PeterDonis said:
This forum has a FAQ with a brief discussion:
https://www.physicsforums.com/showthread.php?t=511170
The Usenet Physics FAQ also has a useful page on this topic:
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/headlights.html
Mathematically, it is obvious that the Lorentz Factor is undefined when v = c; just plug v = c into
\gamma = \frac{1}{\sqrt{1 - v^2 / c^2}}
and observe that \gamma is undefined because its denominator is zero.
"Photons" in SR are objects that move at speed c in all reference frames. Put another way, photons are objects that move on lightlike (or null) worldlines; in a standard spacetime diagram, these are 45-degree lines (because we use units in which c = 1), as shown here:
http://en.wikipedia.org/wiki/Minkowski_diagram
That pretty much covers what I have already observed in my head. I am also familiar with Minkowski diagram. While it helps the observation of sub-luminal up to the luminal, it still leaves me completely lost at understanding the reference frame of an individual Photon.
Would you be willing to advise me on how I should adjust my perception/clean up/fix logical mistakes? Other than the thought experiment doesn't make sense (like in the link.) I have gotten that twice, and to me it seems very reasonable to let it go, but at the same time if one is willing to do things correctly, by acceptable notions of physics one should explore this thought experiment like any other. I find it somewhat frustrating that the people I have talked to don't want to entertain these ideas while applying science and logic. I think there is more to Special Relativity than meets the eye. So given that, please understand, I have been told its "almost useless" thinking about this type of situation.
To better understand the individual properties of light, I have taken to creating a reference frame on an individual photon, traveling in a wave like motion at velocity C.
The wave nature of the photon is assumed sinusoidal, and I base my functions off of Maxwell's Wave Function for Electromag.
Y(X) = A Sin (kx + ωt + ∅)
Such that, A = Amp, K = Wave Number, ω = Angular Freq, t = time, ∅ = Phase.
Now, assuming constant c, and that this Photon is coming from a non-variable / single direction device such as a perfect laser, I managed to conclude a few basic concepts. But first frame relativity would require another observational point. I decided to look at another Photon along this same wave at the same time, and I would call it colinear - or cowave to be more precise. The problem with this is that there is no other perceivable photon in front or behind to an individual photon. Logically, you traveling at the speed of light, the guy behind you is traveling at the speed of the light in the same direction, and the guy in front of you is traveling, again, at the speed of light in the same direction. It is impossible for you to "perceive" anyone else forward or aft. I quickly broke down some math because I need to know if this scenario is logically feasible, otherwise I am spinning my wheels going insane. Thus I looked at the predictability of a Photon with a wave function, and wondered if there was a way of Mathematically showing a Photon could exist in a position of our choosing along this wave.
Y(X) = A Sin (kx + ωt + ∅) is transformed to thus,
Photon Zero: Y(X1) = A1 Sin (k1x1 + ω1t1 + ∅1)
Photon One (Chose by Observer): Y(X2) = A2 Sin (k2x2 + t2 + ∅2)
Considering the approach is for a cowave photon, this reduces the functions complexity.
|A1| and |A2| should be identical at a given time.
K1 = K2, same wave number.
ω1 = ω2, same ang. freq.
t1 = t2, results observed (if only in theory) at identical times.
∅1 = ∅2 should be equal or zero, zeroth for this observation.
Now given that it is wave like motion, I have determine that both Photons are in a constant state of X -> Y -> X vector motion.
This lead me to believe that a Photon is only at velocity C in the X-Direction, when at peak amplitude of the wave motion, either A * 1, or A * -1. So I began contemplating the Photons on sequential peaks.
I broke down the equations further for analysis of Delta X, since X1 cannot equal X2. I.e. observation of two distinct photons.
Knowing how Sine waves function, we know the maximum value determined by this equation is A or -A. Also since the Sine wave intersects the X-Axis, we can also choose a Photon capable of meeting at the x-Axis, at the same time, which correlates to the minimum distance between the two Photons.
So Photon 1, in relation to Photon 0, has a Y(X) = - A, when P0 has A, and a minimum distance of ΔX when A = 0 for both.
Maximum Distance => Opposite Peaks = D = √[(ΔY)^2 + (ΔX)^2]
Minimum Distance => A = 0 = d = ΔX
Maximum Distance => D = √[(2A)^2 + (ΔX)^2] => √[4(A)^2 + (ΔX)^2]
Solving for ΔX requires some basic algebra of the equations above:
Y(X1) = A1 Sin (k1x1 + ω1t1 + ∅1)
Y(X2) = A2 Sin (k2x2 + t2 + ∅2)
Transformed into:
Y(X1) = A Sin (kx1 + ωt)
Y(X2) = A Sin (kx2 + ωt)
Assuming Maximum Distance occurs when A * 1, -1 is reached, we solve for the corresponding Sine function.
1, -1 occurs at Pi/2, 3Pi/2
kx1 + ωt = Pi/2
kx2 + ωt = 3Pi/2
x1 = Pi/2k - ωt/k
x2 = 3Pi/2k - ωt/k
ΔX = X2 - X1 = Pi/K
So we choose a Photon 1, such that ΔX = Pi / k.
Maximum Distance => D = √[4(A)^2 + (Pi / k)^2]
Now the reason why I did the above, was just to show mathematically to myself, that if two Photons are capable of being treated as particles, that I should come up with a logically derived formula even though neither Photon can observe the existence of another.
Given Photon 0, I should be able to find another Photon that exists in this manner.
Now at this given snapshot in time, both of these Photons are traveling at velocity C in the X direction. According to SR formula, the Lorentz Factor is undefined, or more specifically, the relative velocities become undefined with one another. However, Newtonian Physics seems to be more understandable, and perhaps more applicable strangely, as the Newtonian relative velocity is simply zero, which is something I consider to be correct. There is also the oscillation between the Y-Axis in opposing directions in which two Photons are peaking at +/- C in which Special Relativity should would have to be used again. I.e. These two photons traveling at +/-C upon reaching the X-Axis or when ΔX = Pi/K, the velocity of P0 should be in the -Y direction @ C, and P1 should be in the Y-Direction @ C. There is oscillations, and changes, in the Photon's position with respect to one another, its just not in the X-Axis. Two objects will have a relatively zero velocity with respect to one another traveling in identical directions at identical velocities. Co-wave Photons, technically, are the only thing in the universe that would represent a Newtonian Relative Velocity because there is simply no change in reference frame nor velocity (without outside interference.) However, it doesn't seem to work well with a wave function, just a co-linear function.
I have also considered what a single coherent reference frame would probably mean to the Photons and began to think about the concept of time ever so slightly in this reference frame. If there is one unique reference frame for this cowave, it would stand to suggest that the entire wave, every photon would be in the exact same reference frame from start to finish. It could explain why light only deviates in relation to outside factors, there is no internal change to this system. I.e. Light from another Star traveling thousands of years to us is the "same" light emitted from the Star with some external factors influencing it. Unfortunately this idea and many others are just rattling around in my head. I have more "raw" ideas in my head but are almost fantastical without any solid groundwork to begin breaking it down logically what I mean.
I would love some help it determining what I should strive to better understand or fix errors in my logic early on, as I have somewhat become obsessed with this frame of reference. It is almost like I see the entire Universe in my head as a manipulated wave with the basic unit relative to a "Photonic" reference frame. In other words, all observable space is basically Photonic and the smallest possible frame is the Photonic reference frame itself. Explaining why the velocity of light is constant to us, is that its part of the underlying fabric of perceivable space. Again just "fantasy" :D
