measuring and visualizing Planck quantities
The historical record of when fundamental constants were first measured
gives an idea of how accessible the various ones are. The easiest Planck
quantity to measure is the speed of light. It was measured as early as 1676
by the Danish astronomer Ole Römer.
While the earth is approaching Jupiter, events there seem from our point
of view to occur at a slightly increased tempo. A measurable advance has
accumulated by the time the earth is at its closest. While the earth is
retreating from the giant planet, events there seem very slightly too slow,
until, by the time the earth is at its farthest, a measurable retard can be
detected. Römer kept track of the Jovian moon Io over a period of time,
recording when it dove into shadow. He was able to detect and measure the
delicate syncopation in its schedule. From this he deduced that light travels
at a finite speed and calculated the first estimate of that speed. This rhyme
can serve as a reminder:
When Jove at Midnight doth arise
And mount the early morning skies,
Then all his moons appear to race--
Towards him the Earth directs her pace.
But if he stays abed 'til Noon
It seems to slow each Jovian moon.
The Earth retreats--her speed of flight
Is one tenthousandth that of light.
It is said that Römer's result was published in the Journal des
Sçavans, around December of 1676. The Doppler effect, which has a
formal similarity to what Römer noticed, was not described until 1842,
almost two centuries later. It was clever of Römer to grasp the reason
for the advance and retard in schedule. He was the first human to measure a
Planck quantity.
The force is the next most easily measured quantity. Essentially it was
experimentally determined by Henry Cavendish as early as 1798, although his
result was not originally interpreted in the form of a force constant.
Cavendish devised an ingenious torsion balance for measuring the almost
imperceptible gravitational attraction between ordinary-sized objects.
Another way of judging the accessibility of a quantity like
is to imagine how we might
measure it for ourselves--more simply than by Cavendish's method--in an
idealized thought-experiment.
Under weightless conditions, the attraction between two stationary round
objects braced a certain distance apart might be measured with a sufficiently
sensitive force gauge. This would also constitute a measurement of
itself if we could
independently deduce the relationship between
and the measured attraction. We
would need to know a ratio of forces--what fraction of
the measured force is. But with
the balls at the same separation and in circular orbit around each other,
this fraction can essentially be observed from the orbital speed. For our
purposes what is most significant is the combined orbital speed. This is the
sum of the two speeds measured separately, and also the speed of either one
as seen from its partner's point of view. We will use the symbol
to denote the combined speed
expressed as a fraction of the speed of light.
The speed of each, determined separately from a still vantage,
represents a measured share of the combined speed. It will be handy to have a
temporary name for the result of multiplying together the two shares into
which the combined speed is split--when needed this factor will be called the
split. It is a significant part of the picture for us because it indicates
how the mass is distributed between the two bodies.
Regardless of the bodies' sizes and densities, regardless likewise of how
close together or far apart they are, we can tell the attraction between them
simply from their combined speed and from the two fractional shares into
which the combined speed is split. As a fraction of the Planck force, the
pull is equal to the combined speed raised to the fourth power and multiplied
by those two shares.
Seen from earth, the moon travels 3.4 millionths of the speed
of light--3.4 times a common speed for sound in cold air. The combined speed
is 3.4 millionths, and its fourth power is readily found with pocket
calculator. In the case of earth and moon, one body is about 1/80 of the
total mass and the other is about 79/80. These multiplied together yield a
split of about 0.012, which is indicated by the partitioning of orbital
speed. As a fraction of , the
pull between earth and moon is simply
0.012
(0.0000034)
.
The next time you happen to notice that the moon is still on track, you may
attribute this to a definite small fraction of Planck force.
Conversely, if we can measure the attraction between the two objects with
a force-gauge while they are standing still, we can learn the Planck
force itself by dividing that measured force by the two shares and by the
fourth power of . The
following verse can serve as a reminder of this approach to measuring
, here referred to as the
"universal standard weight".
Between a circling pair, the pull,
Divided fourthly by their gait,
And by both fractions of the whole,
Is universal standard weight.
This measures the Planck force in whatever units of force the gauge uses.
If the gauge reads pounds then the Planck force will turn out to be expressed
as a large number of pounds. If the gauge reads force in newtons or in ounces
then the answer will come out to be an even larger number of newtons or of
ounces.
Because I consider direct contact with Planck quantities to be intrinsically
interesting, I will devote additional space to gravitational pull. In the
model just presented, partitioning of speed reflects partitioning of
mass--one is a reversed image of the other. In order to make some numerical
examples more concise, the result of multiplying the two speed-shares
together will temporarily be called the split. If the two speeds
are equal then the split is 0.25, because each share is half the combined
speed and a half times a half is a quarter. On the other hand, if one speed
is 8 times the other then the two shares are 1/9 and 8/9, making the split
1/9 8/9 or about one tenth. For
clarity, the same equation is written here in two styles: verbal and symbolic.
