2.1 The Spacetime Interval

2.1  The Spacetime Interval

 ...The quantity dt is called the elapsed proper time of the interval, and it is invariant with respect to any system of inertial coordinates. 
- Where’s the empirical support for this statement?  Muon decay has been measured when its rest frame is co-moving with the lab frame; this is the 2 msec measurement.
But not when the muon is moving at high speed and altitude as the partial product of a natural cosmic ray collision, nor when the muon is generated artificially in the research accelerators. The measuring equipment in both prior cases would have to be co-moving with the muon, to measure its decay in the rest frame, using its proper time.
To claim the same 2msec decay time at high speeds relative to the lab frame – without decay measurements in the high speed frame, is an unsupported inference…. A guess that the lab frame is equivalent to the high speed frame.  But there is copious evidence to the contrary.
Proton accelerators produce a shower of electrons (aka electron cloud) at the exit tube in CERN experiments.  If the proton high speed frame (v >.99c) were equivalent to the lab frame, then we would not measure any such electron cloud, WHICH GIVES PHYSICAL EVIDENCE OF AN ABSOLUTE DISTINCTION BETWEEN THE LAB AND ANY FRAME MOVING WITH RESPECT TO IT.
To be equivalent physical system,  both frames should see the same phenomenon – either both see no electron cloud, or both see the same electron cloud!  
INERTIAL FRAMES ARE NOT EQUIVALENT IF ONE OF THEM IS THE LAB FRAME!
 
..Now suppose we assess this same physical phenomenon with respect to a relatively moving system of inertial coordinates, e.g., a system with respect to which the muon moved from the spatial origin [0,0,0] all the way to the spatial position [980m, -750m, 1270m] before it decayed.  With respect to these coordinates, the muon traveled a spatial distance of 1771 meters. 
- 1771 m is measured in the lab frame – not the muon’s rest frame!

..Since the advance of the quantum wave function (i.e., the proper time) of a system or particle over any interval of its worldline is invariant,
- Speculation without empirical proof  – see above. …
                                                
                                           dT  =  6.23 microsec.
This represents the time component of the muon decay interval with respect to the moving system of inertial coordinates. 
- If so, confirm this with measurements MADE IN THE MOVING SYSTEM OF INERTIAL COORDINATES.

...Since the muon has moved a spatial distance of 1771 meters in 6.23 msec, we see that its velocity with respect to these coordinates is 284 m/msec, which is 0.947c.
- Which is the speed measured in the lab frame.

...The idea that neutrinos actually do have mass seems to be supported by recent experimental observations, but the questions remains open.
- If E = hf and E = mc2, then the neutrino (and photon) must have mass. 
Then neither travels on null intervals!  
…the quantum state of a system gives (arguably) the most complete possible objective description of the system. 
- Yes, arguably, definitely.

...we assume that two co-moving clocks will keep time at the same instantaneous rate, even if one is accelerating and the other is not. 
- The obvious refutation of this is accelerating to the point of physical destruction of the clock…..
Also, What is the physical process behind the clock’s use as time-keeper?  Gravity?  EM?
Along with the environment this will determine the clock’s validity as chronometer.

...This is just a hypothesis - we have no a priori reason to rule out physical effects of the 2nd, 3rd, 4th,... time derivatives.  It just so happens that when we construct a theory on this basis, it works pretty well. 
- For inconsistent theories, working well proves nothing…
...Another way of expressing this "clock hypothesis" is to say that an ideal clock is unaffected by acceleration, and to regard this as the definition of an "ideal clock", i.e., one that compensates for any effects of 2nd or higher derivatives. 
- What is an example of an ideal clock in reality?

...The point is that these are both assumptions invoked by relativity: (1) the zeroth and first derivatives of position are perfectly relative and undetectable, and (2) the second and higher derivatives of position are perfectly absolute and detectable.  Most treatments of relativity emphasize the first assumption, but the second is no less important.
 · If va,b(t) = -vb,a(t)  then the derivative is relative, not absolute:  Aa,b(t) = - Ab,a(t)
 ·Absolute velocity was detected in the Sagnac & Michelson-Gale exps.
...The notion of an ideal clock takes on even more physical significance from the fact that there exist physical entities (such a vibrating atoms, etc) in which the intrinsic forces far exceed any accelerating forces we can apply, so that we have in fact (not just in principle) the ability to observe virtually ideal clocks. 
· How do we know F = ma  applies in the quantum world of the nucleon?
· Nuclear clocks are sensitive to the environment.

...For example, in the Rebka and Pound experiments it was found that nuclear clocks were slowed by precisely the factor g(v), even though subject to accelerations up to 1016 g (which is huge in normal terms, but of course still small relative to nuclear forces).
- The gamma ray energy was absorbed by the whole lattice;  where do the accelerations up to 1016 g enter into the Rebka and Pound experiment?

...How can an entity have a well-defined velocity and yet have no well-defined rest frame? 
- Because the concept of inertial frame has no existence in reality … an empty definition.

...The only answer can be that the transformation is singular, i.e., the coordinate system moving with a uniform speed c relative to an inertial frame is not well defined. 
- Contradictions cannot be logically well-defined. This is the fundamental contradiction in SRT:
From the SRT 1905 paper:
1. The laws by which the states of physical systems undergo change are not
affected, whether these changes of state be referred to the one or the other of
two systems of co-ordinates in uniform translatory motion.
So -   if a,b,c are 3 different  inertial frames and a is stationary,
Then take Va = 0 and take Vb,a= v1  and  Vc,a = v2, where v1 <> v2
Then   Vb,a <> Vc,a
2. Any ray of light moves in the “stationary” system of co-ordinates with
the determined velocity c, whether the ray be emitted by a stationary or by a
moving body.
So - Vb,a  = c    and Vc,a = c 
Then  Vb,a  = Vc,a
3.   SRT premise 1 contradicts SRT premise 2 ; SRT is inconsistent

... it's clear that for v = ±1 the individual t' and x' components are undefined, but the ratio of dt' over dx' remains well-defined, with magnitude 1 and the opposite sign from v. 
- The ratio of 2 undefined quantities is defined to be c ?? 
Hence, the term ‘mythpages”.