1.2  Systems of Reference
 
There are many theories of relativity, each of which can be associated with some arbitrariness in our descriptions of events.
Indeed there are many relativities of theory and implementation, holding to many conflicting attempts to apply the relativity concept to nature. One must ask what flavor or dialect of relativity is being promoted…  before examining empirical evidence.  Allowing relativity to be left to subjective application opens the door to many post-hoc explanations of experimental results.

For example, suppose we describe the spatial relations between stationary particles on a line by assigning a real-valued coordinate to each particle, such that the distance between any two particles equals the difference between their coordinates. There is a degree of arbitrariness in this description due to the fact that all the coordinates could be increased by some arbitrary constant without affecting any of the relations between the particles.
But this is a property of linear measurement –the reality is not arbitrary.

Thus we have translational relativity for each of the four coordinates x,y,z,t, ….
Time is not a coordinate having the same physical properties as x,y,z, but a parameter that expresses object motion or event occurrence(or lack thereof) within the 3D space.
From this point on, any use of t as a symbol having the same properties as space is unjustified.  

Einstein’s general theory of relativity shows how the laws of physics, suitably formulated, are invariant under an even larger class of transformations of space and time coordinates, including non-linear transformations, and how these transformations subsume the phenomena of gravity.
  any use of t as a symbol having the same properties as physical space is unjustified.  
Even in GR.   Especially in GR.

….we have no basis for asserting that the length of a solid object or the duration of a physical process is invariant under changes in position, orientation or state of motion.
There is also no basis for asserting that time has the same properties as physical space.
Models based on this premise are unphysical and unrealistic,  lacking a link to reality.

..in 1905 Einstein took measuring rods and clocks as primitive elements, even though he realized the weakness of this approach.
Rods and clocks are then the link between SR mathematical models and empirical reality.
Einstein did not establish a physical link to the fundamental concept of mass.
The composition of rods and most clocks is atomic, but light clocks were introduced later, which are not atomic but photonic.  The mechanics of atoms(matter) is different from the mechanics of light(photons).

…. the postulates of the theory are not strong enough to deduce from them equations for physical events sufficiently complete and sufficiently free from arbitrariness to form the basis of a theory of measuring rods and clocks.
Rods and clocks are really additional postulates for testing SR.

 …we are still far from possessing such certain knowledge of theoretical principles as to be able to give exact theoretical constructions of solid bodies and clocks.
In other words  the effect of the 2 SR postulates on the SR test instruments  is unknown. Without this knowledge SR is incomplete.

Whether or not the principles of quantum mechanics are adequate to justify our conceptions of reference lengths and time intervals, the characteristic spatial and temporal extents of quantum phenomena are used today as the basis for all such references.
In other words…. QM principles of space and time are applied to the SR measuring tools , without any independent verification of applicability ….. without justification.

..the most common method of description is to assign absolute coordinates (three spatial and one temporal) to each object,
More violations of the type difference between space and time.

 ..from a strictly mathematical point of view there does not exist a uniform distribution over the real numbers, so this form of representation does not exactly entail the perfect symmetry of position in an infinite space, even if the space is completely empty.
An infinite space?? Perfectly reasonable, but where’s the proof that reality exists in an infinite space?  If reality is finite in extent,  then there’s no objection to absolute 3D coordinates.  

 Any definite set of space and time coordinates constitutes a system of reference.
Objection - Any definite set of 3D space coordinates sequenced by a time parameter constitutes a system of reference, if the system of reference is intended to be used in physical modeling.

….we limit the range of possibilities by requiring contiguous physical entities to be assigned contiguous coordinates. This imposes a definite structure on the system, so it is more than merely a set of labels; it represents the most primitive laws of physics.
We see no need for this condition. The usual need for coordinates that are orthogonal and continuous everywhere is sufficient to guarantee locality in reality mapping into locality in modeling.
 the hypothesis that physical objects have continuous positions as functions of time with respect to a specified system of reference has proven to be extremely useful,
Not a surprise – this is consistent with a realist philosophy of existence/being.

Arguably, we never actually observe fields, we merely observe effects attributed to fields.
The field concept is a useful modeling tool, if the field has a foundation in reality by producing indirectly measurable effects.

Fields also provide a way of maintaining conservation laws for interactions “at a distance”.
If fields have no physical mechanism to explain interactions “at a distance” , then they fail reality and logic tests, for AAAD would be an effect without a cause.

In fact, we might even suppose that the sequence of states of all particles are uniformly parameterized by the time coordinate of our system of reference, but therein lies an ambiguity, because it isn't clear how the temporal states of one particle are to be placed in correspondence with the temporal states of another.
The introduction of an absolute time parameter removes the ambiguity.

We might choose to regard the totality of all entities as comprising a single element in a succession of universal temporal states, in which case the temporal correspondence between entities is unambiguous.
We do so choose.

Given any system of space and time coordinates we can define infinitely many others such that speeds are preserved.
Not clear.  Does this mean  Va,b(t) = -Vb,a(t) ?

..we can then define a reference frame as an equivalence class of coordinate systems such that the speed of each object has the same value in terms of each coordinate system in that class. Thus within a reference frame we can speak of the speed of an object, without needing to specify any particular coordinate system.
We define a reference frame as simply a 3D coordinate system with origin in a physical  object. This allows claims re reference frames to be tested empirically.

...we can adopt velocity additivity as a principle, and this is essentially what scientists had tacitly done since ancient times.
The Galilean principle of velocity addition has a strong intuitive and experimental base.
It is the Sagnac result that  singles out the lab ref. frame as the  physically meaningful reference system for measuring speed.