Tutorial: Relativity - what is a reference frame?

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\(v = c \, \wedge \, K = c^{\tiny -2} \; \Rightarrow \; w = \frac{u + v}{1 + K u v} = \frac{ u + c }{1 + \frac{u}{c}} = c\).
\( K = c^{\tiny -2} \, \wedge \, \begin{pmatrix} \rule{0pt}{20ex} \frac{1}{\sqrt{1 - K v^2}} & \frac{K v}{\sqrt{1 - K v^2}} & 0 & 0 \\ \rule{0pt}{20ex} \frac{v}{\sqrt{1 - K v^2}} & \frac{1}{\sqrt{1 - K v^2}} & 0 & 0 \\ \rule{0pt}{20ex} 0 & 0 & 1 & 0 \\ \rule{0pt}{20ex} 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \rule{0pt}{20ex} \Delta t\\ \rule{0pt}{20ex} c \, \cos \theta \, \Delta t \\ \rule{0pt}{20ex} c \, \sin \theta \, \cos \phi \, \Delta t \\ \rule{0pt}{20ex} c \, \sin \theta \, \sin \phi \, \Delta t \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} \frac{1 + K c v \, \cos \theta}{\sqrt{1 - K v^2}} \Delta t \\ \rule{0pt}{20ex} \frac{v + c \, \cos \theta}{\sqrt{1 - K v^2}} \Delta t \\ \rule{0pt}{20ex} c \, \sin \theta \, \cos \phi \, \Delta t \\ \rule{0pt}{20ex} c \, \sin \theta \, \sin \phi \, \Delta t \end{pmatrix} = \begin{pmatrix} \rule{0pt}{20ex} \Delta t' \\ \rule{0pt}{20ex} \Delta x' \\ \rule{0pt}{20ex} \Delta y' \\ \rule{0pt}{20ex} \Delta z' \end{pmatrix} \\ \quad \Rightarrow \; c^2 ( \Delta t' )^2 - ( \Delta x' )^2 - ( \Delta y' )^2 - ( \Delta z' )^2 = \frac{\left(c + v \, \cos \theta)^2 - \left(v + c \, \cos \theta)^2 }{1 - \frac{v^2}{c^2}} ( \Delta t )^2 - c^2 \sin^2 \theta ( \Delta t )^2 \\ \quad \quad \quad = \frac{\left( c^2 - v^2 \right) \, \sin^2 \theta}{1 - \frac{v^2}{c^2}} ( \Delta t )^2 - c^2 \sin^2 \theta ( \Delta t )^2 = 0 = c^2 ( \Delta t )^2 - ( \Delta x )^2 - ( \Delta y )^2 - ( \Delta z )^2\)

So the experimentally confirmed hypothesis that \(K=c^{\tiny -2}\) and \(K \neq 0\) causes both the velocity composition law to say that the speed of light is the same for a boost in the same direction, as well as the generalization of the Galilean transform to say two events separated by a pulse of light in an arbitrary direction are still two events separated by a pulse of light after an arbitrary boost. This leads to a special case of the invariant interval of relativity being equal to zero in all inertial coordinate systems.

All the questions and claims in the world does nothing to change the value of \(K\), so its important that when one is constructing a model that one does not accidentally use \(K = 0\) when that will materially affect the results.

The core of relativity (Galilean, Special Relativity or General Relativity) is the intellectual exercise of putting oneself in another's shoes to view the physics of the world from another vantage point. If your mental model is using the wrong space-time geometry, your predictions will not match reality. If your mental model isn't rigorous, like the hyperbolic geometry introduced here, then you run the risk of self-contradiction.

 

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Again, that is an impossibility, James. It is simply impossible. What you are saying, literally, is that a light sphere increases its radius at a constant rate as the light sphere travels as a whole in space, supposedly at the same exact rate as the source that emitted the light sphere, in the same direction coincidentally. It is simply impossible for a light sphere to travel as a whole as it grows in size. A light sphere is not an expanding sphere that grows as it travels along in space with the frame of reference that emitted it! The ONLY way that a frame of reference will measure the speed of light to be c is IF the frame has an absolute zero velocity in space. Period!

 

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Motor Daddy: Aren't you assuming \(K=0\) when experiment tells us otherwise?

 

This statement has to assume that space is both fixed and that the velocity of light is relative to space.., correct?

All experimental evidence suggests that the velocity of light, through vacuum or empty space, is relative to the observer and his/her measurement mechanisms and devices, not space itself.

However, even should one assume that space itself were involved, and that it were ridgedly fixed in place, you run into the contradictory evidence, from the Michelson and Morley experiments.., and the solution that Poincare and Lorentz presented.., length contraction in the direction of motion. So even a fixed rest frame for the propagation of light returns the same results in this case as Einstein's alternative, in SR.

This does not seem to be even remotely accurate when other modern tests of GR are taken into account. Both the geodetic and frame dragging effects, go a long way toward supporting the dynamic nature of space... Space is not static or fixed in the background as it seems you contend.

Setting all that aside, the above quote from your post is patently false. Why? Because there have been many many measurements of the velocity of light which agree to within the margin of error of the experimental design and equipement.., and they most if not all were carried out in different inertial frames of reference. Two labs in any two locations on the earth, must under the conditions your statement invokes not only not be at absolute rest, they are not at rest even relative to one another... Yet they reach an incredible degree of agreement on just exactly what the value of \(c\) is.

 

There is one way that the speed of light can be invariant for all inertial reference frames. Answer: the inertial reference frames are made out of naturally existing waves that obey \(c=\lambda f\). Distance in the reference frame is a construct of wavelengths. The progression of time of the reference frame is a construct of frequencies.

