How can the Planck length be claimed to be the smallest length?

  • Thread starter JohnMS
  • Start date
  • #1
JohnMS

Main Question or Discussion Point

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
 

Answers and Replies

  • #2
Chris H. Fleming
On May 5, 1:28 pm, JohnMS <john_m_stan...@yahoo.co.uk> 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



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.
 
  • #3
Igor Khavkine
On May 5, 1:28 pm, JohnMS <john_m_stan...@yahoo.co.uk> wrote:
> Many arxiv papers state that the Planck length
> 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
 
  • #4
J. J. Lodder
JohnMS <john_m_stanton@yahoo.co.uk> 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?


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
 
  • #5
JohnMS
On 6 Mai, 19:19, nos...@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
 
  • #6
Oh No
Thus spake JohnMS <john_m_stanton@yahoo.co.uk>
>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
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)

http://www.teleconnection.info/rqg/MainIndex
 
  • #7
Igor Khavkine
On May 7, 9:20 am, JohnMS <john_m_stan...@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
 
  • #8
Rick
On May 7, 6:20 am, JohnMS <john_m_stan...@yahoo.co.uk> wrote:
> On 6 Mai, 19:19, nos...@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


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
 
  • #9
robert bristow-johnson
On May 6, 1:19 pm, nos...@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.

r b-j
 
  • #10
Chris H. Fleming
On May 7, 10:31 pm, robert bristow-johnson <r...@audioimagination.com>
wrote:
> On May 6, 1:19 pm, nos...@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?
 
  • #11
JohnMS
On 8 Mai, 17:44, "Chris H. Fleming" <chris_h_flem...@yahoo.com> 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?


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
 
  • #12
Igor Khavkine
On May 9, 2:47 am, JohnMS <john_m_stan...@yahoo.co.uk> wrote:

> 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
 
  • #13
donjstevens@aol.com
On May 5, 12:28�pm, JohnMS <john_m_stan...@yahoo.co.uk> 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
 
  • #14
161
0
Planck length - the finest scale?

If the Planck scale were considered as the finest measurement scale; still the general concept of a non-differentiable, no longer smooth manifold, might still be descriptive for a still finer scale. The breaking up of a manifold is just an assumption. After all mathematicians start with the generality of a manifold, and then subsequently add structure such as a metric etc.
 
  • #15
donjstevens@aol.com
On May 7, 11:07am, Igor Khavkine <igor...@gmail.com> wrote:
> On May 7, 9:20 am, JohnMS <john_m_stan...@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). 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
 
  • #16
robert bristow-johnson
On May 15, 5:51 pm, donjstev...@aol.com wrote:
> On May 7, 11:07am, Igor Khavkine <igor...@gmail.com> wrote:
>
>
>
> > On May 7, 9:20 am, JohnMS <john_m_stan...@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).


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
 
  • #17
Reply regarding aliens on planet Zog.

I will redefine the length values so that any advanced people would know their physical value. The L2 value was previously defined as 1/2 of the electron Compton wavelength. The L2 value is also equal to (h/2 mc) where m is the electron mass. The next value needed is L4, defined as (2pi) times the photon sphere radius for the electron mass. In our system, L4 is (2pi) (3Gm/c^2). The L1 value may then be derived from ratio equations as shown.

L1/L2 = L4/L1

L1 = [(L4) (L2)]^1/2 = [(L4) (h/2mc)]^1/2 = (3pi hG/c^3)^1/2

L1 = 2pi (Planck length) (3/2)^1/2

The L3 value is derived from the physical values L1 and L2 with ratio equations. Values L1/L2 will equal L2/L3.

L3 = (L2)^2 (1/L1)

In our units, L3 will be very close to (2pi)^2 (c) (one second). Then the L2 value is equal to [(L1) (L3)]^1/2. The L2 is then equal to 2pi (3pi hG/c)^1/4. If this is precisely correct, the applicable G value must be very close to 6.6717456x10^-11. The ratio equations allow L3 to be developed from physical values.
 
  • #18
John M S, you asked "Is there no hint of a way out?"

A way out was suggested by John A. Wheeler when he proposed that the electron is "--a fossil from -- gravitational collapse". The pattern of equal ratios (below) supports the Wheeler suggestion.

L1/L2 = L2/L3 = L4/L1 = (L4/L2)^1/2 = (L4/L3)^1/3

The ratio L4/L2 is also equal to (3/2)^1/2 times (Planck time) divided by (2pi seconds). These ratios are all equal when the gravitational constant value is 6.6717456x10^-11.

