Sunday, September 7, 2014

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...

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