Chapter 5 Add’l: Understanding the “Holes” in Special Relativity
Despite the experimental confirmations of Special Relativity, our interpretations of it have still left some holes around the nature of light.
This article is a companion piece to my video “The Light Compass: Fixing Special Relativity.” The purpose of this article is not to make a full case for these issues, or to repeat the all the concepts I covered. It is here to allow for a written walk-through of the math and reasoning in the video so that people may review and examine it thoroughly.
“Uh, Hey… You Missed a Spot.” Special Relativity Can’t Be Verified With It’s Own Equations
When I dove into studying Special Relativity (SR) to investigate questions I had around the nature of light, I was stunned to learn that the postulates of SR cannot be mathematically verified with the equations we have used SR to create.
Specifically, I am talking about the postulate: “The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer.”
Of course, that is the way most people describe the postulate. The original postulate is more accurately translated as: “As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.”
The postulates seem fairly straight forward. And indeed, we have since used them to derive a whole world of math and geometry based around it’s principles. Also, as far as we can tell, these formations have held up experimentally and have proven reliable for practical use, such as advanced astrophysics (satellites, space exploration, cosmology, etc) and particle physics.
However, all the equations of SR and those we have derived from it, have still left a hole in our understanding of the relationship between “c” and “light.” In fact, I will show that the equations from Special Relativity do not allow for the “proving” of the above referenced postulate, and the alternatives for doing so open up the possibilities of multiple interpretations.
The problem in a nutshell is this: When we say that the speed of light or “c” is always the same in every inertial frame of reference, does that mean:
- that light always moves at a velocity of “c” away or towards all objects with equal timing in the same frame of reference?
OR
2. that light always moves at “c” from its point of origin in space in an inertial frame of reference?
You see, you can’t have both. Also, #1 implies that light’s velocity is dependent on the observer in the inertial frame of reference, which is basically “magic.” While #2 means that light’s velocity is based on “c” as a geometric dimension of what we commonly call “spacetime” (although, my regular viewers will know I hate that term).
While I am obviously biased toward #2, the point is that since we have never tested light in one direction (all our tests reflect light to test its velocity), we don’t really know the correct answer. Also, the following math will show that whether #1 or #2 is correct, there is still no such thing as a truly independent frame of reference in our Universe.
Attempting to Prove the Postulate of Special Relativity
In the following mental math experiment, we are going to have two frames of reference:
S1) This will be a person (I named him “Bob”) in a theoretical motionless frame of reference. He is standing still on his motionless rock, somehow observing things in the void of space.
S2) This is our rocket ship, traveling 80% the speed of light as measured by S1/Bob.
Here is the setup:
S1/Bob is going to see the rocket ship (S2) shine a beam of light from the front of the ship. The front-most photon of that beam is the focus of our math. That photon is going to travel away or separate from the ship (S2) at the speed of “c” to Bob’s frame of reference.
If Special Relativity is correct, then we should be able to use the math of SR to then show that the from the rocket’s (S2) frame of reference, that photon is also observed as traveling away from the person on the rocket ship at “c” relative to them as well.
NOTE: Keep in mind that in this mental experiment, we are removing the requirement for light to reflect back to/towards the observers to be measured. This is likely impossible to reproduce in real life, but I have proposed a practical experiment to test the principles in my videos.
OK, here we go:
S1/Bob measures how far the light emitted from the rocket ship travels away from or separates from the ship in a period of one second from his perspective. The math for this is very simple:
c = 3*10⁸ m/s (rounded up version)
v = c * 0.8 = 2.4*10⁸ m/s (rocket’s velocity)
t = 1 sec
L_S1 = 60,000,000 meters (distance of separation from Bob’s frame)
NOTE: I made a dyslexic mistake in the video and did (c — v) / t. This does not change the outcome or any of the conclusions, but it is embarrassing none-the-less.
The light will separate a distance of 60,000,000 meters from the rocket ship in one second. This is because Bob measures the rocket ship at 0.8c (80% the speed of light) and the photon at full “c” or 300,000,000 meters a second (we are using the rounded-up version of “c” for simplicity).
