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Basics of Relativity

Below are five tables that summarize basic formulas of Special Relativity. If you want to try them out, fill in the data fields and click the "calculate" button.

Those five formulas are intimately bound. If you would take only one of them into consideration then you could easily demonstrate that special relativity is a nonsense. But if you are using them altogether then everything becomes fluent and perfectly logical.

The speed of light is represented by the symbol c. That's 299,792,458 m/s (in vacuum).


Length contraction: 

Let's consider a rocket. When the rocket stands still compared to you, it has a length of l0. (It may be immobile next to you or parked far away...) When it will be traveling at speed v compared to you, you will see that it now has a shorter length of l. (The diameter of the rocket did not change.) In other words: when the rocket was at rest compared to you, it had a length of l0. Now that it shears at a high speed v compared to you, you will see or measure that it has a shorter length, of l.
Yet a person aboard the rocket will still claim it has a length of l0, because that person stands still compared to the rocket.

 Length of the rocket when motionless compared to you:   m
 Speed it now travels at:   m/s 
the length of the traveling rocket for you:   m
This fact has no feelable impact on everyday life. But let's imagine a 10 km long train. When it stands still, its rear is located at Chasmton and its front is located at Summerville. Now suppose that the train speeds at 100,000,000 m/s. At the moment the mayor of Chasmton sees the back of the train at Chasmton, the mayor of Summerville will see the front of the train about 500 meters before Summerville. The train is shorter.

Reciprocally, people on board the train state that when the back was at Chasmton, the front was already about 500 meters past Summerville. They see the distance between Chasmton and Summerville to be shorter. 
Time dilation: 

If an hourglass lasts t0 second when it is in your hand (or anywhere else, but immobile compared to you), then when it is aboard a rocket that travels at speed v compared to you, you will see that it lasts longer: t seconds.
Yet for any person aboard the rocket, it still lasts t0 second.

 Time an hourglass lasts when immobile compared to you:   s
 Speed of the rocket it now travels inside:   m/s 
the time you see that the hourglass now lasts:   s
One consequence of that phenomenon is that the clocks aboard a satellite run slower. You must take that into account when you make calculations based on the satellite's clock. For example if the satellite tells you "with my telescope I saw that your breakfast took 527.2358801 seconds", this means that if you had measured that time yourself, you would have come out with 527.2358800 seconds (low orbit satellites travel at 7.000 m/s).

Reciprocally, imagine you look with a telescope towards a lamp that lights and dims aboard the satellite. Say you measure the time that the light is on and you come out with a result of 100.000000027 seconds. You phone the satellite and tell him that. He will respond to you: "Ehr, well, no... I measured 100.000000000 seconds. My clock is very accurate you know." If you know about Special Relativity, then you know that this is normal. Neither you, nor the satellite, made an error.
Time shift: 

You put two clocks a distance d apart along a rocket's body. If the rocket now travels at a speed v compared to you and everybody onboard the rocket says that the clocks are synchronized, then you will see that the clocks are not synchronized. The forward clock is t seconds back.

 Distance between the two clocks along the rocket:   m
 Speed that the rocket now travels at:   m/s 
the shift that you see that the front clock now has:   s
That phenomenon is used inside high precision inertial platforms for airplanes. Several clocks are placed apart from each other and a central circuit compares their signals. If one clock seems to hold up then the circuit can calculate that the airplane accelerated in that direction.

