Quick length contraction question please

[rede96]

Assuming 300,000 km/sec for c;

I am in a spaceship which is 300 meters long traveling 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?

 

[bcrowell]

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

 

[Doc Al]

Assuming 300,000 km/sec for c;

I am in a spaceship which is 300 meters long traveling 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?

Yes.

If so, what about length contraction?

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.)

 

[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 traveled 150 meters forward, but my ship would have contracted by 1/2 and hence the toal distance that the light has traveled is still 300m, so therefore the light has traveled 300m in 1 microsecond and I would have measure the right speed for c.

I take it this wrong then?

 

[Doc Al]

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

I take it this wrong then?

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.)

 

[rede96]

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.)

That'll take a bit of thinking about

Thanks.

 

[JesseM]

Ok thanks.

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

I take it this wrong then?

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.

 

[rede96]

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.

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 traveling 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?

 

[Doc Al]

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.

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

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.

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

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

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

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?

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.

 

[rede96]

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.

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?

 

[Doc Al]

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?

Exactly right.

 

[rede96]

Exactly right.

Thank you!