Our story begins a little over one hundred years ago in Bern, Switzerland, where a young man employed as a patent clerk went off to work. He took the electric trolley in each day, and each day he would pass an unassuming clock tower. But today was different, it was special. For today he would pose to himself a question – a question whose answer would set forth a fascinating dilemma.
The hands of the clock appeared to move the same no matter if his trolley was stopped or was speeding away from the clock tower. He knew that the electromagnetic radiation which enabled him to see the clock traveled at a finite speed. He also knew that the speed of the light was incredibly great compared to the speed of his trolley. So great that there would not be any noticeable difference in how he saw the hands of the clock move, despite him being at rest or in motion. But what if his trolley was moving at the speed of the reflected light coming from the clock? How would the hands of the clock appear to move? Indeed, they could not. Or if they did, it would not appear so to him. It would appear as if all movement of the clock’s hands had stopped – frozen in an instant of time. But yet if he looked at the hands of the watch in his pocket, they would appear to move normally. How does one explain the difference between the time of the clock tower versus the time of his watch? And which one was correct?
There was no way for him to know that it would take three years to answer this question. No way for him to know that the answer would eventually lead to the discovery of matter and energy being one
in the same. No way to know that he, this underemployed patent clerk making a simple observation, would soon unearth the answer to one of the greatest mysteries that had stumped every mind that came before his – the very nature of time itself.
Now it might have taken Einstein a few years to develop the answer we now know as the Special Theory of Relativity, but it most certainly took him no longer than a few days to realize that
must be wrong.
Einstein was known for his ‘thought experiments’ – with the light clock being one of them. Let the animation above1 represent his pocket watch. The imaginary clock consists of two mirrors, with a pulse of light that bounces between them. The repeating pattern represents a basic clock, whose time can be calculated by:
T = 2h/c
Where h = the distance between the mirrors, c = the speed of the light pulse and T = time.
It is important to note that this is the way the light clock would appear to him, no matter if his trolley was moving or at rest. Both he and the clock are considered to be in the same frame of reference.
Let the animated image above1 represent the clock tower as his trolley is speeding away. Once the trolley starts moving, he and the clock tower are now considered to be in different frames of reference. To calculate time on this clock will take some basic algebra and geometry2.
If we set time equal to distance divided by velocity, then we can use the Pythagorean Theorem, to get:
c²t² = v²t² + w²
t²(c² – v²) = w²
t²(1 – v²/c²) = w²/c²
Then square root both sides and double the second part of the pulse (opposite side of right triangle, or (w)) to get:
According to this equation, the time between ticks of the moving clock will increase as the velocity (v) increases. Or put more simply – time runs slower for moving objects. Notice what happens when (v) is equal to zero. The equation becomes identical to our original of T = 2h/c. Now notice what happens when (v) is equal to or greater than (c). The equation becomes undefined. Thus the trolley cannot travel faster than or equal to (c), the speed of light. This includes all things with mass, and even gravity itself.
If you keep following this rabbit down the hole, you will find that the speed of light, and all electromagnetic radiation, will always be the same for all frames of reference. This is counter-intuitive, but proven to be true. Keep going further and you’ll end up with one of the most profound revelations in all of human history – E = mc2.
Remember these neat little facts next time you’re hacking away with lasers and wireless modules, and the spooky nature of the laws that govern them.
[Source 1: Time
Dialation Travel Resource Center]
[Source 2: Special Relativity: What Time is it?Michael Fowler, Physics Department, UVa.]
Filed under: Hackaday Columns