The arrow of time can stretch like rubber

Relativistic time dilation makes the universe appear rational

The 1968 movie starring Charlton Heston, Planet of the Apes, used relativistic time dilation to a explain that time had passed on the spaceship far slower than it had for earth. 

The description of time dilation and the equation for calculating the slowing of time as the speed of travel reaches an appreciable fraction of the speed of light can be found in many reference sources. 

The following example on the website of University of Mississippi Department of Physics and Astronomy is headed: 

"Time Dilation"

It turns out that as an object moves with relativistic speeds a "strange" thing seems to happen to its time as observed by "us" the stationary observer (observer in an inertial reference frame). What we see happen is that the "clock" in motion slows down according to our clock, therefore we read two different times. Which time is correct??? 

Well they both are because time is not absolute but is relative, it depends on the reference frame. Let's look at the following classic example. There is a set of twins, one an astronaut, the other works for mission control of NASA. The astronaut leaves on a deep space trip traveling at 95% the speed of light. Upon returning the astronaut's clock has measured ten years, so the astronaut has aged 10 years. However, when the astronaut reunites with his earth bound twin, the astronaut sees that the twin has aged 32 years! This is explained due to the fact that the astronaut is traveling at relativistic speeds and therefore his "clock" is slowed down.

Let's see how we can calculate the time "difference". The equation for calculating time dilation is as follows:

t = t₀ / √ (1-v²/c²) 

where: t = time observed in the other reference frame
t₀ = time in observers own frame of reference (rest time)
v = the speed of the moving object
c = the speed of light in a vacuum

It was puzzling how the equation for time dilation was designed. It remained an "imponderable" question until recently.

Some time ago I noticed the equation was just a representation of Pythagoras' theorem for right-angled triangles. This just added to the puzzle, but didn't help explain how time dilation was related.

What is Pythagorus' Theorem 

• The Pythagorean theorem formula is a² + b² = c².
• It only works for right triangles.
• To solve the Pythagorean theorem, we need to know the lengths of at least two sides of a right triangle.
• The Pythagorean theorem formula can be used to find the length of the shorter sides of a right triangle or the longest side, called the hypotenuse.

 

Accidental discovery on the derivation of the Time Dilation equation

In the triangle below, the hypotenuse has a length c (the speed of light) and one side has a length of v (the velocity) of a body. 

Using Pythoras' theorem for a right angle triangle, the length of the remaining side of the triangle is the square root of the (square of the length of hypotenuse minus the square of the length of the other side). 

That is √ (c²-v²) .

The sin of the angle θ is √ (c²-v²) / c. 

This can be rearranged as √ ((c²-v²) / c²).  

And this can be rearranged to be √ (1-v²/c²)  ... which is the denominator of the time dilation equation.  

 

The Time Dilation equation has a relationship to a simple geometric shape

The following Youtube video gives an explanation for time dilation. The derivation of the relationship is done with analysis of simple geometric shapes, specifically triangles. 

 Somewhat curiously, the author says this proves why we can't travel faster than the speed of light. 

It could also be said that no matter how fast we travel, the measured speed of light will always be the same. The time arrow will shrink or stretch to ensure this result. If it did not, the world around us would look very confusing.