Simulation - Relativity

Speed of Light

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Speed of Light Measurement

This simulation illustrates the theoretical ideas behind the speed of light experiment.

Instructions:
Drag the detector to where the photos are exiting in the end to measure the speed of light (see calculations below).
Set the angular velocity of the rotating mirror to zero to see how the photons travel. You could drag on the rotating mirror to change its angle directly.
As long as the rotating mirror is at rest, the photons always exit at zero angle.
Now turn on the rotating mirror and adjust the "speed of light" (something you cannot do in real life!) and see how it affects the angle of they exit the instrument.
The higher the speed of light, the smaller is the angle of exit.

Explanation:
When the angular velocity of the rotating mirror is zero, a photons travels the same way back, returning to the detector at zero angle of exit.
When the mirror is rotating, a photon who travels to the circular mirror and comes back to the rotating mirror will find the mirror at a different orientation from before due to the finite amount of time it takes to run to the edge and back to the middle. As a result, the photo no longer can return the same way and will exit at a non-zero angle. The lower the speed of light, the more time it allows the mirror to rotate, and the larger is the final angle. Therefore by measuring the angle one can deduce the speed of light.

Remarks:
In the actual experiment, the circular mirror will need to have an opening (not shown in the simulation for simplicity) on the right for the photon to escape.
The "speed of light" in the simulation is very low compare to the actual speed of light, so in practice the rotating mirror will need to rotate at an extremely high rate to achieve just a tiny deflection. Usually the angle of exit is so close to zero that one needs to use a microscope to observe it!

The Relativity of Simultaneity

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The Relativity of Simultaneity

Two events occurred simultaneously from the perspective of the top observer.
See if they are also simultaneous for the bottom observer based on the light signals from the two events.
Note that the simulation is in the the perspective of the top observer. Therefore due to the length contraction the bottom ship would actually be longer than the ship at the top if they are both at rest.

Time Dilation

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Relativity: Time Dilation

Adjust the velocity and see how time flows at different rate in the two light clocks.
Note that the photons always travel at the speed of light $c$.

Length Contraction

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Length Contraction

Change the velocity to observe length contraction.
You can also drag the two objects together to compare their sizes.

The Necessity of Length Contraction

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The Necessity of Length Contraction

The view at the top shows two photons bouncing inside an L-shaped ship. The two photons always gets back to the corner at the same time.
The view at the bottom is the same ship viewed from a moving frame. If length contraction is disabled (so the horizontal arm does not contract), the two photons will be coming back out of sync, leading to contradiction.
Therefore length contraction is necessary to perserve the local observation (made at the corner) that the two photons always arrive back at the same time.
Although it may appear to you the photons are moving at different speed, in fact all photons are always traveling at the same speed. The illusion arises because the photons are moving on top of a moving object, thus making some appear to travel faster than others.

The Ladder-Garage Paradox

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Relativity: Ladder and Garage Paradox

Instructions:
A ladder is too long to fit inside the garage. Can you speed up the ladder to shorten it (length contraction) so it can fit inside the garage at least momentarily?
It appears the plan works... but now click on "Switch Observer" to see how the situation appears from the ladder's perspective. It looks like the plan not only failed, but in fact made matter worse by shortening the garage even more!
So does the ladder fit inside the garage or not when it is moving at high speed?

Explanation:
The key is the relativity of simultaneity, i.e. the idea that "at the same time" is a relative concept.
From the garage's perspective, at some point the doors were both closed "at the same time", trapping the ladder inside. However, what happened at the same time according to the garage did not happen at the same time according to the ladder. In other words, from the ladder's viewpoint the doors were never both closed at the same time.

Even more detailed explanation (if you are interested!):
In relativity the temporal ordering (what happened first) of events is relative. Define two events, LC for the closing of the left door, RO for the opening of the right door. From the garage's perspective, LC happened before RO, so the ladder was momentarily trapped. From the ladder's perspective, RO happened before LC, so by the time the left door closes, the right door is already open, therefore "untrapping" the ladder. See if you can observe the ordering of these events in the simulation.