the earth as a threepenny planet
Gravitational length is a concept closely allied with Planck quantities and
can be used to define the Planck set. At this point what is offered is a
condensed introduction for readers who already have some preparation. The same
material will receive a leisurely treatment later.
The gravitational length is essentially a way of looking at the mass of a
body. In the earth's case, this length is roughly the thickness of three
pennies. Here it is depicted rather approximately:
. The
gravitational length of the sun is roughly a mile. By "roughly" I
mean within ten percent--if more accuracy were needed at this point I would say
nine tenths of a mile. Both lengths will be expressed more exactly as soon as
there is need for it. Referring to the earth as a three pennywidth planet and
the sun as a mile star is intended to evoke the notion of gravitational length
in a way that implicates the Planck length. Where it seems cumbersome to write
out "gravitational length" in full, I am prepared to write
"gravilength" for short. It will sometimes be convenient to write the
gravitational length symbolically either as L or using some more
specific notation such as L
and L
.
The next table lists a few of the things which gravitational lengths tell
us. More conventional formulations depending on the Newtonian constant G and on
inertia, denoted M, are shown for comparison.
| application |
version using gravilength |
version using inertia |
| light bending angle |
4L/r |
4GM/c r |
| orbit speed |
(L/r)  |
(GM/cr)
|
| escape speed |
(2L/r)
|
(2GM/cr)
|
| acceleration |
L(c/r) |
GM/r |
| black hole radius |
2L |
2GM/c |
| once-around pendulum |
L |
GM/c |
| stationary orbit speed |
( L/c)
|
(GM/c)
|
| earth-sun pull |
 LL/r |
GMM/r |
| rest energy |
L
|
Mc
|
In this table speeds are given as fractions of the speed of light. The light
bending angle is given in radians and is the angle by which light is bent while
passing within a distance r of center. Escape speed and acceleration due
to gravity are for small bodies located at a distance r from center.
Orbital speed is for a small body in a circular orbit, r being again the
distance from center. The ratio L/r is basic to several of these
applications and one instance of it will soon be evaluated: the ratio of the
earth's gravitational length to its radius. The outline letter
which you
see in the table stands for the Planck force--the fact that a body with
gravitational length L has rest energy equal to
L is
one connecting gravitational length with Planck quantities. "Black hole
radius" refers to the normal model of a non-rotating black hole. One with
mass equal to the sun's would have a 1.8 mile radius (twice the nine tenths
mile mentioned earlier). A black hole with mass equal to the earth's would have
a radius twice the earth's threepenny gravitational length. A stationary orbit,
such as a communication satellite might follow, is one matching the rate
(radians
per unit time) at which the central body turns.
measuring gravitational length
The combined speed and separation of two things in circular orbit around
each other reveals their combined gravitational length. If one of the two is
small enough for its contribution to the combined length to be neglected this
immediately indicates the gravitational length of the central body. The way
this approach works is that if the orbiter is a distance r from center
and traveling at a fraction
of the speed
of light, then the central body gravilength is r. This is particularly easy to
apply to the earth's nearly circular orbit around the sun. The speed fraction
is 
, and
consequently its square is
, while the
distance is some 90 million miles. Multiplying this distance by the square of
the speed fraction gives 0.9 mile, indicating the sun's gravitational length.
By taking a little extra trouble with the definitions, this approach can be
generalized to cases where the orbit is elliptical and also where the two
bodies are comparable, instead of one dwarfing the other. By arranging to
measure both orbital speeds separately it can also be extended to determine the
two gravitational lengths separately as well as their combination.
In normal sealevel gravity a pendulum whose length is L goes back and
forth in the time it would take light to go once around the earth at
sealevel--it is a "once-around pendulum" in this sense. When located
a greater distance r from earth center, its period would be the time
light needs to travel in a circle with that radius. If we are prepared either
to adjust for or ignore some minor effects of the earth's rotation and
irregular shape, this offers an elementary means of determining the earth's
gravitational length. Observing satellites in orbit around the earth is clearly
another option.
connecting gravitational length and Planck quantities
One connection between the gravitational length and the Planck quantities
has been already suggested: the fact that the Planck force
is the
proportion relating gravitational length to rest energy. A body's gravitational
length L and its rest energy E are related by E=L. Indeed this relation
can serve to generalize the idea of gravitational length and define it for
bodies whose rest energies are known but which are too small to allow their
gravitational lengths to be measured in the ordinary way. The equation
=
is a special case
of the foregoing: the Planck length
is the
gravitational length of a body whose rest energy is the Planck energy
.
A further connection between gravitational length and Planck quantities
occurs in the definition of the Planck energy
. The Planck
energy is uniquely determined by the fact that its associated gravitational
length and its associated wavelength are equal. Every quantity of energy has
such a pair of lengths associated with it but
is the only
energy for which the two lengths coincide. Moreover the Planck length
is likewise
determined as this common length--a meeting of wavelength with gravity. There
will be more discussion later, for now we simply note a connection between
gravitational length and the Planck quantities. The version of wavelength used
is cyclelength divided by 2
.
where gravilengths are known more accurately than inertias
The gravitational length (whether as L or as Lc) is what
astronomers measure when they measure mass. It is known with considerable
precision and is the form of mass appropriate for subsequent calculation if
there is a need for accuracy. Once gravitational length has been measured it
may be converted to the less certain form of inertia by means of the Newtonian
constant G, and expressed in grams or kilograms of inertia. Because of the
uncertainty with which G is known, the earth's inertia in grams is provided
with considerably less precision than the gravitational length. If inertias are
to be used subsequently for calculating orbits they must first be multiplied by
the Newtonian G (a second opportunity for error) to get the information back
into relevant gravilength form. The intervening conversion into inertia units
has added needless uncertainty.
