Lorentz Transformation Factor (γ)

My Derivation
In 1905 Albert Einstein proposed two basic principles that apply for all Inertial (Non-accelerating) systems.

1. The speed of light (c) is the same for all observers.
2. The Laws of Physics remain constant in all such systems.

So simple! Doesn't seem so radical, does it? But the implications are very far reaching. Let's explore further.

We have some event in spacetime. We have two systems (S and S') moving relative to each other at a constant velocity (v). We will generate two equations that describe the event, one for each system.

In the S system the x coordinate of the event at (x,t) can be described as:
x = x' + vt'     (1)

And in the S' system the x' coordinate of the event at (x', t') can be described as:
x' = x - vt     (2)

Now we have two descriptions of the SAME event which are BOTH true from their respective viewpoints. To make our observations valid for BOTH systems we must introduce a 'conversion factor' that we will call γ (lowercase Greek gamma). We must therefore re-write the above equations as:
x = γ(x' + vt')     (3)
x' = γ(x - vt)       (4)

We now employ some basic algebra to solve for γ in the above system of two related equations (3 & 4). We multiply the *LHS of equation (3) by the *LHS of equation (4), and equate the result to the product of the *RHSs of equation (3) and (4). This gives us:
xx' = γ2(xx' + xvt' - x'vt - v2tt')       (5)

We agree that the speed of light (c) is constant so:
x = ct       (6) x' = ct'       (7)

Substitutting these values into equation (5) we get:
c2tt' = γ2(c2tt' + vctt' - vctt' - v2tt')       (8)

Dividing both sides of equation (8) by tt' we get:
c2 = γ2(c2 - v2)       (9)

We are almost there. We divide both sides of the equation by (c² - v²). This gives us:
γ2 = c2 ⁄ (c2 - v2)       (10)

which can be reduced to:
γ2 = 1 ⁄ ( 1 - v2 ⁄ c2 )       (11)

Taking the square root of both sides of equation (11) we get γ which is known as the Lorentz Transform Factor :
γ = 1 ⁄ √(1- v² ⁄ c²)       (12)

This important factor can be used to calculate relative Mass, Length, and Times of any two systems moving in a constant velocity relative to each other. It is indispensable in the scientific community. At relative velocities to which we are accustomed and which are very much less than the speed of light (c), the factor (γ) is essentially 1 and no significant observeable effects occur (see the table below). However at relative velocities approaching the speed of light the factor (γ) becomes very significant and the effects cannot be ignored. Such are the speeds, for instance, of electrons zooming about within atoms.
Currently we can use large powerful particle accelerators to bring small particles like electrons and protons up to enormous velocities and observe the effects. The Lorentz Transform Factor has NEVER been shown to be in error. The longest Linear Accelerator is the Stanford Linear Accelerator Center built in 1966 in Menlo Park, California. The world's biggest Particle Accelerator is the CERN Super-Collider in Switzerland. CERN was originally built in 1954 and now houses the Large Hadron Collider (LHC) which is 27 km (16.8 miles) long.

🎦 CERN 4 minute Video

Length Contractions & Time Dilations
Other measurements are affected as two systems move relative to each other. It turns out that Length and Time are modified in the moving system. If we let Lo be the rest length of a moving rod, and To be the origin time in both systems (t=0), then to calculate the length Lm in the Moving system, and the elapsed time Tm in the Moving system we employ the Lorentz Transform Factor appropriately thus:
Lm = Lo(1- v²/c²)       (LENGTH IS CONTRACTED in the moving system by this relationship)

Tm = To(1- v²/c²)       (TIME IS DILATED in the moving system by this relationship)

* LHS, Left Hand Side       RHS, Right Hand Side