The ratio of forces is split beta fourth.
= split
One might picture situations in which the combined orbital speed is
10
,
so that
is
10
.
Referring to a metronome set at tempo moderato to beat once per tick, the
pair can be imagined moving one span with each beat--or ten beats to the
pace. This speed might be appropriate for a double asteroid (one has been
photographed during a recent fly-by). For definiteness, suppose that the two
masses, and therefore also the two speeds, are in the ratio of one to eight
so that the split is nearly 0.1. In that case, split
must be about
0.110. Using
10
stone for , a simple calculation
shows the force of attraction to be one million stone. The pull, in other
words, is on the order of a thousand tons.
On the other hand suppose the combined orbital speed is
10
,
so that
is
10.
This is ten metronome beats to go the width of a finger--ten tick to travel a
tenth of a span. Two solid piano-sized or refrigerator-sized objects might
have that speed if they were orbiting within arms-reach of each other. More
massive pairs could also display the same speed were they orbiting at wider
remove. Suppose the two bodies are equal so that the split is 0.25. In that
case, split
must be
0.2510.
Using
10
stone for F, a simple calculation shows the force to be 0.025 stone. That is
about one ounce, something measurable with a postage scale.
The first person to count electrons was G.J.Stoney in 1874, and he did it
in order to estimate the electron charge in terms of a conventional unit like
the "coulomb". Some forty years earlier Michael Faraday had determined the
amount of charge required to liberate a gram of hydrogen by
electrolysis--there was then no way of quantifying charge other than in
conventional bulk terms. Stoney surmised the existence of a natural unit of
charge, to which he later gave the name "electron". He supposed that in
electrolysis it took one natural unit of electricity to liberate each atom of
hydrogen--so that by estimating the number of hydrogen atoms in a gram he
could form an idea of how many "electrons" corresponded to Faraday's bulk
quantity of charge. By dividing Faraday's quantity by the estimated number of
electrons which comprised it, Stoney arrived at the first human expression of
the electron charge.
Today researchers can virtually count electrons as they go past. It is
possible for a low-temperature quantic device combining the Josephson and von
Klitzing-Hall effects to measure a steady current by emitting a signal whose
cyclic frequency is twice the rate electrons are flowing through the device.
A billion cycles generated in some interval of time would indicate that
during the same interval half a billion electrons passed through.
Now that charge can be dispensed in numbers of electrons, the conventional
unit could simply be a code-name for some large definite multiple of the
elementary charge. Indeed this situation has been reflected in developments
at the International Bureau of Weights and Measures: particularly by the
Bureau's 1990 electrical standards. According to these standards, which have
not yet been adopted as official units, the coulomb is simply an alias for a
certain exact multiple of the natural charge. Using the new standards it is
no longer possible to "measure the charge of the electron" and essentially
this is because an exact multiple of the electron has become the unit. The
use of natural units, proposed by Stoney in 1874, appears to be happening in
this case as well as in the category of speed.
Planck quantities constitute a deep structure in nature, so perhaps one
should not expect them to be obvious or readily accessible. But some of them
are not entirely elusive. I mentioned earlier that the historical record of
when fundamental constants were first measured can give an idea of how
accessible the different ones are. The easiest Planck quantity to measure is
the speed of light--measured in 1676 by Römer. The Planck force was
experimentally determined by Cavendish as early as 1798, although not
identified as such. The natural unit of force was actually first identified
by Stoney in 1874, at the same time that he estimated the electron charge.
Stoney estimated the natural unit of length as well, but his definition was
of a length which is shorter than
by a factor of
. In my judgment the length
is representative of what is
least accessible in the Planck set. Harder to define and imagine than such
quantities as the speed, the force, and the charge, it was perhaps for that
reason slower to elicit interest. Aside from Stoney's, the first estimate was
by Max Planck in 1899. Even though Planck was highly regarded he had a
difficult time getting his colleagues interested in the length. The question
occurring to me concerns a thought-experiment--suppose you have already found
c, and and e, what is the most
rudimentary and direct way for you to get hold of
?
I hinted earlier at one approach, in a passage quoted from the section on
"reaching the billion mark" dealing with an imagined youngster.
The Planck length
is the geometric mean of your
gravitational length and your associated wavelength--one is a billion times
and the other is a billionth of
. Throughout your life the
Planck length will always be the geometric mean of those two lengths.