In Astrophysical Quantities, a standard source of astronomical
data, the earth's gravitational length appears as Lc: the gravit multiplied by a
factor of c.
Because the speed of light is exact in modern units, this c factor is itself exact and does
not effect the precision with which the length is known. In other words the
c factor can
be regarded as a formality and excised without loss of information.
Evidently it makes no logical difference whether one uses L or
Lc--one being an exact multiple of the other makes them equivalent as
scientific data--but the two quantities are of different types. One is a length
and the other is of a type without conventional name or commonly understood
meaning. For this reason it seems to me more intuitive to use the length
consistently.
sample calculations with gravitational length
The length of a pendulum varies as the square of its period, and that of a
"once-around" pendulum is the earth's gravitational length L.
With pendulums, the length varies as the square of the period. So the length of
a "four-times-round" pendulum is 16L. In the case of the
earth, 16L is long enough so that you could actually make a pendulum of
that length and time it.So measuring 16L for the earth is a do-able home
experiment, say with a button tied onto the end of a length of thread. Light
travels four times around the earth in a semisecond--so once you have adjusted
the pendulum to have a period of one semisecond you can take its length to be
16 times the gravitational length of earth. If you divide the pendulum length
by sixteen you have, in a rough sense, an experimental measurement of the
gravitational length of the earth.
A metronome set to 111 or thereabouts will beat approximately once per
semisecond--on the metronome here, this setting is in a range labeled tempo
moderato. To measure the gravitational length of earth using a small weight
tied on the end of a thread, adjust the length so that the pendulum motion
matches the metronome and divide the length by 16.
Because units like the pennywidth and mile have exact metric equivalents, we
may use any desired degree of precision available in metric contexts. To the
extent that having a few extra digits of precision might be appropriate, the
earth's gravitational length L is 2.744 pennywidths. Stacking three
coins does give a good idea of the actual length, but for some applications
this further accuracy is helpful.
The ratio L/r appeared basic to several of the items listed in
the earlier table and one instance of it might be worth calculating: namely the
ratio of the earth's gravitational length to its radius. The earth's equatorial
radius is 3.94 thousand miles. Considering that a pennywidth is a billionth of
a thousand miles, and that 2.744 divided by 3.94 is about 0.70, evidently
L/r is approximately 0.70 billionths. Let's see what we can
calculate about the earth from this:
Multiplying L/r by four immediately tells the angle by which
grazing light is bent--2.8 billionths of a radian. Furthermore,
2L/r is 1.4 billionths, which has 37 millionths as its square
root. This means escape speed from the earth's surface is 37 millionths. A
millionth of the speed of light corresponds approximately to the speed of sound
in the cold air at ordinary cruising altitudes. So escape speed can be thought
of as 37 times this cold air sonic reference speed: as "Mach 37" if
you wish to interpret it that way. As a further illustration, L/r
divided by 7 is 0.10 billionths, which has 10 millionths as its square root.
This means that circular orbit speed at 7 earth radii from center is 10
millionths--ten-fold faster than cold air sonic. This "Mach 10" speed
is roughly that of communication satellites in stationary orbit.
The ratio L/r has a physical meaning which has not been
mentioned. In the case of a small body located a distance r from main
body center, it expresses the kinetic energy needed for escape as a fraction of
the small body's rest energy. For something on the earth's surface to get
loose, it needs to be given 0.70 billionths of its own rest energy. L/r is a
kind of binding energy fraction, and for all of us here on the earth's surface
this fraction is 0.70 billionths of our rest energies.
gravitational length and baryon number
The protons and neutrons in an ordinary object comprise nearly all the
weight and are referred to as baryons (from the Greek word for
"heavy"). There is an interesting number 13 billion billion=13
10
which comes into
play here. The gravitational length of a proton is much shorter than the Planck
length--in fact it is
. As long as
we do not want too much precision the same number will do for the neutron--that
being about the same mass. By contrast, the associated wavelength of either
particle (the so-called Compton wavelength) is much longer, namely 13 billion
billion .
Evidently an ordinary object which happens to have
for its
gravitational length must comprise 13 billion billion baryons--13 billion
billion protons and neutrons. A seven-year-old whose gravitational length is a
billion times the Planck length must comprise 1310
. As a rule of thumb, to find the
number of baryons in a thing one multiplies its gravitational length, expressed
in Planck terms, by 13 billion billion. This can just as well apply to the sun,
or to a mile star (one whose gravitational length is 10
), as to the
billion
youngster featured earlier. The number of baryons in a mile star--alpha
Centauri is an example of one--is 13 billion billion multiplied by 10. The baryon number of the
sun--with its 0.9 mile gravit--is 90% of this.