is to
imagine how we might measure it for ourselves--more simply than by Cavendish's
method--in an idealized thought-experiment. 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
(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 then the answer will come out to be an even larger number of newtons.
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 111 times a minute or roughly twice a
second, the pair can be imagined moving one span with each (semisecond)
beat--needing ten beats to go one 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 semisecond ticks 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
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
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
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
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 about twice a second, 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 0.98
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 semisecond--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.
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 0.14
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 1.1
3
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