Earth Is One Meter Away From The Sun (Length Contraction)

Length Contraction

Move at 299 792.457 999 999 99 km/s and the Sun will appear to be 1.222 meters away from Earth.

In this post we will discover why this happens, see how length and time are related, understand what length REALLY is and look at a paradox that seems to arise.

So far special relativity has changed the way that we understand time. We have seen that time can slow and that the idea of simultaneity is relative to observers. Special relativity also has implications for space.

Speed = Distance/Time

Since the speed of light is invariant and we know that time can change; distance must also change to keep the ratio of distance/time the same.
This is known as Lorenz Contraction or Length Contraction.

We will see that distance is also relative and that there is no such thing as an absolute distance between two points in space.
We will also discover that the idea of length is reliant on time and that the two are closely related.
That means a fast moving observer could claim that the sun is only 1 meter from Earth and it is perfectly reasonable to accept this claim!

Below is an illustration of this. As speeds increase, the length of the ball becomes shorter and shorter.

We have team train and team platform.
Team platform are stationary on a platform watching team train go by on a train.
Team train can measure the length of the train easily using a tape measure. Team train walk to the back, put one end down, walk to the front and then read off the length.

Team platform, however, can't use a tape measure because the train is moving fast. Team platform instead have a stopwatch, and press start when the front of the train passes and stop when the back passes.
Team platform can then calculate the length by doing the speed of the train x the time taken for the train to pass (assuming they know the speed of the train).

Team platform measure a shorter length than team train.

So why do they measure different lengths?

Team train have read these blog posts and understand time dilation. They know that time is going slowly for team platform since team platform move relative to them.
This means that the time that team platform record on the stopwatch is low.
Length = speed of train x dilated time
and so you can see the length will be shorter according to team platform.

So now we know both lengths are different we want to know who is correct?
Both are correct. Both observers are in inertial reference frames and their methods of measurements are perfectly legitimate. The length of the object just depends on the speed of it.

What is length?

Length is a concept that we have to rethink. To measure the length of an object you have to measure the position of the front and back at the same time. You can't put one end of the tape measure at the back of a moving train, wait 10 seconds then put the other end at the front. This would give a much longer length. You must measure both at the same time. But we now know that simultaneity is relative. This means that two observers moving relative to each other measure different lengths.

A Second Approach

We can also think about another approach by team platform which gives the same result.
Team platform lines up next to the track all holding clocks and makes sure all the clocks are in sync.
The front man waits till the train reaches him and then shouts "Now!"
At this moment someone in the line will be in line with the back of the train and this person shouts "Now!".
Team platform can then measure the distance between the two people who shouted "Now!" and will have the length of the train.
We will call the moment both people shout 12:00.

Again the length measured by team platform will be shorter .This time the relativity of simultaneity illustrates why.

We know that clocks at the back of the direction of motion are ahead of clocks at the front (think back to the light hitting the back of the train before the front).
Team train sees themselves as stationary on a train while team platform rush by in the opposite direction to the movement of the train. Team platform's clocks must all be out of sync. Team train see the clocks of people at the back of the motion (towards the front of the train) as ahead and the clocks at the front of the motion (towards the back of the train) as behind. This means the shouts of "Now!" come at different times. The front man shouts first but since the second person is towards the front of the motion (back of train) he has not reached the 12:00 yet as he is behind in time. By the time he is at 12:00, the back of the train has moved more towards Person A.

This means team train says
"Of course team platform measure a shorter length. The second shout of "Now!" came too late and by then the train had moved making their measurements short!"

Team platform can also explain why team train measure a longer length.
Team train will measure the length using spray paint. Two members will stand at the ends of the train. At 12:00 they will spray their cans onto the platform. Team platform will then be able to see the distance between the two paint marks and determine what length team train see the train as. When team platform see this happen it is clear to them why team train measure a long length. The back of the train is ahead in time compared to the front of the train. This means that when the member at the back makes his mark at 12:00 it isn't 12:00 at the front of the train yet. The member at the front makes his mark some moments later when it is 12:00 at the front.

Team platform say "Of course team train measure a longer length. The member at the back of the train pressed his mark, the train then travelled a distance and then the member at the front pressed his mark!"

What We See

The lengths we see in everyday life are the rest lengths or proper lengths of objects. This means that this is the length of the object when you are at rest relative to the object. These lengths are just as legitimate as any other length that you might measure the same object to be at.
This is because length is determined by the speed of an object.
This idea is very counter intuitive as the speeds we see in our lives are no where near the speeds at which these effects are noticeable.

Derivation

Gamma is always greater than or equal to one and so L' is always less than or equal L.

Lengths measured by observers moving relative to an object is always less than or equal to its rest length.

A Paradox?

Objects shorten along the length of motion. This is obvious when we consider what happens if the train goes into a small tunnel. In one frame will the train shorten in height and be able to fit in the tunnel while in the other it stays at its rest height and smashes into the tunnel? No. Lorenz contraction only occurs parallel to the direction of movement. The train shortens in length but its height will remain unchanged.



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Check out my other posts on special relativity! (Link to all the posts)

The Relativity Of Simultaneity

How To Slow Down Time (Time Dilation)

How To Travel To The Future

How Galilean Relativity And The Lorenz Maxwell Equations (The 2 Postulates) Contradict

The Postulates Of Special Relativity

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