This is very important to understand that Relativity is a logical and fluid system. Before he understood this clock phenomenon, Albert Einstein even thought of suicide. Take for example the speeding train between Chasmton and Summerville. How is it on Earth possible that at the very moment that the back of the train is at Summerville, people aboard the train tell that the front of the train is 500 meters past Chasmton and people of Chasmton tell that the train is 500 meters before Chasmton? Nonsense! Well it becomes logical if you take into account that the clock in front of the train runs back a little (as seen by people on the ground) (people aboard the train see the clocks to be perfectly synchronized). So, if a voyager at the back of the train says "at 4 pm exactly, the back was exactly at Summerville", the driver in front of the train will respond: "well at 4 pm exactly, we were 500 past Chasmton." That's quite normal, since the driver's clock runs back a little (that's what people of Chasmton and Summerville see, when they look through the train's windows and look at the clocks inside the train). But the people aboard the train see that their clocks are synchronized, whatever experiment they perform. For example they can light a flash bulb just in the middle of the train and the driver at the front and the voyager at the back press their clocks' button to synchronize just when they see the flash. Or they can both walk towards the middle of the train, meet and synchronize their clocks, then go back to their positions. Or they can use the train's phone system, send each other a top signal and take into account those signals take some time to travel back and forth. People aboard the train tell for sure that their clocks are synchronized. But... when they look at the people of Chasmtom and Summerville and their clocks, then they see that the clocks of Chasmton run back a little. So they can tell them: "Well, we understand that you believe that the train's front was 500 meters before Chasmtom. That's because the clocks of Chasmtom run back a little compared to those of Summerville. But when people of Chasmton and Summerville check that out, they will come out with the result that their clocks are perfectly synchronized. Nobody has the privilege of absolute synchronicity. The voyagers inside the speeding train perfectly synchronized their clocks. The people on the ground perfectly synchronized their clocks. But when one group looks at the clocks of the other group, it sees them as being desynchronized. That's the world of Special Relativity. If the train brakes and stops, then its clocks will no more appear to be synchronized. If the driver and the voyager walk towards the center of the train and compare their clocks, they will realize that the driver's clock runs back a little. Suppose that they synchronize their clocks, go back to their positions and start the train again up to a speed of 100,000,000 m/s, then the people of Chasmton and Summerville will see that the clocks aboard the train are synchronized. But if the driver and the voyager meet once again in the middle of the train, they will see that now the driver's clock runs forwards a little. The people of Chasmtom and Summerville know why. They saw it happen. Clocks traveling at high speed run slower. When the voyager walked towards the middle of the train, his walking speed was added to that of the train. So his clock ran even slower. While the driver's clock ran a little faster, since his walk speed was subtracted from the train's speed. When they meet in the middle, the driver will tune his clock back a little, to synchronize with the voyager's clock. Now they will see that their clocks are synchronized, even if they go back to their positions and check it by any means. But people of Chasmton and Summerville see that the driver's clock runs back a little compare to the clock of the voyager. And so on...
Speed contraction: 

If a rocket travels at a speed v compared to you and a snooker ball is made to roll on the rocket's flour towards the front and the passengers inside the rocket say that the ball rolls with a speed vm compared to them, then vs is the speed of the ball compared to you.

 Speed of the ball measured aboard the rocket:   m/s
 Speed that the rocket travels at:   m/s 
the speed of the ball for you:   m/s
This formula is one way to check the fact that it is not possible to attain the speed of light. Suppose a canon is build to fire shells at 200,000,000 m/s. A rocket travels at a speed of 200,000,000 m/s compared to you and carries the canon. When it fires a shell forwards, you will see that the shell travels at 276,805,111 m/s, which is less than c.
Mass increase: 

If an object has a mass m0 when it is motionless compared to you, then it will have an increased mass m when it moves at a speed v compared to you.

 Weight of the rocket when motionless compared to you:   kg
 Speed that the rocket now travels at:   m/s 
the weight that the rocket now has for you:   kg
That phenomenon explains why you are several tens of kilograms heavy. Indeed, would the quarks that you are made of be motionless, then you would weight less than a kilogram. But they turn around each other at a speed very close to the speed of light. That makes them become very heavy. Most of your weight is due to that relativistic phenomenon.


Galilean Relativity states the following:

Special Relativity adds the following:

In order to keep this rule true, the Universe squeezes every objects. And changes the speed of every clock.

General Relativity adds the following:

This implies once more that objects are squeezed and that the speed of clocks changes. For example, clocks on Earth run a little slower than clocks on the Moon. Even space itself is bend and curved...

General Relativity is shipped with an impressive calculation system. It can compute out just any situation: accelerating or turning laboratories, electromagnetic or gravitational fields, encounters with big masses... Yet it can be quite difficult to handle.

That statement of objects, clocks and space being squeezed by the Universe is just a way to express the reality. There are other ways round to describe the reality; other points of view, that come out with the same physical results. Be open to them.

Maybe you conclude that the Universe simply cannot behave like a Tex Avery cartoon and that the content of texts like this one must be erroneous. First of all, in a Tex Avery cartoon the characters squeeze and expand at random. While in the Universe, it happens a very accurate, mathematic and reproducible way. Second, this all has been tested out thoroughly. From the faintest atomic phenomenon up to stars, till now everything obeys the formulas of Relativity. (Yet there seems to be a problem with the universe as a whole. It seems to expand in a way that may be not compatible with General Relativity.)

Eric Brasseur  -  March 12 1998  till  June 1 2011