How can the Planck length be claimed to be the smallest length?
JohnMS
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
Chris H. Fleming
The length scale you describe is the radius in which you would have to
fit the Compton wavelength of the particle for quantum gravitational
effects to come into play.
You provided a formula that gave a length scale smaller than the
Planck length. The question is, does this formula describe anything
physical?
For the electron this length scale is 10^-57 m
For comparison, the electron's classical radius is 10^-15 m
I do not believe any laboratory experiments have confined an electron
to the degree you require.
Igor Khavkine
> is the smallest measureable length.
Most of the time, this is merely a heuristic. The Planck length comes
up as the scale beyond which quantum gravitational effects become non-
negligible. This is shown using dimensional analysis, much in the same
way as the Bohr radius, beyond which the full quantum mechanical
description of the Hydrogen atom cannot be neglected. Note that the
Bohr radius was derived before a modern quantum mechanical treatment
of Hydrogen became available. A similar statement can be made about
the Planck length.
Current established theories neither require nor propose a minimal
length. However, tentative models proposed for quantum gravity
sometimes assume some kind of discreteness or granularity of space-
time at the Planck scale. Thus, the nature of the Plank length as the
smallest measurable one should be considered as one of the hypotheses
assumed by these models. While there are arguments for the validity of
this hypothesis, like all others, it must be subject to experimental
verification.
If you want a more detailed discussion of how to resolve your
"paradox", you'll have to specify which model of quantum gravity you
are assuming.
Hope this helps.
Igor
J. J. Lodder
The Planck length comes from dimensional analysis.
If we agree to have c = \hbar = G = 1 and dimensionless,
then lengths must be expressed in multiples of the Planck length.
By it's very nature dimensional analysis
has nothing to say about numerical factors.
Neither does it stipulate that these factors must be of order one.
To go beyond dimensional analysis (and get the factors)
you need a physical theory.
One that explains for example why the electron mass
is so small in terms of the Planck mass.
Since this is sadly lacking we do not know
whether or not statements like:
gravitational length = order 10^{-20} Plancks
do or do not have physical meaning.
Purely from dimensional analysis
there is nothing wrong with them,
hence no paradox.
Best,
Jan
JohnMS
> To go beyond dimensional analysis (and get the factors)
> you need a physical theory.
> One that explains for example why the electron mass
> is so small in terms of the Planck mass.
>
> Since this is sadly lacking we do not know
> whether or not statements like:
> gravitational length = order 10^{-20} Plancks
> do or do not have physical meaning.
> Purely from dimensional analysis
> there is nothing wrong with them,
> hence no paradox.
String theory is often claimed to have a minimum length
(the Planck length or near it), so does loop quantum gravity,
so do many other approaches. The theory-independent approaches
argue convincingly that lengths below the Planck length
cannot be measured by any known procedure or device.
The electron mass is also well known, and
R=2GM/c^2 is not in doubt, as electrons
have gravitational effects. It seems difficult to say
that R=2GM/c^2 is wrong
for electrons.
So in practice there IS a paradox, because
electrons are known experimentally to
have a gravitational length smaller than the
Planck length.
Is there no hint of a way out?
John
Oh No
fundamental length scale. This should be regarded as a hypothesis for a
model, not as an established fact. It is well below the scale of
measurable lengths. My own research hypothesises that L=2Gm/c^3 is a
fundamental time. Of course, I think that is more promising. :-)
Regards
--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)
Igor Khavkine
> String theory is often claimed to have a minimum length
> (the Planck length or near it), so does loop quantum gravity,
> so do many other approaches. The theory-independent approaches
> argue convincingly that lengths below the Planck length
> cannot be measured by any known procedure or device.
Unfortunately, neither string theory nor loop quantum gravity enjoys
the status of an experimentally verified theory. The theory-
independent approaches are precisely the ones that use dimensional
analysis, as discussed previously by J. J. Lodder and myself.
> The electron mass is also well known, and
> R=2GM/c^2 is not in doubt, as electrons
> have gravitational effects. It seems difficult to say
> that R=2GM/c^2 is wrong
> for electrons.
What would it mean for the above paragraph to be wrong? You've defined
a length scale R. You can write down any length you want and call it
R, no-one will argue with you solely on that basis.
> So in practice there IS a paradox, because
> electrons are known experimentally to
> have a gravitational length smaller than the
> Planck length.
I see. And what is this experiment that has measured the gravitational
length of the electron? None of the modern particle physics
experiments have probed lengths that are even withing a few orders of
magnitude from the Planck length, not to mention beyond it.
Hope this helps.
Igor
Rick
Paradoxes point out a flawed argument.
The relationship reality -> mathematical structure is one
to many.
The problem here is an assumption that the relationship
(a particular mathematical structure) -> reality is one
to one. There in lies the rub.