This is perfectly normal, and within all current expectations of physics and SR. Now we need to see if the rocket ship will see the light travel away at a full speed of “c” from its own frame of reference:
First, we need to compensate for time dilation. Using the standard time dilation equation, we find that for one second passing for S1/Bob, 0.6 seconds pass for the S2/rocket ship.
For these, we are going to use gamma (γ) for simplicity:
So time dilation here will be:
t_S1 = 1 second (Bob’s time)
t_S2 = 0.6 seconds (Rocket ship’s time)
OK, now we need to adjust for length contraction. This part confuses some people because we are going to INCREASE the length for the S2/rocket ship frame of reference. Some may think it should decrease as things get “compact” at near light speeds to outside observers. However, as we are comparing the distance light separated to the INERTIAL frame of reference of S2, we make it longer. This is because everything in the S2 frame would perceive everything in that frame as normal, it’s actually S1/Bob who would be originally seeing the S2/rocket ship as compact.
So, we use this equation:
L_S2 = 100,000,000 meters
Now we take the S2 length, and the S2 time to see how fast the photon is moving from the rocket ship’s frame of reference:
C_S2 = 1.67*10⁸ m/s
As you can see, our new calculation for “c” is a little over half that of our reverence speed for “c” of 300,000,000 m/s. So what’s wrong?
Well, as I said, currently there is no accepted method for this kind of “reverse engineering” of SR. We have to accept this postulate as true to then form our other equations in the first place. Therefore, the traditional method of answering the question “How far did the light travel from the rocket ship’s/S2 frame of reference?” is to do the following:
c = 3*10⁸ m/s
t_S2 = 0.6 sec
L_S2 = 180,000,000 meters
This gives the length for “c” to travel in 0.6 seconds to match “c” so…
L_S2 = 180,000,000 meters
t_S2 = 0.6 sec
c = 3*10⁸ m/s
And, of course it does, because that’s circular reasoning. This is the math equivalent of “Light moves at ‘c’ because light moves at ‘c.’”
There is a missing ingredient here. In SR, we have the following main equations:
- The Lorentz Transformation
- Time Dilation
- Length Contraction
- Relative Addition of Velocities
As you can see from my above example, I only used three of them. The problem is that “Relative Addition of Velocities” only works when comparing the velocities of two moving objects (reference), and our S1 observer, Bob, is not moving. This means half of the equations will all be zero, so they are useless here… almost.
I have discovered that you can apply the same thinking in relative addition of velocities to modify our length contraction equation in the absence of a moving comparison. It looks like this:
L_S2 = 180,000,000 meters
As you can see, by adding this modifier, we get the same result as the “traditional” method for finding the length. Don’t worry, as long as your S1 reference has a time comparison of one second, the equation works for any S2 velocity. Further modifications can be made for any amount of time.
The fact that this equation works already illustrates a rather fascinating relationship between “c” and these reference frames: When starting from a motionless (zero ‘c’) frame, all other frames share in built-in relationship to “c” as a direct percentage of velocity. This means that all reference frames are actually dependent, not independent. There is a continuity from one frame to another.
But Wait, There’s More! A Second Approach to Proving Special Relativity
Instead of trying to modify length contraction, we are going to imagine an “arm” extending out from the S2/rocket ship with a mirror at the end to reflect the light back. The length will be 100,000,000 meters in the inertial frame of the S2/rocket ship, and therefore 60,000,000 meters to S1/Bob, matching our initial distance of separation.
As we already calculated, it will take a photon one second to reach the mirror from S1/Bob’s point of view, and 0.6 seconds to reach the mirror from the S2/Rocket ship’s point of view. This brings us back to the 1.67x10⁸ m/s for the measured speed of light from the S2 perspective.