Whatever the two lengths become they will always be related to each other by
in exactly this way. This is a
special case of something more general which is true even for a proton or for
an individual quantum of light.
The gravitational length is just one instance of the "energy length"
associated with any quantity of energy--the distance
would have to push to produce
it. Since we already have access to
we can always tell the length
from the energy. The Planck length resides in every material particle and
quantum of light as the geometric mean of its associated "energy length"
(which we can write simply as L) and its wavelength (which we can
write as 
). Another way to say
this is that the Planck area
can be found simply by multiplying these two lengths together.
To take an example, it happens that the wavelength
of a proton is 13 billion
billion . (Technically this is
called the proton's Compton wavelength and is measurable by X-ray scattering
methods developed by Arthur Holly Compton in the 1920s.) On the other hand,
the proton's energy length L can be independently determined and is
equal to divided by 13 billion
billion. In view of the cancelation, it is easy to see that multiplying
L and gives
.
You could visualize this by imagining that L and
define a rectangular area
L, and that
is the side of a square with
the same area. This is true whatever the particle happens to be.
To get hold of all we need
to do is get simultaneous access to both the lengths and find their geometric
mean. It is clearest to imagine doing this with some color of light. Then
both the energy and wavelength have intuitive meanings and can be measured
using the simplest of instruments. So our approach to finding
will be by measuring L
and for photons of some pure
color. Here, for emphasis, are several ways to state the basic premise:
L =
=
With any bit of light, the geometric mean
of its wavelength and its "energy length"
is always equal to
The wavelength of some color
of light (technically cyclelength divided by
2
) can be readily measured with
a diffraction grating. A photoelectric tube can be used to find the "energy
length" L.
In a photoelectric tube there is a gap between two facing
pieces of metal--I will picture them as metal plates with wires attached to
them leading out of the evacuated glass. Light shines on one of the plates
and stimulates a flow of current across the gap. The wire leads allow us to
measure the current and to set up an opposing voltage V between the plates.
If the opposing voltage is high enough it can stop the flow of
current--roughly speaking, the bluer the light the higher the opposing
voltage must be.
The explanation offered by the 25-year-old Einstein in 1905 was
essentially that shorter wavelength, or "bluer", light consists of more
energetic individuals. Each jumping electron is supplied by a single photon
with the energy it needs to cross--energy equal to Ve: the gap voltage
multiplied by the electron's charge. Since individual photons supply this
energy, we can infer the energy carried by photons of some pure color of
light by measuring how high the gap voltage can be and still allow current
stimulated by that color to flow. Then
will tell us the associated
length. Apart from corrections relating to the type of metal used, and how
tightly its electrons are bound to it, the limiting voltage V is
related to the length L by Ve = E =
L.
Sensations of color in the retina depend most directly on photon energy
rather than wavelength. We are associating a standard wavelength or "vacuum"
wavelength with each energy, because the actual wavelength varies with the
medium. Light's speed in air is slightly less than the standard
speed--although so nearly the same that the difference is commonly
ignored--and the slowing shortens wavelengths ever so slightly. It is the
vacuum wavelength which is denoted here by
and which is in a definite
relation to the energy.
Along these lines, by being more energetic than visible light, ultraviolet
light can cause things to happen which visible light does not. Being "bluer"
than blue, photons of this kind can cause fluorescence in special dyes,
ionization, chemical reshuffling, and genetic damage. Higher frequency or
shorter-wavelength quanta of light generally carry larger amounts of energy,
and provides a graphic way to
relate wave properties to quantic energy.
You might say that current concern about the ozone layer is evidence of a
role that the Planck length plays in everyday life. But even though
plays a role, it seems less
accessible to measurement than other Planck quantities. One indication of its
relative inaccessibility is that in approaching
we implicitly made use of two
more readily measurable quantities: e and
. The former helped us interpret
the opposing voltage in a photoelectric tube as the energy which an electron
requires to jump across the gap. The latter helped us interpret this energy
as a length. The Planck force and charge can be measured without knowing the
length, but I cannot see how to discover the length without relying on
knowledge of the other quantities.
Although less accessible to measurement than some others, the Planck
length and time are abundantly present in everyday life. I am going to
recall, with a new emphasis, a couple of things mentioned in the earlier
section on "involvement with everyday experience".
The force between unit charges is simply equal to
divided by the square of the size of the separation (expressed as a
multiple of ). The emphasis
here is on the participation of Planck length. At molecular level a common
separation is a billionth of a span. This readily translates to
10
--in Planck terms this size of
separation is
10
and its square is
10
.
