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You will see the proofs in a short while, but beforehand,
...let's see how Einstein ingeniously formulated the time dilation.
Time it takes for the light to reach the ceiling relative to the train is t'.
y is the distance that light takes in t' time, with constant speed c.
Time it takes for the light to reach the ceiling relative to the ground is t.
z is the distance that light takes in t time, with constant speed c.
x is the distance the train takes, with the speed v, relative to the ground.
Since I've determined the distances x, y and z, in terms of c, v, t and t'...
...in the Pythagorean Equation, I can start simplifying.
Our aim is to leave term 't' that represents the time passed on the ground,
...alone on one side of the equation.
First, I move the term vxt² to the right side of the equation.
I take these terms out of paranthesis since there is multiplication sign between them.
I do the same for the left side.
When I see the minus sign between the terms on the right side,
...I take them in t ² parenthesis.
I take c ²-v ² term to the left side of the equation leaving t ² on the right side.
Our equation is mostly simplified now.
But if we want to reach Einstein's formula, we have to further simplify it.
Because Einstein believed nature must have very simple rules.
In a fraction, when we divide numerator and denominator by the same number,
...result will be the same. So why don't we divide each side by c ² arbitrarily?
Then the c ² terms with multiplication sign between,
...will cancel each other out and only t ² will remain on the numerator.
We cannot do the same on the denominator since there is subtraction there.
But we know this representation will be equivalent to this.
We also know c ² / c ² is 1, so remove it too.
There is one last thing left; taking the square roots of both sides.
I can take the square root of the numerator and denominator separately...
...instead of the whole.
So, as the square cancels the square root... This square cancels this one…
Voilaa!
Our formula is ready! Now let's try this on an example.
Damla and Serap are twins of age one.
We put Damla on a spaceship and let her travel for one day, at a speed close to that of light.
How old will Serap be, when Damla comes back the next day?
Unbelievable, isn’t it? But what’s happening here is the plain truth.
Time dilation and length contraction are basic properties of our physical world.
It’s impossible for us today, to make such kind of space experiments,
...but all the experiments made until now with high speed particles...
...are in 100% coordination with the theory of Einstein.
...are in 100% coordination with the theory of A. Einstein.