" the universe is, so the smaller the Planck
length is.  Of course, Planck's constant and the gravitational
constant are actually very SMALL, so the Planck length is really
small.  I don't have a calculator on hand but I seem to recall that
the Planck length is about 10^{-33} meters.  This is WAY smaller than
the length scales I was talking about before - RIDICULOUSLY smaller. 
That's why we haven't seen any (obvious) signs of quantum gravity
effects, and why it will be so hard to do any quantum gravity experiments.  

Note that in all the previous three examples a length scale was proportional
on the inverse of a mass - in particular, the electron mass.  The
Planck length is peculiar in that it does not depend on a mass in this
way.  Of course, it depends on the gravitational constant, which has a
lot to do with mass!  In fact, the combination of gravity, relativity
and quantum mechanics sets a mass scale - the Planck mass - as well as
a length scale.  The Planck mass is huge (by particle physics standards) so the
Planck length is puny.

A rough way of understanding the Planck length is as follows.  
Every mass determines a Schwarzschild radius - that is,
the radius of the event horizon of a black hole having that mass.  Now
this is curious, in that I've been saying repeatedly  that a mass
scale sets an inverse length scale, but the radius of a black hole
isproportional* to its mass.  Of course, this is dimensionally possible 
in that the gravitational constant involves units of mass.   
We'll work out the Schwarzschild radius of a given mass in a minute. 
But also every mass determines a Compton wavelength, as I explained earlier:

L_{compton} = hbar/mc						1)

We can then work out how big a black hole we need for its Compton
wavelength to equal its Schwarzschild radius!  This sort of black hole
will have mass about equal to the Planck mass, and radius about equal
to the Planck length.  

What does this mean?  Well, remember that the Compton wavelength of a
particle is the length scale at which quantum field theory becomes
very important in describing it.  So the Planck length is the size of
a black hole for which quantum field theory becomes very important. 
Hawking has predicted that black holes of any size emit radiation due to
quantum-field-theoretic effects - the bigger the black hole, the less
radiation.  His calculations treat the black hole classically and only
use quantum field theory in treating the electromagnetic radiation. 
For a black hole about as big as the Planck length one would expect
this approximation to break down drastically.  

To be picturesque, we can say that if we have a black hole about the
size of the Planck length, and we try to locate it to an accuracy
equal to its radius, the Heisenberg uncertainty principle makes the
the momentum of the black hole so poorly known that there may be
enough energy around to create another black hole of that size!  I
warn the reader to take this with a massive grain of salt, since there
is no good theory of this sort of thing yet - much less any
experimental evidence.  But people have sharpened this sort of
thought experiment and seen that things get awfully funny at the Planck
length.  By analogy with particle physics, one might expect processes
involving virtual black holes to be very important at this length
scale.  Hawking and others have written interesting but papers on
reactions induced by virtual black holes... but I would not take
their predictions too seriously yet.

Okay - let's start with Newtonian gravity:

force = -Gm_1m_2/r^2

and remember that we can write the potential energy as

V = -Gm_1m_2/r

Now let's say we have a little particle with mass m_2 in the field of
a big thing with mass m_1, and let's compute its escape velocity. 
That's the velocity for which its kinetic energy plus the potential
energy above is zero, i.e.:

m_2v^2/2 = Gm_1m_2/r

Calling m_1 simply m, we get

v = (2Gm/r)^{1/2}

Black holes were in fact predicted before general relativity simply by
noting that the escape velocity can become larger than c, so that
light cannot escape!  Since we are being deliberately sloppy in these
articles, let's use that idea to guess the Schwarzschild radius of a
black hole of mass m.  We get

c = (2Gm/r)^{1/2}

or 

r = 2Gm/c^2							2)

I don't have any books at hand at the moment, but I know this isn't
too far off, assuming I did the algebra right.  I believe it's a bit
too low, due the nonlinearity of general relativity.  It's just one of
those constant factors that we are blithely ignoring here.  I'm shocked 
that even I kept the "2" around above!

Okay, so now set the Schwarzschild radius - 2) - equal to the Compton
wavelength - 1) - and forget that darn "2", getting

m^2 = hbar c/G

for the Planck mass.  Plugging that into formula 1), we get
the Planck length: 

l = (hbar G/c^3)^{1/2} 

as expected.

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