To find the force in such a case one simply divides
by
10,
which comes to something on the order of one billionth of a stone. Tiny
precise fractions of the Planck force are indeed at work in muscle and nerve,
and the Planck length is involved in determining them.
The force between currents in parallel linear wires, on a segment whose
length is half the separation, is equal to
multiplied by the sizes of the two currents, rated in Planck terms.
Again note the emphasis, which points to the involvement of Planck time. A
fairly common current in appliance motors is
10
e/
, and its size in comparison
with the Planck current e/ is
therefore
10.
If parallel currents are both of that size, calculating the indicated force
simply means multiplying
by
10
.
In a technical sense the preceding discussion is elementary: a
concise summary would consist in simply reiterating that the Coulomb constant
is equal to
/e.
My concern is with achieving a sense of familiarity.
These two examples were initially offered to illustrate how the Planck
force participates in everyday experience--through life chemistry, muscle
contractions, nerve impulses, the running of the dishwasher, and so on. But
the two examples also show a joint participation of Planck quantities such as
and
.
The Planck power can be visualized as Planck force pushing at Planck
speed: as the force pushing at
the speed of light. If you snap your fingers in tempo moderato, which is to
say once per tick, with each snap you can imagine the force pushing 100
thousand miles and delivering the energy embodied materially in 100 thousand
stars. (These are mile stars, some 10% more massive than the sun, but
generally similar to it. Alpha Centauri is an example of such a star.) By
supplying the rest energy embodied in their material existence the force can
be imagined as bringing them into being--with each snap of the fingers it
creates a hundred thousand stars. The force is moving at the speed of light
and leaving a trail of stars.
Using the fact that the gravitational length of material inside a circular
orbit is
r,
the gravitational length of our galaxy has been estimated to be 9-12 light
days. It was mentioned earlier that this figure is based on the reputed
existence of material near the edge of the galaxy, or at least outside the
main concentration of mass, with r = 50,000 light years and
= 0.0007-0.0008. Admittedly
involving guesswork, this leads to a straightforward
r
estimate of 9-12 light days. For definiteness suppose it is 11 light days. It
would accordingly take Planck power 11 days to create the galaxy from
nothing. Since we know how fast the power can create stars, this gives an
idea of how many mile stars would be needed to equal the galactic mass.
Doppler measurements of the microwave background show that the galaxy is
moving at about Mach 1800 ( =
0.0018) relative to the background. The result of multiplying 11 days by
0.0018 is about 30 minutes. This is how long the Planck force would have to
push to get the galaxy moving at its present speed, starting from standstill.
There is also the obvious question of how far the force would push during
those 30 minutes.
The kinetic energy associated with the galaxy's motion relative to the
background can be expressed as a fraction of galactic rest energy. This
kinetic fraction is
,
in other words it is half of 0.00182, or 1.6 millionths. That part of 11
light days is around 300 thousand miles, which is how far the Planck force
would need to push. It gives some idea of the galaxy's vast energy of
motion--equivalent to the energy materially embodied in 300 thousand stars
similar to alpha Centauri.
At the earth's average distance from the sun, a square mile in vertical
sunlight receives a fraction
10
of Planck power. The approximation is remarkably close: within two percent.
If you are on a hilltop overlooking a sunlit tract of land, the power you see
is on the order of
10
. That much light on an
absorptive surface exerts a force of
10
--more if the surface is
reflective--so the light you see is exerting something like a stone
(2.7 pounds) of force.
Were more accuracy appropriate, we could say that a square
mile at the earth's average distance from the sun receives
0.9810
of sunlight. This means that an
absorptive square mile is pushed with a force of
0.9810
by the light shining on
it.
Small fractions of Planck power can be visualized as light, although the
whole is too great to imagine that way. As just suggested,
10
is comparable to the sunlight
on a tract of land. It is also true that
10
is comparable to the entire
light of the sun. If you have a fist-sized stone in hand and are raising it
at one billionth of the speed of light--in other words by one span per
tick--then your output is essentially equal to
10
. That is because you are
pushing with a force of
10
at a speed of
10
c. The overall radiant power of the sun is roughly equal to the geometric
mean of Planck power and its humanly scaled cousin, which is the rate you are
performing work by raising the stone. Planck power is to the sun as the
sun is to lifting a stone.
(The approximation to the geometric mean is within 6%. In case
more accuracy is desired,
expressed in the humanly scaled units of power is
1.00410,
the sun's output is
1.05810,
and the irradiance on a square mile is 0.986 billion.)
