Epistemological derivation of Special Relativity

So let's take a look at how Special Relativity falls out from the epistemological arguments. Do not confuse this as the full analytical defense of the argument; for that, it is best to follow the details represented in the book.

To understand what is the significance of this type of derivation of Special Relativity, just keep a close eye on the fact that none of the arguments presented is dependent on any type of speculative nature of the universe, or require any kind of experimental result of any type. Time to do some thinking!

First significant fact is commented at page 69; The equation (2.23), which represents the four universal epistemological constraints of an explanation as a single equation (See this post for some more comments about those constraints), is a linear wave equation of fixed velocities (besides the interaction term; the one with the Dirac delta function).
(Note; Page 69 appears to contain a typo; it refers to equation (3.29), when it should be (2.23))

Equation (2.23):
$$\left \{ \sum_i \vec{\alpha}_i \cdot \vec{\triangledown}_i + \sum_{i \neq j} \beta_{ij} \delta(\vec{x}_i - \vec{x}_j)   \right \}\Psi=\frac{\partial}{\partial t}\Psi = im\Psi$$

This equation represents fundamental constraints that any self-consistent explanation / world-view / theory must satisfy, but it does not in any way define what kinds of elements the explanation consists of. I.e. it doesn't imply any assumptions as to what the world is; any kinds of defined objects can be represented with the associated notation (defined in the opening chapters).

In practice this means any self-consistent explanation can be fully represented in such a way that it directly satisfies the above equation.

The fact that the equation happens to be a wave equation of fixed velocity simply means any self-consistent explanation can be fully represented in a form where all of its defined elements move at constant fixed velocities.

The second significant fact is that the equation (2.23) can be taken as valid only in one particular coordinate system; the "rest frame" of whatever is being represented. That is to say, you cannot represent your solution in terms of moving coordinate systems without employing some kind of transformation.

Third fact; if an explanation has generated object definitions in such a manner that the "rest of the universe" can be ignored when representing those objects, it implies the same equation (2.23) must also represent a valid constraint for representing a single object. To quote myself from the Schrödinger post;

Note further that if it was not possible - via reasonable approximations or otherwise - to define microscopic and macroscopic "objects" independently from the rest of the universe, so that those objects can be seen as universes unto themselves, the alternative would be that any proposed theory would have to constantly represent state of the entire universe. I.e. the variables of the representation would have to include all represented coordinates of everything in the universe simultaneously.

In other words, if an explanation contains object definitions where objects can be represented independently from the rest of the universe, then there must also exist a valid transformation between the rest frames of those individual objects, in such a manner that the equation (2.23) preserves its form as it is.

Philosophically, if each object are truly independently represented, it is the same thing as saying that there is no meaningful way to define a universal rest frame; at least not in terms of the dynamic rules defined by the explanation.

And the fact that the equation (2.23) preserves its form means it can be seen as a wave equation of fixed velocity inside the rest frame of any individual object. This should start to sound familiar to those who understand Relativity; we are fast approaching the fact that the speed of information can be defined as isotropic across reference frames,  because it is already guaranteed that a valid transformation mechanism exists, that gives you that option. Lorentz' transformation is exactly such a valid mechanism, and can be employed here directly.

But what about this business of every element constantly moving at fixed speed? (Let's call it C)

Remember, the notation defined in first chapters contained the concept of imaginary $\tau$ (tau) axis, defined to ensure no information is lost in the presentation of an explanation. It is a feature of the notation and has got only epistemological meaning. By its definition of being an imaginary axis created for notational purposes, it is meaningless what position the objects get along $\tau$. Or to be more accurate, the probability of finding an object at specific location is not a meaningful concept. But the velocity along $\tau$ is meaningful, and plays a role in representing the dynamics of the explanation. It is certainly possible to represent explanations without this concept, but the equation (2.23) was defined under a terminology that requires it.

And since we are using it, it just means objects that are at rest in $(x, y, z)$ space (if we wish to use 3 dimensional representation of the world), will be moving at velocity C in $\tau$.

On the other hand, anything moving at velocity C in $(x, y, z)$ space, implies 0 velocity along $\tau$, which is rather interesting in the light of the definition of "rest mass" defined in page 57. Directly related to the velocity of the object along $\tau$. So during Schrödinger deduction, we already reached a point where any defined object can be identified as having energy, momentum and mass exactly as they manifest themselves in terms of modern physics (including all of their relationships), via simply appropriately defining what we mean by those terms. And now we have reached a point where any object moving at velocity C  in $(x, y, z)$ space cannot have any mass. Not because world happens to be built that way, but because a meaningful definition of mass yields that result.

