Quick length contraction question please

1. Jul 30, 2010

rede96

Assuming 300,000 km/sec for c;

I am in a space ship which is 300 meters long travelling at 0.5c. A beam of light passes me and I want to measure how long it took to travel the length of my ship.

As c is the same in all frames of reference, I would work this out as distance (300 m) divided by speed (300,000 km/sec) which = 1 microsecond.

Is this correct?

If so, what about length contraction?

2. Jul 30, 2010

bcrowell

Staff Emeritus
Yes, this is correct. The ship is not length contracted in your frame of reference, because it's not in motion relative to you.

3. Jul 30, 2010

Staff: Mentor

Yes.

As far as you are concerned, your ship is not contracted. (But the frame in which you are moving at 0.5c will see your ship as contracted. They will compute a different time for the light to traverse your ship.)

4. Jul 30, 2010

rede96

Ok thanks.

I would have said (in my ignorance!) that in the 1 microsecond it took the light to pass me, I would have travelled 150 meters forward, but my ship would have contracted by 1/2 and hence the toal distance that the light has travelled is still 300m, so therefore the light has travelled 300m in 1 microsecond and I would have measure the right speed for c.

I take it this wrong then?

5. Jul 30, 2010

Staff: Mentor

Yes, it's wrong. With respect to yourself, you're not moving. (You are moving with respect to the other frame, but who cares?) So no forward motion to worry about and no length contraction.

But viewed from the frame in which you are moving at 0.5c, you do travel forward and you are length contracted. (The length contraction is about 1/1.15, not 1/2.)

6. Jul 30, 2010

rede96

That'll take a bit of thinking about

Thanks.

7. Jul 30, 2010

JesseM

You might want to take a look at the example I posted here showing how when you take into account length contraction, time dilation and the relativity of simultaneity, two pairs of observers can both agree that the same light beam has a speed of c, even though they are each using rulers and clocks at rest relative to themselves to define "speed" in terms of distance/time.

8. Jul 31, 2010

rede96

Thanks JesseM. I've read your post and it made me think of something else I was trying to get my head around.

Would you mind explaining it using a different hypothetical, which I have put below please?

Assuming 300,000 km/sec for c...

I am on a spaceship travelling at 0.6 c relative to you. My spaceship is 300 meters long.

Attached horizontally to the side of my spaceship in the direction of my motion relative to you is a simple light clock also 300m long, where a beam of light bounces back and forth off the two mirrors at either end. This is in a clear glass tube so you can see the light beam in your frame of reference.

I measure the time it takes the light beam to travel one length or one ‘pulse’ as 1 microsecond, therefore I know c is 300,000 km/sec and that a million pulses are one second.

From your frame of reference, you would measure the length of my ship as 240m. (Using gamma-factor of 0.8) and hence measure my light clock as only 240m.

I am also assuming that my clock would appear to run slower from your frame of reference.

So how then can we both measure the same speed c for the light in my light clock?

EDIT: Doh! I think it just came to me. I think you would measure the light as taking 800 nano seconds (1 microsecond x 0.8) and hence still get 300,000 km/sec for the light in my clock. Is that right?

Last edited: Jul 31, 2010
9. Jul 31, 2010

Staff: Mentor

You measure each one-way traversal as taking 1 μs; the round trip takes 2 μs.

Right, in that frame the length of your ship is 240 m. (gamma = 1.25)

The round trip for your light clock would take 2.5 μs as measured in the other frame.

Not exactly. The other frame would measure that it takes 2 μs for the light to traverse your light clock when the light is moving in the same direction as you travel (realize that you are moving, so the light has a greater distance to travel, even though your ship is shorter), and only 0.5 μs for the reverse trip.

10. Jul 31, 2010

rede96

I wanted the other frame to be able to measure c by using the light in my clock on any one-way traversal.

So I guess, if measuring in the direction of travel, the other frame measures 2 μs. In order to calculate c at 300,000 km/sec the distance measured would have to be 600 m. Which I guess would be the length of the ship measured 240 m + 360 m travelled

In the other direction, the other frame measured 0.5 μs so d = 150 m. So agian I guess this would be the 90 m I would travel in the 0.5 μs taken off the overall length of 240 m.

Is that right?

11. Jul 31, 2010

Staff: Mentor

Exactly right.

12. Jul 31, 2010

Thank you!