# Length contracting between objects

1. Nov 21, 2014

### Scheuerf

If two objects start moving near the speed of light at the same speed, is the length between the two objects contracted?

2. Nov 21, 2014

### Staff: Mentor

The length is always contracted. I think that you are probably more interested in whether or not it is constant. That depends on the timing of the acceleration.

3. Nov 21, 2014

### Scheuerf

That confuses me though. How is the space between moving objects contracted? Wouldn't this mean that an objects position after it is contracted is dependent on the movement of other objects?

4. Nov 21, 2014

### A.T.

Length contraction is between different frames, not between before and after they start to move. Whether the distance changes over time in some frame depends on how you accelerate them as observed in that frame.

5. Nov 21, 2014

### Staff: Mentor

The distance between two objects certainly depends on the motion of both objects. This is the same in relativity as in Newtonian mechanics.

6. Nov 21, 2014

### Scheuerf

http://imgur.com/c2Zra2o,rWLcnTx#0

I uploaded a couple of images here. Rocket A is in the same location in both pictures. If both rockets are moving near the speed of light in both pictures, does the length contraction between the two rockets cause rocket A to reach the second planet first in the second picture relative to an observer at rest relative to the planets?

7. Nov 21, 2014

### Staff: Mentor

The event of rocket A arriving at planet 2 and the event of rocket B arriving at planet 2 are necessarily timelike separated. This means that they occur in the same order in all frames.

8. Nov 21, 2014

### Jonathan Scott

If two separate bodies accelerate from rest in exactly the same way starting simultaneously in a given frame, their separation in space as seen in that frame remains constant. If they were joined by a string, the Lorentz contraction would break it. Attention was called to this potentially surprising fact by John Bell (of the Bell Inequalities) and it is known as "Bell's Spaceship Paradox".

9. Nov 22, 2014

### ghwellsjr

The way to answer questions like this is to use the Lorentz Transformation and draw the results on a diagram. Otherwise, it is very difficult to imagine what Length Contraction is.

Here is a spacetime diagram depicting two objects separated by 5 feet that are stationary in a frame:

The dots represent 1-nanosecond ticks of time for the two objects. We call these events. Now we can transform the coordinates of all these events using the Lorentz Transformation to a frame in which the two objects are traveling at 60% of the speed of light. The speed of light is 1 foot per nsec:

You can see that the distance between the two objects is 4 feet. If you look at the last event for the blue object at the Coordinate Time of 5 nsecs, you will see that at that same time, the Coordinate Distance to the red object is 4 feet (from 3 feet to 7 feet). That is what Length Contraction is--their separation or distance according to one frame where the objects are moving compared to their separation or distance according to the frame in which they are at rest.

But you asked what happens between two that start moving. Here is a spacetime diagram to show one such example of that where they both start moving at 60%c according to their original rest frame:

As you can see, their separation remains the same, 5 feet. But if we transform to the frame in which they end up at rest, we see that their separation expands:

But if you remember our definition of Length Contraction, it is their separation in the frame in which they are in motion, the second diagram up, compared to the frame in which they are at rest, the one immediately above and we see that it is 5 compared to 6.25 which has the ratio of 0.8, the same ratio before when we compared 4 feet to 5 feet.