Time is imaginary

Time is imaginary, therefore, time squared is a negative number. Pythagorean theorem, in relativity theory, is called spacetime. And spacetime is invariant in all frames of reference.

\[ict\]

\[(ict)^2=-c^2t^2\]

Here are the well known Lorentz Transforms, I coded them in to functions.

#Lorentz Transforms
x.prime <- function(x,v,t){ 
 (x-v*t)/sqrt(1-v^2/c^2) 
}

t.prime <- function(x,v,t){ 
 (t-v*x/c^2)/sqrt(1-v^2/c^2) 
}

Let’s say in the first frame movement distance is 100,000km, time passed is 0.3 sec. Speed relative to the second frame is 200,000km per sec. So in the second frame we would see movement distance to be 5.366563e+04km, and time passed to be 1.043498e-01 sec.

x1 = 100000
t1 = .3

v = 200000
c = 300000 

x2 = x.prime(x1,v,t1)
t2 = t.prime(x1,v,t1)
c(x2,t2)
## [1] 5.366563e+04 1.043498e-01

Now we apply Pythagorean theorem. We learned from school it was a squared + another squared. But squared time is negative, therefore, we need to use squared - another squared. Also after taking square root, we get spacetimes of the two separate frames, and they are identical 43588.99.

\[s^2=x^2+(ict)^2=x^2-c^2t^2\] \[s=\sqrt{x^2-c^2t^2}\]

s1 = sqrt(x1^2 - c^2*t1^2)
s2 = sqrt(x2^2 - c^2*t2^2)
c(s1,s2)#invariant
## [1] 43588.99 43588.99