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

Many arxiv papers state that the Planck length
is the smallest measureable length.

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?

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.

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?

Many researchers in quantum gravity assume Planck length as a

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.

On 6 Mai, 19:19, nos... at (no spam) de-ster.demon.nl (J. J. Lodder) wrote:

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

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.

On May 6, 1:19 pm, nos... at (no spam) de-ster.demon.nl (J. J. Lodder) wrote:



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.

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.

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

On May 7, 9:20 am, JohnMS <john_m_stan... at (no spam) yahoo.co.uk> wrote:

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

On May 7, 11:07am, Igor Khavkine <igor... at (no spam) gmail.com> wrote:



On May 7, 9:20 am, JohnMS <john_m_stan... at (no spam) yahoo.co.uk> wrote:

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

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).