Note that this is in sharp contrast to common perspective, where mass is seen as a fundamental ontological thing that objects just have. Here, it is intimately tied to how all the associated concepts are defined. It simply means that anything moving at velocity C must have no mass, by the definition of mass, energy, momentum, C, and a host of other definitions that these definitions require.

From page 71 onward the Special Relativistic time measurement relationships are demonstrated simply via defining how a photon oscillator operates (representing a clock, or internal dynamics of any macroscopic object), under the terminology established thus far. It should not be very difficult to follow those argument to their logical conclusion.

Just in case the drawn diagrams create some confusion, here are simple animated versions;

A stationary photon oscillator (a photon and two mirrors) is defined as:

All the elements are represented with "infinite length" along $\tau$ because the position in $\tau$ is not meaningful concept. The velocity of all the elements is fixed to C, but orthogonal to each others.

When the same construction is represented from a moving coordinate system, it looks like this;

The "self-coherence reasons" alluded to there are exactly the reasons why Lorentz transformation must be employed (see the beginning of this post).

So this is effectively a simple geometrical proof of time relationships being expected to have exactly the same form as they have in Special Relativity, but having absolutely nothing at all to do with any speculative ontology of the world (such as space-time or any ontological nature of simultaneity), or even with Maxwell's Equations per se. 

None of the arguments so far have made any comments about how the world is; everything is revolving around the ideas of what kind of representation can always be seen as valid, under appropriate definitions that are either forced upon us due to self-consistency requirements, or always available for us as arbitrary terminology choices.

So none of this could possibly tell you how the world really is. There is merely scientific neutrality in recognizing in what sense we really don't know how things are; we just know how things can be represented. It can also help us in recognizing that there really are all kinds of ways to generate the same predictions about the universe, which is to say there are different valid ways to understand the universe.

Next destination will be slightly more challenging to analyze; General Relativity...

Some background on Special Relativity

Alright, time to talk little bit about how Special Relativity falls out from the analysis (see the book on the right). This one is in my opinion quite a bit easier to understand than the deduction of Schrödinger, at least if the reader is familiar enough with the logical mechanisms of Special Relativity.

The first signs of relativistic relationships arose in Chapter 3, during the deduction of Schrödinger Equation (see the discussion relating to the $mc^2$ term on page 55). Chapter 4 contains a rather detailed presentation of the explicit connection between Relativity, and general epistemological requirements. If you have followed the arguments through Schrödinger, Relativity falls out almost trivially. In fact, someone who is familiar enough with the logical mechanisms behind Special Relativity, might already guess how things play out after reading just the first couple of pages of the chapter.

Unfortunately most people have a rather naive perspective towards Relativity, more revolving around ontological beliefs of space-time constructions, rather than the actual logical relationships that the theory is composed of. The space-time construction is just a specific interpretation of Special Relativity, and if the reader is under the impression that this interpretation is necessary, then the logical connection between fixed velocity wave functions to relativistic relationships might not be too obvious.

Because the space-time interpretation is so common, I think it's best to first point out few facts that everyone having any interest in understanding Relativity should be aware of.

In my opinion, one of the best sources of information on Special Relativity is still Einstein's original paper from 1905. Because unlike most modern representation, that paper was still reasonably neutral in terms of any speculative ontologies. In the paper, the relativistic relationships are viewed more as requirements of Maxwell's equations of electromagnetism (For instance, the moving magnet and conductor paradox is mentioned in the very opening) under the fact that one-way speed of light cannot be measured (something that was well understood at the time). Relativistic space-time is not mentioned anywhere, because that idea arose later, as Herman Minkowski's interpretation of the paper. Later, space-time became effectively synonymous to Relativity because the later developments, especially General Relativity, happened to be developed under that interpretation.

The history of Relativity is usually explained so inaccurately, filled with ridiculous myths and misconceptions, that it seems almost magical how Einstein came up with such a radical idea. But once you understand the context in which Einstein wrote the paper, the steps that led to Special Relativity start to look quite a bit more obvious. I would go so far as to say pretty much inevitable.

When Einstein was writing the paper, it was well understood within the physics community that the finity of the propagation speed of information (speed of light) led to the impossibility of measuring the simultaneity of any two spatially separated events. If the events look simultaneous, it is not possible to know whether they actually were simultaneous without knowing the correct time delays from the actual events. To know these delays, you would have to first know the speed of light.

