This posting is due to a conversation I had recently about the problems to understand modern physics. Especially for quantum mechanics and the behavior of the small world, they say that nobody can understand what is happening. One professor famously said at that start of the first lesson in quantum mechanics: „I am now the only person here who does not understand quantum mechanics, and at the end of the course you all will not understand it.“ I am not a physicist. Consequently, I have no problems with spooky actions as long as they happen to reflect reality.
What we do here is much more elementary. We do a thought experiment, trying to understand the first milestone in modern physics, the special theory of relativity by Einstein. It is based on the fact that the speed of light is measured the same in all directions. We would expect it to be faster if we move towards the light source and slower if we move away from it. But it is not. The first evidence of this fact has been shown in the Michelson-Morley experiment.
I should mention that the following approach is very elementary. I did not check for sources. But the „Gedankenexperiment“ was a sharp tool used by Einstein himself. So, ours is sure to have been made before.
We are going to use basic stuff. So, we assume our system is moving relative to another system, which we fix as „absolute“ for the moment. Light moves with constant speed in that absolute system. Now, we measure time and distance in our moving system using light. We imagine a square of distance d with one side parallel to our movement, where the distance is the actual distance as measured in our absolute system. The question is, what we get in our system.
Assume we send rays of light perpendicular to our motion and reflect it back. From the moving view, it just looks as if the light is sent out perpendicular and comes back. We would think on the moving system that the time it takes is
t = \frac{2d}{c}where c is the speed of light and d is the distance of the mirror. But from the absolute viewpoint, we see the light travelling along a triangle. The sides of this triangle are formed by the distance we travel with speed v until the light comes back, and twice the path of the light. With Pythagoras, taking of half our travel with speed v, the distance d and half the path of the light, we get
\left(\frac{t_ov}{2}\right)^2 + d^2 = \left(\frac{t_oc}{2}\right)^2This leads to
t_o = \frac{2d}{\sqrt{c^2-v^2}}The time it takes for the light be reflected as measured on the absolute system is
\delta_v = \frac{t_o}{t} = \frac{1}{\sqrt{1-(v/c)^2}}longer. At v=c/2, time runs about 15% slower.
This looks like a computation which applies to light bouncing only. But it turns out that everything we do is indeed related to light. An established fact is that all clocks in moving system do indeed run slower as seen from the outside. An astronaut travelling fast and returning would have aged less. A nuclear active particle has a longer half time when it moves with high speed.
If we send the ray forward in the direction of our motion and reflect it in distance d, we move away from the light source as seen from the absolute system and towards it after reflection. The first time thus satisfies
t_1c = d + t_1v \quad\Longleftrightarrow\quad t_1 = \frac{d}{c-v}Similar on the way back, so that the total time is
t_o = \frac{2dc}{c^2-v^2}This is measured in the absolute system. In the moving system, the time runs slower. We get
t = \frac{t_0}{\delta} = \frac{2d}{\sqrt{c^2-v^2}} Assume that we measure the distance d on the inside using the forward beam of light, assuming it travels to the mirror and back with the speed of light c. The result is
\tilde d = \frac{tc}{2} = d\delta_vThe distance to the mirror appears longer.
A consequence of our thinking in terms of an absolute reference which you might have observed already is that the light coming from the side will appear to be slightly tilted. This is called aberration of light. It has been observed in lights from stars which appear different when the earth moves to either rectangular side to the observation. In fact, it was used by Bradley to compute the speed of light with an accuracy of a few percent.
A similar Gedankenexperiment can be used to explain, why light bends in an accelerated system. It was the idea of Einstein that an accelerated system cannot be told from a system under the influence of gravity by any internal experiment. In fact, the light just bends the same way.