The physics community will cry foul: inertial reference frames are mathematical devices, not physically real!!! Please excuse me for injecting a physical/ontological cause into a mathematical disccusion of special relativity.

 

You would do well to read the OP. Reference frames are "made out of waves"? Holy crap.

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If given a choice between an ontological characteristic of the vacuum, and a math model, most physicists will choose the math model over the ontological characteristic...even when it doesn't make sense.

Good luck with that!

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Conceptually simple is a useful characteristic. But the quantum vacuum and the space-time continuum are not made of meter sticks and clocks. When particles collide at the LHC, it doesn't litter the inside with sticks and clocks.

I wanted to come up with a way to unify QM and GR (or at least SR) in some ontological and intuitive way. I noticed that the speed of light, c, in a vacuum, is an important constant in QM and GR. But it doesn't necessarily refer to light; then again, there are lots of characteristics that could be explained by an all pervasive medium, if I picked the right kind.

So I took the two postulates of SR, I took the wave-function, from QM, of a plane wave \(\psi = e^{i\omega t}\), and I made it into an aether medium (a light bearing medium). I simply argue that wave-functions, a math solution to solve the Schrodinger equation, are ontologically real. In empty space, they behave like the electromagnetic spectrum. These aether waves are the reason why permitivity of free space, permeability of free space, and the speed of light exist.

When you shine a light, light energy propagates through the aether waves. When the light is gone, the aether waves are still there; they are undetectable. The distance between two points in space has an ontological existence because of the wavelengths of these aether waves. Particles are just a group of excitations across a range of aether wave frequencies.

 

That is exactly what Mazulu does. He has no concern about consistency. For instance, he claims that the aether waves create time and space. How can a frequency create time or space? Seems to be "ontologically" impossible. But he does not care. Such problems do not slow him down. As far as reference frames go, he had to be schooled on that by (I think it was) Arfa Brane. Mazulu claimed that reference frames had to be objectively real. That they were real objects or at least attached to real objects. Eventually (after consulting wikipedia) he recanted his belief but this shows that he is not worth arguing with. The fact that after his reference frame boo boo with Arfa, that he has surfaced in this thread shows his arrogance and shallow understanding of Physics.

 

James, You're wrong. Light speed is not relative to the source that emitted the light. Light speed is relative to the the point it was emitted from, and the time of light travel, which determines the meter(s) traveled per time interval.

Now please, stop repeating Einstein's laughable second postulate and start by telling me how you determine a meter in a reference frame. You seem to be avoiding the issue.

 

The length of the path that light travels in space is relative to the frame of reference of space (ie, the radius of the light sphere in all directions from when and where it was emitted), and has nothing to do with different frames of reference of material objects that move slower than light. Those frames of reference travel relative to the light sphere, not the other way around.

This is the only mention of meters in your opening post:

Rather lacking in scope for me, how about you?

Can you explain in more detail the "somehow" that you speak, of ensuring that each stick has length 1 meter? That would be rather fantastic of you!!!

So, are you saying that if light is sent from the rear of a ship, and moves along the sticks of the reference frame of the ship, that a meter in that frame is when the light reaches a clock along the sticks that reads 1/299792458 of a second, and that every distance in that frame is a meter if it takes light 1/299792458 of a second to travel from point a to point b in that frame?

 

"From where and when it was emitted", eh? According to which observer?

Really that's where your logic goes off the track. The rest is just wrong; an observer can be a material object, and light travels 'distances' according to observers.
There is no frame of reference for the light sphere itself, so nothing "travels relative to" a light sphere. That part is just your imagination.

 

It seems to me that the discussion of light spheres is a distraction and divergence from the OP. While the concept of frames of reference become important to the discussion of light spheres, light spheres in and of theirselves do not add to the definition of a frame of reference...

On the issue of light spheres it often seems to me that the term or reference to a light sphere is not well defined, in these discussions. It may be my misunderstanding of some aspects of the discussion, but it seems that what a light sphere IS, is often being confused with what I would understand a "light horizon" to be.

Correct me if I am wrong or misunderstanding the context, but;

When I think of a light sphere it seems to me to be the spherical shell of light which was emmitted at some definite point in time and space and expands outward from that event. While GR may lend some dynamic properties to that expansion, from a purely SR perspective, the point of origin of that sphere of light or that expanding shell of photons originates at one point in time and space, does not move. Even while the object from which the shell of photons originated does move in space. Objects emitting light continously, while moving through space, are creating an infinite number of light spheres, the center of each being located at a specific point in time and space.

A light horizon is not the same thing. It does move with the observer.

Both may be loosely described as spheres. One is formed from the photons emitted from a single event and the other from photons which were emitted by many events. Each photon observed in an observer's light horizon, would actually be one photon in a light sphere whose origin is somewhere and time in the observer's past.

No one can observe a light sphere, because they can only see the one fragment of the sphere, the part that is part of their light horizon. Everyone can observe a light horizon.

This discussion should be split off of the OP, a thread of its own, or that portion of the OP that is addressing a description or definition of frames of reference, should maybe be duplicated and locked as a resource for further and other discussions.

 

Who said anything about a "light horizon"????

 

That is the current definition of a meter on earth. We chose to define the meter that way because it was the way that varied the least of all other methods. In Motor Daddy theory, the current definition of a meter on earth should reveal that the earth is moving. Meter sticks would not match light measurements in all directions. Unless I misunderstand your argument?