Can an Odometer Measure Relativistic Distance?

[ghwellsjr]

Is this what you mean?

The green lines are worldlines at 0.99c and each is a one-year interval.

But I'm going to need some help with the rest of this:

If I did the first step correctly, can you copy the diagram and draw in the next step or all the remaining steps would be even better. I just am not grasping what your are saying.

Your picture is not what I meant for the congruence. I meant to take your red world line, and displace it a tiny bit down and to the left for each new world line of the congruence (that we show in a diagram; one assumes there is a mathematical description of the continuous infinity of non-intersecting world lines).

I still have no idea what you are talking about. Remember, I'm trying to understand your statement from post #15 that the inertial Earth's measurement of the accelerated star as having traveled 0.8 light-years has some meaning. I'm wondering specifically if it means something like what I illustrated in my previous post where I can transform to a different rest frame and show a distance of 2.4 or 4 light-years but in this case it would be 0.8 light-years. Of course I understand that we can have a conveyer belt traveling at some speed that makes an odometer read 0.8 light-years during the time that the star is traveling to the Earth but that seems contrived to me.

 

[PAllen]

Thanks, great diagrams, as always. One further point obvious from the drawings, is that for the officers world line portion from Earth to the star (the part where both odometers are functioning), as of passing the star:

- the prisoner progress odometer reads: 2.02 light years
- the Earth - star odometer reads 2.4 light years

both applying to same officer spacetime path.

 

[PAllen]

I still have no idea what you are talking about. Remember, I'm trying to understand your statement from post #15 that the inertial Earth's measurement of the accelerated star as having traveled 0.8 light-years has some meaning. I'm wondering specifically if it means something like what I illustrated in my previous post where I can transform to a different rest frame and show a distance of 2.4 or 4 light-years but in this case it would be 0.8 light-years. Of course I understand that we can have a conveyer belt traveling at some speed that makes an odometer read 0.8 light-years during the time that the star is traveling to the Earth but that seems contrived to me.

Imagine a second star (B) whose world line goes vertically in your diagram upwards to the point (3,-3.75). Then this world line turns left going .8c, reaching Earth world line at (0,0). Between these two world lines, you can fit in many similar 'marker' world lines. That is an example of a non-inertial congruence.

Now simply imagine that the passage of B is is when Earth turns on their odometer to measure travel distance from B to the original star. They find .8 light years. Note that in the Earth frame, the coordinate distance between B and A (original star) after both are moving, is, indeed, .8.

You could also draw the frame in which B and the original star are at rest after their sudden acceleration. They would be 4/3 light years apart in this frame, would see Earth take 5/3 years to get from B to A, during which 1 Earth year would go by. And if you compute the Earth odometer in this frame, of course you still get .8 for measure of B to A travel by earth.

 

[ghwellsjr]

Thanks, great diagrams, as always. One further point obvious from the drawings, is that for the officers world line portion from Earth to the star (the part where both odometers are functioning), as of passing the star:

- the prisoner progress odometer reads: 2.02 light years
- the Earth - star odometer reads 2.4 light years

both applying to same officer spacetime path.

Thanks, that was actually my intended point but I neglected to make it.