1.3  Inertia and Relativity

He [Galileo] believed, for example, that circular motion was a natural state that would persist unless acted upon by some external agent.
-True for a natural aether vortex….

..the roughly circular motion of the Moon around the Earth might suggest the existence of a force (universal gravitation) acting between these two bodies, but it could also be taken as an indication that circular motion is a natural form of unforced motion, as Galileo believed.
-A log caught in a river eddy requires no force to explain its motion…. It floats in the water medium and follows its flow. This is natural and unforced motion.

By indirect reasoning, the natural philosophers of the seventeenth century eventually arrived at the idea that, in the complete absence of external forces, an object would move uniformly in a straight line, and that, therefore, whenever we observe an object whose speed or direction of motion is changing, we can infer that an external force – proportional to the rate of change of motion – is acting upon that object.
This is the principle of inertia, the most successful principle ever proposed for organizing our knowledge of the natural world. Notice that it refers to how a free object “would” move, because no object is completely free from all external forces.
-If an object is carried by the flow of its medium (log in a river) is this motion of a free object? If so, the object follows the medium’s flow and direction, which is not necessarily uniform nor a straight line.

From this resulting set of ideal states of motion, it is necessary to identify the largest possible "equivalence class" of relatively uniform and rectilinear motions. These motions and configurations then constitute the basis for inertial measurements of space and time, i.e., inertial coordinate systems. Naturally inertial motions will then necessarily be uniform and rectilinear with respect to these coordinate systems, by definition.
-Again, the motion of objects in aether can be force-free and inertial, yet curvilinear and non-uniform.
 
Each thing...continues always in the same state, and that which is once moved always continues to move...and never changes unless caused by an external agent...  all motion is of itself in a straight line...every part of a body, left to itself, continues to move, never in a curved line, but only along a straight line.
- What does left to itself mean? Taken out of the medium/aether, or left in it?
 
1) Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by the forces impressed upon it.
2) The change of motion is proportional to the motive force impressed, and is made in the direction   of the right line in which that force is impressed.
-That must include aether forces in the net force.

3) To every action there is always opposed an equal and opposite reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
-Action and reaction forces must include aether.

...the [inertia] principle also implies the equivalence of uniform motion in all directions in space.
-If the aether is at rest …

Corollary 5 of the Newton’s Principia states:
The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forwards in a straight line without circular motion.
- Yes, if space means aether… It seems Newton did recognize a mechanical aether.

Our concepts of uniform speed and straight paths are ultimately derived from observations of inertial motions, so the “laws of motion” are to some extent circular.
- More so, the occurrence of uniform linear motion never occurs in the cosmos; on earth it is either articially produced or only approximate.

Thus the “laws of motion” are true by definition. Their significance lies not in their truth, which is trivial, but in their applicability. The empirical fact that there exist systems of inertial coordinates is what makes the concept significant.
-An interesting admission – inertial systems are tautologies – self-fulfilling concepts. As noted before, inertial systems are rare in nature. There are very few pure systems of inertial coordinates….


...the principle of relativity asserts that for any material particle in any state of motion there exists an inertial coordinate system in terms of which the particle is (at least momentarily) at rest.
- But are there any other systems with the same inertial motion?

...the third law implies (requires) that if the spatial origin of one inertial coordinate system is moving at velocity v with respect to a second inertial coordinate system, then the spatial origin of the second system is moving at velocity v with respect to the first.
That is,  Va,b(t) = -Vb,a(t)

....that it is also necessary to specify the loci of constant temporal position, and this is achieved by choosing coordinates in such a way that mechanical inertia is isotropic. (This means the inertia of an object does not depend on any absolute reference direction in space, although it may depend on the velocity of the object. It is sufficient to say the resistance to acceleration of a resting object is the same in all spatial directions.)
- All this is abrogated by the existence of a space with properties that are anisotropic , like aether winds. Where aether is at rest and uniformly dense, inertial isotropy holds, as above.

...the physically meaningful "relative velocity of two material bodies" is best defined as their reciprocal states of motion with respect to each others' associated inertial rest frame coordinates.
-This only holds for point particles – for extended objects the origin of the ICS is ambiguous.

....we are not free to arbitrarily adopt this [Galilean] or any other transformation and speed composition rule for the set of inertial coordinate systems, because those systems are already fully defined (up to insignificant scale factors) by the requirements for inertia to be homogeneous and isotropic and for momentum to be conserved.
- In other words, we are ignoring the possibility that space can have properties, like motion, which would add to the speed the speed of aether(= space).

Of course, inertial isotropy is not the only possible basis for constructing spacetime coordinate systems. We could impose a different constraint to determine the loci of constant temporal position,
- Such as the existence of aether.

....we will find that mechanical inertia is generally not isotropic in terms of the resulting coordinate systems, so the usual symmetrical laws of mechanics will not be valid in terms of those coordinate systems (at least not if restricted to ponderable matter). 
-The laws of mechanics must be modified to include interaction of matter and aether. Experiments like Sagnac demonstrate the existence of aether and the need for modifying the laws of mechanics and EM to satisfy the aether experiments.

Such coordinate systems, while extremely awkward, would not be logically inconsistent.
- Of course not.

....since physics consists of identifying and understanding the symmetries of nature, the option of disregarding those symmetries does not appeal to most physicists.
What has priority – what appeals to physicists or the experimental proof of the scientific method?

The laws by which the states of physical systems undergo changes are not affected, whether these changes of state be referred to the one or the other of two systems of [inertial] coordinates in uniform translatory motion.
- Einstein’s proposition has been disproven by the absolute frame discovery in Sagnac-type experiments. The laws by which the states of physical systems undergo changes are not affected …. If the absolute lab frame is used.

...the class of coordinate systems that Einstein was trying to identify (the inertial coordinate systems) are those in terms of which inertia is homogeneous and isotropic, so that free objects move at constant speed in straight lines, and the force required to accelerate an object from rest to a given speed is the same in all directions.
-This only holds in situations where the aether is fixed and at rest in the absolute frame.