We can now see that L4 and L1 are essentially the same. With gravitational collapse, the L2 photon wavelength is blue shifted to the L1 wavelength. When the twin effect of gravitational length shortening (to match time dilation) is included, the length that is observable from a distance is L4. There is no known reason why these ratios should apply to electrons if electrons are not gravitationally confined (gravitationally collapsed) particles.

Don Stevens
 
  • #19
Many persons have viewed these posts, so it may be appropriate to add some information.

Much of the work needed, to show that a self-confined, single wavelength photon, has the fundamental properties of an electron was done by J. G. Williamson and M. B. van der Mark. Their paper "Is the electron a photon with toroidal topology?" (1997) shows that the electron charge can arise from the toroidal topology of the photon path, in combination with the photon electric field. A left-handed flow produces an electron while a right-handed flow produces a positron. Explanations are provided for half-integer spin, gyromagnetic ratio and de Broglie wavelength. When gravitational collapse is included in this concept, the new model defines the quantized electron mass value and defines a specific relationship to the Planck mass.

electron mass = (h/4pi c) (c/3pi hG)^1/4

Don Stevens
 
  • #20
3,962
20
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
Hi,

I have always felt that Planck units are trying to tell us something fundamental but that view is often undermined by frequent mentions that are is nothing physical or fundamental about the Planck length. In other words the Planck length does not represent a minimum distance. It is easy to see how a discrete Planckian coordinate system falls apart. Imagine a grid of coordinates of based on units of one Planck length. A diagonal can not be made of a whole number of units. The circumference of a circle with radius of one Planck unit is not a whole number of Planck units. Even if we "square off" a circle at the Planck scale to give it a "circumference" of 4 Planck units the diagonals are no longer a whole number. One way out of this dilemma is to forget about the discreteness of distance at the Planck scale and consider the Planck unit of time or its inverse (frequency) as being the more fundamental minimum unit. With time as the discrete unit the problem of Planck circles or diagonals disappears if no longer insist on distance as being discrete. The wavelength of a photon still comes in discrete units of Planck length but that is side effect that is completely determined by the discreteness of time and the constancy of the speed of light. Knowing the frequency of a photon (which is always a discrete inverse multiple of time units) determines the discreteness of its other physical qualities such as energy and momentum. You can also think of the temperature and mass of a Planckian black hole in terms of a characteristic frequency.

You may be familiar with Peter Linde's solution to the Zeno paradoxes which was that time is smooth and continuous and that there is no such thing as an instant of time. He did not make it clear whether he meant that there is no such thing as interval of zero time or no such thing as a minimum discrete interval of time. My suggestion of a worthwhile path of investigation is that distance is smooth and continuous but the quantum discreteness comes about by discreteness of a minimum time interval. I believe it is harder to find processes that can be proven to happen in less than a Planck interval of time than it is to find examples that seem to contradict the limit set by the Planck interval of distance or examples of at the Planck scale that do not appear to be quantisized as whole units of Planck length. Retaining the fundamental discreteness of the Planck time interval while rejecting the discreteness of length solves some of the geometrical problems at the Planck scale while retaining it discrete quantum nature.
 
  • #21
Hi kev,

Yes the Planck units are trying to tell us something. If the electron photon sphere circumference is physical and 1/2 of the electron Comptom wavelength is physical, then the L1 photon wavelength that is the square root of the product of these values is clearly physical also. [L1 = 2pi (Planck length) (3/2)^1/2]

Add to this the observation that the ratio (L1/L2) is equal to the square root of the ratio (3/2)^1/2 (Planck time) divided by (2pi seconds). This implies that the time dilation factor at the photon sphere circumference for any gravitationally collapsed mass has a fixed value.

The L1 wavelength is a critical size because this photon has energy that is defined either by the gravitational constant or the Planck constant.

E = hc/L1 = L1 (c^4) (1/3pi G)

When the two energy values are equal, the L1 wavelength is (3pi hG/c^3)^1/2. This tells us that there is clearly something physical or fundamental about the Planck length.
 
  • #22
3,962
20
Following on from post#20, it can be shown that most of the Planck units are not fundamental minimums or maximums. For example the Planck mass Mp is not a minimum mass as the mass of electron is many orders of magnitude smaller than Mp. For that reason if you consider the electron to be a Planck mass black hole then its Schwarzschild radius is many orders of magnitude smaller than the Planck length Lp. The Planck energy (Ep) is not a fundamental minimum as it has aproximately the energy content of a lightning bolt and there are clearly many examples of energy values that orders of magnitude smaller. It might however be considered as the maximum energy value that a single photon could have, because any higher value would require the photon to have a frequency that is greater than the Planck frequency. The Planck charge (Qp) is also not a minimum value as the elementary charge of an electron or proton is smaller.

