Re: How can the Planck length be claimed to be the smallest length?
- From: donjstevens@xxxxxxx
- Date: Sat, 10 May 2008 19:18:45 +0000 (UTC)
On May 5, 12:28�pm, JohnMS <john_m_stan...@xxxxxxxxxxx> wrote:
Many arxiv papers state that the Planck length
is the smallest measureable length.
On the other hand, the gravitational length
L=2Gm/c^2
associated with every electron or proton
is 19 or 22 orders of magnitude smaller
than the Planck length.
Nobody seems to doubt either
of the two statements.
What is the exact answer to this paradox?
One can imagine at least 3 solutions:
1 - Lengths of objects can be smaller than L_Planck,
but not positions.
2 - Lengths can be smaller than L_Planck if
one makes many measurements and then makes
a statistical average.
3 - There is an uncertainty relation between
length L and position x:
� � � � �L x > L_Planck^2
There might be other answers. What is the
canonical answer by researchers to this question?
Thanks!
John
Hello John; Let us suppose that the gravitational length (radius) of
the electron is L: where L = 3Gm/c^2. This is the photon orbit radius
for the electron mass. Next, suppose that the shortest meaningful
distance is Planck length times the square root of (3/2). The
circumference is 2pi (radius). A circumference is (2pi) (Planck
length) (3/2)^1/2. This value is (3pi h G/c^3)^1/2. When a photon with
energy equal to the mass energy of one electron plus one positron is
gravitationally blue shifted to the wavelength (3pi h G/c^3)^1/2, the
size reduction factor is (L/L)^2 rather than (L/L). This is because
distance is shortened to match time dilation. The observable length
will then be equal to the photon orbit circumference, 2pi (3Gm/c^2),
while the radius is 3Gm/c^2. This is discussed in "Talk:Black hole
electron", Wikipedia.
Don Stevens
.
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