Sunday, January 15, 2017

Simple Derivation Of Special Relativity

Once I had read somewhere that Einstein claimed speed of all objects in the Universe, when measured in 4 dimensions, always equal to c (speed of light)!
(The object can be anything from subatomic particles to stars.)
(This also implies if the speed of an object in space increases toward c then its speed in time must decrease toward 0 and vice versa.)
It seemed farfetched at first but then I realized if assumed to be true it leads back to Special Relativity (SR).
Meaning it is a direct consequence of SR and if SR is accepted to be true then so it must accepted to be true also.

Let's start with assuming it to be true:
(Vx^2+Vy^2+Vz^2+Vt^2)^(1/2)=c {4D speed equal to c}
Let's reduce 4 dimensions to 1 space and 1 time dimensions:
(Vxyz^2+Vt^2)^(1/2)=c
Or just:
(Vx^2+Vt^2)^(1/2)=c {Vx: speed in space; Vt: speed in time}
then:
Vt=(c^2-Vx^2)^(1/2) {and also Vx=(c^2-Vt^2)^(1/2)}

Since Vt:0 to c (also Vx:0 to c as a consequence of above)
then Vt/c:0 to 1
and if deltaT'=alpha*deltaT
where deltaT':traveler time and deltaT:observer time
then
alpha=Vt/c=(c^2-Vx^2)^(1/2)/c

If alpha is squared, simplified, square-rooted:
(c^2-Vx^2)/c^2 -> 1-Vx^2/c^2 -> (1-Vx^2/c^2)^(1/2)

Continuing from above:
deltaT=deltaT'/alpha=(1/alpha)*deltaT'
if gamma=1/alpha=1/(1-Vx^2/c^2)^(1/2)
which is Lorentz Factor of Special Relativity!

(Also notice that the equations of space and time are the same meaning time is a dimension similar to dimensions of space.)

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