I claimed that the Planck time interval Tp is a fundamental minimum time interval and this corresponds to a maximum fundamental frequency f = 1/Tp

All other Planck units can be defined or derived from the f and/or the fundamental physical constants:
c (speed of light),
k (Boltzman's constant),
e (elementary charge),
£ (permitivity of free space)
h (Planck's constant / 2pi)

The gravitational constant (G) is not in the fundamental list as it can be defined in terms of the other constants and f:

G = c^5/h/f^2

Taking f as the base Planck unit the derived Planck units are:

Mass = h f/c^2
Length = c/f
Temperature = h f/k
Area = c^2/f^2
Momentum = h f/c
Energy = h f
Volume = c^3/f^3
Density = h f^4/c^5
Force = h f^2/c
Power = h f^2
Pressure = h f^4/c^3
Charge = 2 e Pi £

The Planck charge (Q) is not a fundamental minimum as it is defined in terms of the elementary charge e and the permitivity of free space (£) which are possibly more fundamental. The fine structure constant (@) can also be defined in terms of the other constants as @^2 = e^2/(4Pi£hc)

The Planck charge Q is used in the remaining derived units as a convenience:

Voltage = h/f/Q
Resistance = h/Q^2
Current = Q/f

Ref http://en.wikipedia.org/wiki/Planck_units
 
  • #23
Hi kev

You said (in post 22) the Planck energy "-- might however be the maximum energy that a single photon could have--".

An argument can be made that the maximum energy a single photon could have is less than the Planck energy. The photon labeled L1 (in post 21) has energy equal to (2/3)^1/2 times the Planck energy. This photon has the energy needed to produce two mass particles, each with a photon sphere circumference equal to the photon wavelength. This photon energy density is enough to produce "limit" space curvature so photon energy has reached a limit.

With this limit, the maximum energy photon has the wavelength 2pi (Planck length) (3/2)^1/2.
 
  • #24
3,962
20
Hi kev

You said (in post 22) the Planck energy "-- might however be the maximum energy that a single photon could have--".

An argument can be made that the maximum energy a single photon could have is less than the Planck energy. The photon labeled L1 (in post 21) has energy equal to (2/3)^1/2 times the Planck energy. This photon has the energy needed to produce two mass particles, each with a photon sphere circumference equal to the photon wavelength. This photon energy density is enough to produce "limit" space curvature so photon energy has reached a limit.

With this limit, the maximum energy photon has the wavelength 2pi (Planck length) (3/2)^1/2.
Hi Don,
I read somewhere that the "gravity" of a matter dominated universe scales as x^(2/3) while a radiation dominated universe scales as x^(1/2). An empty universe scales as X^(1-1/n) where n aproaches infinity. The result is that for a given energy density, matter in the form of pure energy causes a stronger curvature than the the energy equivalent in the form of mass. I am not sure how that comes about, but if it is true it might have to be taken into account when calculating the photon orbit radius.
 
  • #25
Hi kev,
There is a long history of speculation regarding electron quantized mass and electron size so this will probably continue.

Each time a new piece of this puzzle is found to be an exact fit with known electron properties, I feel more convinced that the gravitational confinement concept is correct.

When the photon wavelength 2pi (3pi hG/c)^1/4 , labeled L2, is 1/2 of the electron Compton wavelength, the electron mass is (h/2c) (1/L2). The gravitational attraction force between two electron mass particles separated by the distance !/2 Compton wavelength is interesting because the G value cancels.

F = G m^2 (1/L2)^2

F = G (h/2c)^2 (1/L2)^2 (1/L2)^2

F = G (h/2c)^2 (1/2pi)^4 (c/3pi hG)

F = (h/12 pi c) (1/2 pi)^4 = 3.761704x10^-47

This attraction force is also defined as shown below.

F = G (9.1093819x10^-31)^2 (1/1.2131551x10^-12)^2

This force is equal to 3.761704x10^-47 (as found earlier) only when the G value is 6.6717456x10^-11. With this G value, the length 2pi (3pi hG/c)^1/4 is 1.2131551x10^-12 meters. This piece of the puzzle has a good fit.
 

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