It is difficult to avoid an acquaintance with the Planck heat capacity k =
/
,
one reason being that we are often too warm or too cold. Within a narrow
range, we humans are remarkably sensitive to small differences in
temperature, and when we are cold the Planck heat capacity is what determines
how much energy will be required to warm us up. In this regard all the
particulars are well-known. My point is merely that the fundamental heat
capacity we constantly experience is the energy-to-temperature ratio
belonging to the Planck set.
For each atom in a block of metal the heat capacity is
approximately 3k and to a large extent that is also how it is with your body.
Your body is chiefly water and the heat capacity of liquid water averages 3k
per atom. You might prefer to say that for water the heat capacity is 9k per
molecule, but since there are three atoms in a molecule that amounts to the
same thing. The 3 in the 3k is simply the number of spatial dimensions. The
subject of heat capacity is marvelously complex and this is a grave
oversimplification, but indeed the heat capacity of liquid water is very
nearly 3k per atom--nor are those of the human body and of a block of metal
very different. In a wide variety of media, the atom is temporarily anchored
more or less in one place and has the opportunity of restrained or springy
motion in 3 dimensions. Its heat capacity contains one copy of the Planck
standard k for each dimension of springy motion.
Except for the fact that and
are inconveniently large, we
could speak of heat capacity in those terms and say that to warm me up by
means providing 3
for each of my atoms. That
is just how the per atom heat capacity works: it tells how much heat you need
to supply per atom to warm up by some amount. Because
and
are awkwardly big we use
decimal scaledowns:
10
and
10.
The latter is essentially what we have been calling a degree and, although we
have no name for the tiny energy it would be easy enough to find one.
Accordingly, to warm me up by
10
means providing 3 of those delicate
10
energy amounts (whatever they might be called) for each of my atoms.
Bound or springy motion is especially good at absorbing energy
because work can go into stretching and compressing the bonds as well as into
the motion itself. By contrast, a spatial dimension of unrestrained movement
is only half as good as one that is restrained. So the heat capacity in
helium gas, where the molecules simply consist of single unrestrained atoms,
is 3/2 k per atom. That is 1/2 k for each spatial dimension the helium atoms
sail around in. For oxygen and nitrogen molecules--little dumb-bells--there
are 5 degrees of freedom: 3 dimensions for travel and 2 for tumble (end over
and end around). So the heat capacity of the air is 5/2 k per
molecule.
Although we do experience the Planck energy-temperature ratio
/
as a heat capacity, it is widely recognized as more than just that. The ratio
k provides a standard connection between temperature and energy telling us
the energies corresponding to much small scale behavior at each temperature.
One instance of this involves the thermal radiance from a warm pavement or
redhot piece of metal. If the quanta in a thermal batch of light are counted
and ranked by energy, there will be one particular bit, I call it the septan,
which is just over one seventh of the way from the bottom in this ranking. If
more precision is needed, the septan is found 14.7% from the bottom.
The septan energy is kT. If
T
stands for the
temperature at the sun's surface, the septan energy in sunlight must be
kT. The invisible warmth
from the walls of a room is virtually a scale model of the blaze of sunlight
or the glow from a forge--superficial differences affect emission slightly,
but the shape is fundamentally the same and only the septan is different.
Indeed if the septan is used as an energy scale along the x-axis, the
shape (by now a familiar one) is simply
. The septan energy kT at
some temperature T could be said to serve as a keynote in the
glow at that temperature, or as the energy scale appropriate to that
temperature. What has been said here about k, conventionally known as the
Boltzmann constant, is routine except possibly for the observation that k =
/,
and even that goes back to Max Planck's 1899 definition of the "natural unit"
of temperature.
The temperature of the sun's surface is
0.1410,
or
0.40710
if greater precision is needed. Consequently the septan energy in sunlight is
0.410
.
The average photon energy in a thermal batch of light is 2.7 times the
septan, or 2.701kT if more accuracy is needed. Parenthetically, the
number 2.7 which appears here arises as 3 multiplied by the ratio of two
Riemann zetas, in other words as 3
. This leads to the following
mantissa-calculation, and a figure of
1.110
for the average energy of a
photon from the sun:
3
0.4 = 1.1
To be visible a photon should be roughly twice as energetic as the
sunlight average--it must be in the range from 1.5 to 2.5
10
. Photons with energy 1.5 are
seen as red and those with energy 2.5 are seen as violet. Nearly all visible
sunlight is consequently between four times and six times the septan, four
times septan being red and six times septan being violet. By adding two
percentages in the diagram, you can tell that about 15% of the photons in
sunlight are visible. They represent a percentage of the energy in sunlight
which is considerably higher than 15% because individually they have higher
energies than average.