Here’s the most important thing to keep in mind now: Special Relativity is direction/vector dependent. Length contraction only applies in the direction the S2/rocket ship is moving. So now, once the photon reflects off the mirror and travels back, we need to calculate its velocity without length contraction, only time dilation will apply. So here’s how that looks:
As the photon travels to the mirror we use the original length and time dilation to calculate the measured speed from the rocket ship’s frame:
C_S2-> = 1.67*10 m/s
As the photon travels back we need to start from S1/Bob’s frame, so first we reincorporate the motion of the rocket ship traveling towards the photon. As the rocket ship and the photon are now traveling towards each other, we now add them together:
c = 3*10⁸ m/s
v = 2.4*10⁸ m/s
t_S1 = 1 second
C_S1<- = 5.4*10⁸ m/s
Now, calculate the time it took from S1/Bob’s frame for the photon to reach the S2/rocket ship:
L_S1 = 60,000,000 meters
C_S1<- = 5.4*1⁰⁸ m/s
t_S1<- = 0.11 seconds (1 is repeating, Medium.com won’t let me underline)
Next we adjust for the same time experienced by the rocket ship/S2:
t_S2<- = 0.066 seconds (6 is repeating)
Now, keep in mind that while the photon’s velocity is calculated without length contraction, the OBSERVED distance traveled from the S2/Rocket Ship’s frame is still 100,000,000 meters there and back. This is because from the S2 frame, there is no inherent awareness of the length contraction. So, the S2/Rocket ship will still conclude, without any external information, that the photon still traveled at “c:”
L_S2 = 100,000,000 meters
L_S2 *2 = 200,000,000 meters
t_S2 = 0.6 seconds
t_S2<- = 0.066 seconds (6 repeating)
t_S2 + t_S2<- = 0.666 seconds (6 repeating)
C_S2<-> = 3*10⁸ m/s
This result shows that by incorporating information from the relative frames, we can show that light OBSERVED (human perspective) might not be the same as light MEASURED (equipment), due to the difference in timing vs. information carried by the light. We have learned we can get the same result with…
Traditional Understanding
c <-> = 3*10⁸ m/s (same speed both directions)
OR
Incorporating Frames of Reference
c -> = 1.67*10⁸ m/s (S2 speed towards the mirror)
c <- = 1.5*10⁹ m/s (S2 speed back from the mirror)
This leaves us with a conundrum: The observed consistent velocity of light might in fact be due to the reflection/re-emission of light, not a “magic” property of universe where light adjusts its speed dependent on the observer. So how do we resolve this?
NOTE: The Independent Frame of “c” is From the Dimension, not Light
In this mental experiment, it is critical to acknowledge that the independent reference frame of light I am assuming is a consequence of “c” not light. It is the limit imposed by “c” as a dimension that creates the independent frame. If light is “slowed down” in a medium, that independent frame is changed has to incorporate the inertial frame of the medium.
This distinction has implications for the conditions by which we have to test this proposal. It can only be confirmed by measuring photons in a vacuum, unimpeded by any medium.
How to Incorporate Information Between Frames of Reference
There are some people who might look at the situations I am proposing and say, “This is all pointless because the information exchange required between frames is impossible.” And from a pure math standpoint, that is true. However, we don’t live in math on paper, we live in a real Universe.
A rocket ship in space has no way to tell it’s going 0.8c the speed of light without external context, but most circumstances do have information carried with them. For example, it could deduce its speed from the length contract observed from surrounding objects, but only if it already knew what those lengths were “at rest.” It could know it’s going 0.8c if it accelerated there from a position “at rest,” bringing the information with it.
The problem with our thinking on Special Relativity is that we have become so focused on theoretically “perfect” inertial frames of reference, we have forgotten that those are virtually non-existent in the real world. In the case of the possibilities, I am outlining here, there may, in fact, also be information that is incorporated in all moving frames by testing light in one direction only, with no reflection. That experiment is what I proposed in my video on The Light Compass.
The purpose of this entire discussion is because if contextual information is relevant to inertial frames of reference, in regard to the measured speed of light, it supports the argument that “c” is actually a geometric dimension of our Universe. This is the first major testable claim that results from my proposed approach to physics under the Unifying Theory of Dimensional Geometry and Interaction.
