Newton's Universe
---------------
In Newton's day and for a century after there was no application
for gravity in the world except within the solar system. It had
everything known to move around. The stars beyond were a sort of
background against which the planets &c were gaged for position and
motion. The real character of the stars was indeterminate, altho there
were many philosophical and theological ideas. Newton could believe
that the world beyond the solar system was populated with luminous
bodies, each having a definite mass, ranging on and on into infinite
depths. Already the new telescope was unveiling stars far fainter than
could be discerned by eye and these were plausibly far more renote
than the naked-eye stars. With each increase in telescope strength
more and fainter stars were seen, giving a vivid impression that the
universe is indefinitely deep and thick with stars.
If these stars had gravity toward each other, why did they not
collapse together or exhibit orbital motion? Newton felt that they wre
arrayed uniformily thruout space such that in opposite directions
around any one there were equal numbers of stars pulling at it. Thus
for every star the gravity toward one direction balanced that in the
opposite direction and the whole enchillada remained stationary.
(Orbital motion in binary stars and proper motion in individual stars
was still far in the future.)
Early Problems
------------
A moment's consideration reveals that there is a very serious
difficulty with Newton's world. Not only does the gravity cancel out
in opposite direction around a star but the separate strength of the
two gravity forces is infinite! Stars range outward for infinite
distances thereby putting an infinite number of stars on the two
opposite sides. The gravity of the two sets of stars is infinite.
There is a tug-of-war of incredibly massive scale in all directions
around every point in the universe!
Should there be even the merest imbalance in the summed up masses
of stars the forces become unbalanced and the central star runs off in
chaotic motion. This upsets the mass for other points and soon the
whole world is in utter upheavel as stars spiral and career all over
the place. The balance of mass must be absolutely exact or else.
In Newton's time there was no solution. He lived when Euclid was
simply the geometry and space was an externality and gravity was an
instantaneous action-at-a-distance.
Newton seems to have been troubled by this situation as evidenced
by correspondence he carried on with Bentley. But in the main
astronomers sort of ignored it and simply accepted that the universe
is infinite in extent yet remained stable and flat. Even the discovery
of proper motion and binary stars and even open clusters didn't
disturb astronomers greatly. These effects were very localized and for
stars far apart the forces still netted out.
Impossible World
--------------
In the climate of Einstein physics we can see that the world of
Newton is actually impossible. We can not have a world populated by
gravitational bodies in a Euclid space with constant scalefactor.
2*k*c^2 / R^2 = 4 * pi * gamma * rho0 / R^3 - (q + 1) * H^2
4 * pi * gamma * rho0 / R^3 = (q + 1) * H^2+2 * k * c^2 / R^2
= 0 for a constant scalefactor and k = 0 for a Euclid geometry.
So
4 * pi * gamma * rho0 / R^3 = (q + 1) * 0^2 + 2* 0 * c^2 / R^2
= 0 + 0
= 0
which is false. The leftside term has all nonzero factors. The
universe can not be both Euclid and static under self-gravitation.
Either k or H may be zero, but not both together.
Zollner in 1872 is probably the first to offer a fix. He used the
newly emerging nonEuclid geometries. In Riemann geometry space is
spherical. So the farther away one goes from a given point the
incremental mass engulfed around that point tends to a finite limit.
Ergo, the stars deploy a finite mass around any point and the gravity
of these stars is finite. Zollner had no way to appreciate that a
curved space is itself, as Einstein much later demonstrated, the
equivalent of gravity. One could not have both curved space and Newton
gravity at the same time.
Gravitation under Newton
----------------------
We review the concept of gravitation in Newton physics. Newton
declared that gravitation was an innate property of matter. Matter in
turn lives in an external and absolute reference frame in space and
time. Gravitation percolates, as it were, from the matter into the
surrounding space and time. This gravity acts on other bodies. Orbits,
trajectories, collisions, and so forth are described against the
external reference frame in which the bodies sit or move.
A very popular and easy means to visualize gravitation in Newton
physics is thru the notion of fields. We think of body #1 as
establishing in space and time a field of gravity around it. At some
point #2 remote from the body the field attains a certain strength to
act on other bodies placed at that point. The gravity field strength
produced at point #2 by the body #1 is
G{1/2} = -gamma * m1 / r{1 / 2}^2
gamma is the Newton constant that doctors up the sense, units, and
values so they balance. Its value is 6.672E-11n.m^2/Kg^2. The French
brackets tell the sense of the calculation: from point #1 to point #2.
In English we can read them as 'G of 1 on 2' and 'r from 1 to 2'. The
negative signum reminds that the action of the field is TOWARD point
#1 while the distance is measured FROM #1.
When an other body is placed in the field, it is acted on by a
force from, or is accelerated by, the field in proportion to its own
mass m2
F{1/2} = (-gamma * m1 / r{1/2}^2) * m2
-gamma*m1*m2/r{1/2}^2
This is Newton's law of universal gravitation. The 'universal' he
asserted comes from the premise that each and every bit of matter in
the whole universe has the property of gravitation. In his day the
only things beyond the Earth he could test his theory on were the the
Moon, planets, and the comet of 1680. Not until the discovery of
binary stars a century later did Newton physics extend to deep space.
Newton's signal achievement was the clear distinction between
force and mass. These quantities were totally fuzzy before his day. In
fact in his own Principia Newton never actually uses terms comparable
to today's 'mass' or 'force'. Not even in the original Latin.
Definition of Force
-----------------
Newton and physicists today treat mass as the fundamental property
of matter and force as an incidental feature depending on the peculiar
situation that mass finds itself. We define separately the unit of
force and Newton did so by his second law of motion
F = m * 2der(r,t)
The unit of force is that force which on one unit of mass produces one
unit of acceleration.
[unit of force] = [kg]*([m]/[s^2]}
= [kg] * [m]/[s^2]
= [kg.m/s^2]
[newton] = [kg.m/s^2]
This is named in honor of Newton. One newton is the force producing
one meter/second^2 of acceleration on a one kilogram mass.
In the oldstyle system of measures the force is the prime entity,
as the force of gravity on a unit of matter and the mass within that
matter is the derived unit. The old 'pound' was the unit for the force
of gravity and we used to deal with 'pounds' of matter. But soonest it
was discovered that the gravity field strength varies over the Earth,
even if only slightly, and it is altogether different in outer space,
this system of measures was junked.
Two Interpretations of Field Strength
-----------------------------------
The field of gravity pulls on an other body with a certain force.
It also accelerates that body. That is, the measure of the field
strength at point #2 may be either the acceleration or the force of
attraction caused at that point. Consider the law of gravitation.
F{1/2} = -gamma * m1 * m2 / r{1/2}^2
= [n]
= [Kg.m/s^2]
= [kg]*[m]/[s^2]
[n]/[kg] = [m]/[s^2]
= -gamma * m1 / r{1/2}^2
= G{1/2}
In other words, the gravity field strength is either expressed in
meter/second^2 of acceleration or newton/kilogram of force on a unit
mass. Astronomers exploit this equivalence routinely in their work in
celestial dynamics.