Measuring Tape, Speed of Light & Time Dilation

[Xavier Cross]

Had a thought the other day. If a spool of a measuring tape was on a vehicle traveling close to the speed of light away from the Earth, with the spool anchored to the Earth, due to time dilation, would an observer on the vehicle see the spool unraveling faster than the speed of light?

 

[Drakkith]

No, the spool would either rotate at less than the speed of light, or it would break. Note that time dilation affects the passage of time on the moving spool, not its velocity.

 

[Xavier Cross]

So the vehicle could only move at less than half the speed of light relative to the Earth, due to the speed of the fastest forward moving edge of the spool, and an observer on the vehicle would see it as unspooling at close to the speed of light. Ironic that if a second vehicle with the same initial mass left the Earth at the same time, it would outpace the one with the unspooling tape, even though the slower one has less weight.

 

[A.T.]

Note that time dilation affects the passage of time on the moving spool, not its velocity.

A cyclic process, like the rotation of the spool, will surely be subject to time dilation.

 

[A.T.]

would an observer on the vehicle see the spool unraveling faster than the speed of light?

No, kinetic time dilation is symmetrical. So in the frame of the vehicle a full "revolution" of a spool on Earth takes longer than in the frame of the earth.

I put revolution in scare quotes, because what actually happens in the vehicle frame is not a normal rotation, but what is described here:
http://www.spacetimetravel.org/tompkins/node7.html

And when you add the optical effects, due to signal travel time it gets ever weirder, depending on the viewing angle:
http://www.spacetimetravel.org/tompkins/node8.html

 

[Drakkith]

A cyclic process, like the rotation of the spool, will surely be subject to time dilation.

Will it? Interesting.

 

[A.T.]

Will it? Interesting.

Consider a clock right next the rotating object. If they are synched in their rest frame, they must by synched in any frame.

 

[Dale]

So the vehicle could only move at less than half the speed of light relative to the Earth, due to the speed of the fastest forward moving edge of the spool, and an observer on the vehicle would see it as unspooling at close to the speed of light. Ironic that if a second vehicle with the same initial mass left the Earth at the same time, it would outpace the one with the unspooling tape, even though the slower one has less weight.

This is not correct. The vehicle could still move arbitrarily close to c. Given a finite material strength, there is a limit to how fast you could go before the spool and rope would break, but in principle that is not .5 c.

 

[Drakkith]

Consider a clock right next the rotating object. If they are synched in their rest frame, they must by synched in any frame.

What?

 

[A.T.]

What?

What is unclear? Clock and rotating object are close and at relative rest. The clock ticks every time the rotating object completes a cycle. This is a frame invariant fact. So if in some other frame the clock is dilated, then so is the duration of one cycle.

 

[Drakkith]

What is unclear? Clock and rotating object are close and at relative rest. The clock ticks every time the rotating object completes a cycle. This is a frame invariant fact. So if in some other frame the clock is dilated, so the duration of one cycle.

Oh, okay. I had no idea what you meant at first.

Here's a question for you. If you put a clock on the rotating object prior to the beginning of rotation, will it be synced with the clock at rest after it spins up?

 

[A.T.]

If you put a clock on the rotating object prior to the beginning of rotation, will it be synced with the clock at rest after it spins up?

Depends where on the rotating object you attach the clock.

But I didn't mean a clock that is attached the rotating object. Just one that is at rest to the center of rotation. The scenario is that you have a spool attached to the Earth, and a clock standing on the Earth. If the clock is dilated then so one cycle of the spool.

 

[Drakkith]

Got it.

 

[Ibix]

This is not correct. The vehicle could still move arbitrarily close to c. Given a finite material strength, there is a limit to how fast you could go before the spool and rope would break, but in principle that is not .5 c.

I see how you came up with 0.5c, OP. The "bottom" of the spool must be at rest with respect to the "spooled out" rope; the axle is moving at the same speed as the vehicle; the "top" is moving at twice the speed of the vehicle. Speed cannot exceed c, so the vehicle's velocity cannot exceed 0.5c - right?

Unfortunately, velocities don't add that way in relativity. If the vehicle is moving at a speed u with respect to the spooled-out rope then, in the vehicle's rest frame, the top of the spool must be doing u and the bottom -u. The relativistic velocity addition formula in this case (u=±u, v=u) tells us that, in the rest frame of the spooled out rope, the bottom of the spool is at rest (obviously) but the top is doing 2u/(1+u²/c²). The velocity of the rim actually varies at different positions round the spool.

 

[A.T.]

The velocity of the rim actually varies at different positions round the spool.

That is true even in classical mechanics. The key here is how it varies:

http://www.spacetimetravel.org/tompkins/node7.html