Special Relativity Rebooted — Part 1
Velocity Addition Rebooted
Maxwell’s Electromagnetic Wave Equation is traditionally read as a three-dimensional wave equation containing a fixed speed of light. An alternative way to read the mathematics is that
Maxwell’s 3D Wave Equation contains a Fixed 3D Speed.
NeoClassical Relativity (NCR) is based on this interpretation of Maxwell’s mathematics. ©
In previous posts, I introduced the two postulates of NeoClassical Relativity.
The first postulate is that: The Speed of Light is a three-dimensional combination of the Speed of Oscillation and Speed of Linear Travel. Mathematically, this postulate can be written as the “3D Speed Equation”
Equation (1) ______________ c² = u² + v²
{Where u is the speed of oscillation in two spatial dimensions, and, v is the speed of travel in the third.}
The second postulate states that: The Oscillation-Length of a 3D Cycle is a Fixed Quantity. Mathematically, this postulate leads to the “3D Length Equation”
Equation (13) ____________ (λ-c)² = (λ-min)² + (λ-v)²
{Where, λ-c is the cycle-length, λ-v is the wave-length, and λ-min equals 2π times the Planck Length — which equates to a length of 1.0155 x 10⁻³⁴ meters.}
In this first part of a three-part reboot of Special Relativity, I will address the concept and nature of “Velocity Addition”…
Einstein’s work on Special Relativity was his resolution of a fundamental conflict between the principle of “relativity”, and the “absolute” nature of the speed of light (c).
The classical principle of relativity, which was first articulated by Galileo Galilei in 1632, basically says that the speed of a thing is relative to the speed of the observer. This is something everyone pretty much recognises: if you are driving on at 60 miles per hour, and a car, coming in the opposite direction, is also doing 60, then it will seem to you like the oncoming car is travelling at 120 mph.
We are all used to this idea of relative speeds and adding velocities. The trouble is that it doesn’t appear to work with light.
If you are standing at the side of the road and a car, with its headlights on, goes past you at half the speed of light, does that mean that the light from the headlights is actually travelling at one and a half times the speed of light from your point of view?
Einstein said “No. The speed of light is not relative to anything, it is absolute. The light you see always travels at c.”
If this is so then, how do we add velocities?
Einstein’s Velocity Addition
Einstein formulated his Theory of Special Relativity based on Two Postulates.
- The Laws of Physics are the same for all observers in any state of “Uniform Motion”.
- It is a Law of Physics that the Speed of Light is a Fixed Quantity.
The strength of Einstein’s theory is primarily predicated on the fact that Einstein was able to derive, in a few short pages, all the same results that had taken his predecessors many years of long and complicated work. From two simple principles, Einstein derived the Lorentz Transformation Equations; and it was from these equations that Einstein ultimately derives his “Velocity Addition Equation”.
However, for the purposes of this discussion, the important thing to note is that this Velocity Addition Equation has not been derived directly from first principles, but from the Lorentz Transformation Equations.
I will address the validity of the Lorentz Transformation Equations in Special Relativity Rebooted — Part 3, but in the meantime, I will now derive, from first principles, an alternative Velocity Addition Equation for NCR...
Derivation of Velocity Addition
Okay, let’s start by recalling
Equation (7) _____________ v = f (λ-v)
Equation (8) _____________ c = f (λ-c)
And when we take the ratio of v/c, the frequency (f) cancels out and we are left with
Equation (28) ____________ v/c = (λ-v) / (λ-c)
And we can rearrange this to be
Equation (29) ____________ v = (λ-v/λ-c) (c)
And using equation (13) we can write we can rewrite this as
Equation (30) _______ v = ((λ-v) / SQRT {(λ-min)² + (λ-v)²}) (c)
This equation tells us that the velocity that any observer sees is dependent solely on his observation of lambda-v (because λ-min and c are constants).
Now using this formulation of velocity, let’s examine the velocity of a thing from the point of view of two different observers.
Crucial to this analysis is that, in NCR, every velocity (v) expressed in terms of (c) has an associated wave-length (λ-v), as the table below shows
Observer one see a thing moving in his stationary frame of reference. The thing has a velocity (v1) and an associated wave-length (λ-v1).