Perhaps there is no hard reality for a minimum length, or
Oh No's divide it by c to dimensionaly come up with a time.
That does not mean it can't have some value in a model based
representation. Just do not confuse the model with the singular
notion of reality.
Rick
robert bristow-johnson
>
> The Planck length comes from dimensional analysis.
> If we agree to have c = \hbar = G = 1 and dimensionless,
> then lengths must be expressed in multiples of the Planck length.
>
> By it's very nature dimensional analysis
> has nothing to say about numerical factors.
> Neither does it stipulate that these factors must be of order one.
> To go beyond dimensional analysis (and get the factors)
> you need a physical theory.
> One that explains for example why the electron mass
> is so small in terms of the Planck mass.
because the Bohr radius is
a_0 = ((4 pi \epsilon_0) \hbar^2)/(m_e e^2)
which is
a_0 = (m_P/m_e) (1/ \alpha) l_P
where m_P is the Planck mass, l_P is the Planck length, and \alpha is
the Fine-structure constant (which is not a particularly huge or tiny
number).
it's been said that the reason gravity is so weak is really because
the masses of particles are so small ( m_e <<<< m_P ). but the reason
that particle masses are so small is the same reason that the size of
atoms are so big ( l_P <<<< a_0 ) which is another what of saying that
the Planck length is so small (compared to the sizes of atoms or even
particles contained therein).
but, out of ignorance, i dunno why anyone says that it's the smallest
length. it's a quantity, we can always define a length much smaller.
but such a teeny length might not be in the ballpark of any physical
thing. hell, maybe not even the Planck length is comparable to any
physical thing. but i think the reason that the Planck length is tiny
compared to the radius of an atom is the same reason the Planck mass
is huge compared to that of an atom.
r b-j
Chris H. Fleming
wrote:
If gravity quantizes around the planck scale, then below that scale
one does not have the convenience of a classical metric. Without a
metric, how do you define length?
JohnMS
> If gravity quantizes around the planck scale, then below that scale
> one does not have the convenience of a classical metric. Without a
> metric, how do you define length?
If one takes 10^24 atoms of silicon in a single
crystal (around 1 kg)
it is undisputed that the whole object has a
measureable gravitational length.
The crystal bends space-time around it and attracts
other masses;
that is easy to measure. And there is a definite
metric in our environment.
It is also undisputed that all atom masses
in the crystal essentially add up
(the crystal binding energy can be neglected here).
Since the gravitational length of the silicon
crystal is defined as R=2GM/c^2,
it is very hard to avoid saying that every silicon atom
has a gravitational length given by the
same formula, this time using the atomic mass.
However, the gravitational length calculated
in this way for one atom is much smaller
than the Planck length. (about 10^18 times smaller).
So it does seem that much smaller lengths than a Planck
length have a physical meaning...
John
Igor Khavkine
> If one takes 10^24 atoms of silicon in a single
> crystal (around 1 kg)
> it is undisputed that the whole object has a
> measureable gravitational length.
> The crystal bends space-time around it and attracts
> other masses;
> that is easy to measure. And there is a definite
> metric in our environment.
>
> It is also undisputed that all atom masses
> in the crystal essentially add up
> (the crystal binding energy can be neglected here).
> Since the gravitational length of the silicon
> crystal is defined as R=2GM/c^2,
> it is very hard to avoid saying that every silicon atom
> has a gravitational length given by the
> same formula, this time using the atomic mass.
It is true that a kilogram of silicon has a measurable gravitational
field. This field corresponds to space-time curvature of order 1/L,
where L is some length. We know for a fact that these gravitational
effects are very weak, which in turn implies that L must be very
large. When we measure gravitational effects due to this hunk of
silicon, it is L that we measure, not the R that you've defined above.
R would be the size of the hunk of silicon if it were dense enough to
become a black hole. Since it is not a black hole, we have another
demonstration that R is irrelevant to the physical situation.
In short, your paradox is avoided because, no matter how small R is,
it never comes up as an experimental measurement; only L does. And L
is of regular macroscopic proportions.
Hope this helps.
Igor
donjs...@aol.com
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
donjs...@aol.com
Hi Igor; Some photon wavelength equations relate the Planck length to
the electron mass.
L1/L2 = L2/L3
L1 = (L2)^2 (1/L3) = 2pi (3/2)^1/2 (Planck length)
Where L2 is the photon wavelength with energy equal to the mass energy
of one electron plus one positron and the wavelength L3 is (2pi)^2 (c)
(one second). The L2 length is then equal to [(L1) (L3)]^1/2. When L2
is defined, the electron mass energy can have only one quantized
value.
Don Stevens
robert bristow-johnson
so, if human beings decided to use a different unit of time than a
second, L3 (and then L1) would come out to be a different physical
value?
i have trouble imagining that physical reality gives a rat's ass what
unit of time we humans happen to use. or the aliens on the planet
Zog.
r b-j