But that leads directly to another fact which was also well understood back then. That one-way speed of light cannot possibly be measured. In order to measure the speed, you need two clocks that are synchronized. Which is the same thing as making sure the clocks started timing simultaneously.

That is to say, you cannot synchronize the clocks without knowing the one-way speed of light, and you can't know the one-way speed of light, without synchronizing the clocks. Note that clocks are going to be by definition electromagnetic devices, thus there must be an expectation that moving them can affect their running rate. The strength of the effect is expected to depend on the one-way speed of light in their direction of motion. Which just means you can't even synchronize the clocks first and then move them.

So here we have ran into a very trivial circular problem arising directly from finity of information speeds, that cannot be overcome without assumptions. It wasn't possible then, it's not possible now, and it will not be possible ever, by definition. This problem is not that difficult to understand, yet we still have people performing experiments where they attempt to measure one-way speed of light, without realizing the trivial fact that in order to interpret the results, we must simply assume some one-way speed of light. That is exactly the same thing as measuring whether water runs downhill, after defining downhill to be the direction where water runs.

As simple as this problem is, it does not exist in the consciousness of most people today, because they keep hearing about all kinds of accurate values for speed of light. It is almost never mentioned that these measurements are actually referring either to average two-way measurements (use a single clock), or to Einstein convention of clock synchronization (the convention established in his paper; assume C to be isotropic in all inertial frames, and synchronize the clocks under that assumption).

Next important issue to understand is how exactly Einstein's paper is related to the aether theories of the time. The physics community was working with a concept of an aether, because it yielded the most obvious interpretation of C in Maxwell's Equations, and implied some practical experiments. Long story short, the failure to produce expected experimental results led Hendrik Lorentz (and George FitzGerald) to come up with an idea of length contraction affecting moving bodies (relating to variations to the propagation speeds of internal forces), designed to explain why natural observers could not measure aether speeds.

The significance of this development is that the transformation rules Lorentz came up with survive to this day as the central components of Einstein's theory of Special Relativity; that is why the relativistic transformation is called Lorentz transformation.

In terms of pure logic, Lorentz' theory and Special Relativity were effectively both valid. The difference was philosophical. For anyone who understands relativity, it is trivial to see that Lorentz' theory can be seen as completely equivalent to Einstein's in logical sense; just choose arbitrarily any inertial frame, and treat that as the frame of a hypothetical aether. Now any object moving in that frame will follow the rules of Lorentz transformation, just like they do in Special Relativity, and all the other consequences follow similarly. For natural observer, everything looks exactly the same.

When Einstein was thinking about the problem, he had Lorentz' theory in one hand, and the fact that one-way speed of light / simultaneity of events cannot be meaningfully measured on the other hand. It doesn't take an Einstein (although it did) to put those together into a valid theory. Since the universal reference frame could be, and would have to be, arbitrarily set - as far as any natural observer goes - it is almost trivial to set C to be isotropic across inertial frames, and let the rest of the definitions play out from there.

So getting to Special Relativity from that junction is literally just a matter of defining C to be isotropic, not because you must, but because you can. Setting C as isotropic across inertial frames is exactly what the entire paper about Special Relativity is about. Note that the language used in the paper is very much about how things are measured by observers, when their measurements are interpreted under the convention defined in the paper.

While Lorentz' theory was seen as a working theory, it was also seen as containing a redundant un-observable component; the aether. Thus Einstein's theory would be more scientifically neutral; producing the same observable results, but not employing a concept of something that cannot possibly be observed by its own definition.

And just judging from the original paper itself, this is certainly true. But there is great irony in that then Einstein's theory would get a redundant un-observable component tacked to it within few years from its introduction; the relativistic space-time.

A typical knee-jerk reaction at this point is to argue that relativistic space-time is a necessary component by the time we get to general relativity, but that is actually not true. General relativity can also be expressed via different ontological interpretations; some may be less convenient than others depending on purpose, but it is certainly possible.

Another reaction is to point out that different ontological interpretations of Relativity do not fall under the category of physics at all. This is true; but it is also true when it comes to relativistic space-time.

There really are some very solid epistemological reasons for the validity of relativistic time relationships, that have nothing to do with neither aether concepts, nor relativistic space-time concepts, or any other hypothetical structure for the universe. "Epistemological reason" means purely logical reasons that have got nothing to do with the structure of the universe, but everything to do with how we understand things, and those reasons are what Chapter 4 is all about.

I will write a post more directly related to the arguments of Chapter 4 very soon.