Observer two is an external observer, and he too sees the thing moving, but he also sees the frame of reference moving. The thing has velocity (v1) and an associated wave-length (λ-v1). The frame of reference has velocity (v2) and an associated wave-length (λ-v2).
Now, if we put (λ-v1) into equation (30) we will get the velocity that the internal observer sees; and if we put the combined length (λ-v1 + λ-v2) into equation (30) we will get the combined velocity that the external observer sees.
Thus we can see that equation (30) can be adapted into an equation for the addition of two velocities (v1 and v2)
Here we have an equation where the combined velocities can never be greater than (c) because (λ-min) ensures that the value below the line is always greater than the value above the line.
And this is the crucial insight that leads to the resolution of the velocity addition problem.
The resolution of the velocity addition problem is, NOT to add velocities, it is to ADD THE WAVE-LENGTHS.
With equation (31) we have a formula for velocity addition in terms of wave-lengths; however, it would be convenient to have this same formula expressed in terms of velocities relative to (c).
We already know that any wave-length (λ-v) can be expressed in terms of (λ-min)
Equation (14) _________ λ-v = (v/u) λ-min
So we can use this formula to calculate values for lambda-v1 and lambda-v2, each in terms of (λ-min)
____________________ λ-v1 = (v1/u1) λ-min
____________________ λ-v2 = (v2/u2) λ-min
And when we substitute these values into equation (31), λ-min cancels out top and bottom and we are left with
This is the Velocity Addition Equation for NCR, and if differs significantly from Einstein’s Equation.
Comparing Equations
Let’s now compare NCR’s equation for velocity addition to Einstein’s equation, using a worked example.
Say v1 is half the speed of light and v2 is three quarters the speed of light.
Firstly: using NCR’s Equation (32)
____________ v1 = .5c
____________ u1 = .866c
____________ v1/u1 = (.5c/.866c) = .5774
And
____________ v2 = .75c
____________ u2 = .6614c
____________ v2/u2 = (.8c/.6c) = 1.134
Thus
____________ (v1/u1) + (v2/u2) = 1.711
And next
___ SQRT{ (1)² + ((v1/u1) + (v2/u2))² } = SQRT{ 1 + (1.711)² }
___ SQRT{ (1)² + ((v1/u1) + (v2/u2))² } = SQRT{ 1 + 2.929 }
___ SQRT{ (1)² + ((v1/u1) + (v2/u2))² } = SQRT{ 3.929 }
___ SQRT{ (1)² + ((v1/u1) + (v2/u2))² } = 1.982
And therefore
____________ v1+2 = ( 1.711 / 1.982 ) c
____________ v1+2 = .863c
And secondly: using Einstein’s Equation
___________ (v1) + (v2) = (.5c) + (.75c)
___________ (v1) + (v2) = 1.25c
And
___________ 1 + (v1)(v2)/c² = 1 + (.5c)(.75c)/c²
___________ 1 + (v1)(v2)/c² = 1 + .375c²/c²
___________ 1 + (v1)(v2)/c² = 1.375
Therefore
___________ v1+2 = (1.25c) / (1.375)
___________ v1+2 = .909c
Who Is Right?
So the different equations give different results, so which one is right?
Well, the usual thing to do in such circumstances is to experimentally test these predictions. However, in the absence of experimental results, and from a purely rational perspective, I believe that the Velocity Addition Equation of NCR is the correct formulation for three primary reasons.
- Firstly this formulation of adding wave-lengths makes intuitive sense — something that is generally missing from the traditional understanding of Special Relativity.
- Secondly, Einstein’s formulation was not from first principles, but rather a concocted formulation derived from The Lorentz Transformation Equations (which are also concocted formulations and that have been around in various forms since 1887).
- Lastly, NCR is ultimately a gateway to Quantum Mechanics, (which implies that this formulation has deeper explanatory power).
© Kieran D. Kelly
This is Part 1of 3. Part 2 is here. Part 3 is here.
This is Post #5 in the series on NeoClassical Relativity Theory
