Calculating Distances in Special Relativity - Bob's Question

[bobbles22]

Hi there,

I think I understand how, using Lorentz transformations, I can find relative speeds of objects within different frames of reference. However, I'm struggling to transfer that understanding to distance. I'm sure that the distance as measured by observers within two different references must be different, but if I know the relative speeds, and I have a distance as measured by one observer, how do I translate that into distance as measured by the other observer?

I'm not sure if I should be looking for an understanding by complicating this or simplifying it. Maybe I'm missing something obvious here.

Many thanks for your help.

Bob

 

[Doc Al]

Hi there,

I think I understand how, using Lorentz transformations, I can find relative speeds of objects within different frames of reference. However, I'm struggling to transfer that understanding to distance. I'm sure that the distance as measured by observers within two different references must be different, but if I know the relative speeds, and I have a distance as measured by one observer, how do I translate that into distance as measured by the other observer?

I'm not sure if I should be looking for an understanding by complicating this or simplifying it. Maybe I'm missing something obvious here.

Many thanks for your help.

Bob

You can use the Lorentz transformations to convert time and distance measurements between events in one frame to that of another. For example:
$$\Delta x = \gamma(\Delta x' + v\Delta t')$$
If the primed frame measures a distance, that usually means that the "events" happen at the same time in that frame, so $\Delta t' = 0$ and the LT reduces to the simple 'length contraction' formula.

 

[ghwellsjr]

Hi there,

I think I understand how, using Lorentz transformations, I can find relative speeds of objects within different frames of reference. However, I'm struggling to transfer that understanding to distance. I'm sure that the distance as measured by observers within two different references must be different, but if I know the relative speeds, and I have a distance as measured by one observer, how do I translate that into distance as measured by the other observer?

I'm not sure if I should be looking for an understanding by complicating this or simplifying it. Maybe I'm missing something obvious here.

Many thanks for your help.

Bob

I guess I don't understand what the problem is. If you know the speed of an object according to a frame of reference and you know its position in that frame at some particular time, then you know its position at all other times and you can transform those positions at those times (which is what events are) according to the LT and determine its position at all times according to that new frame. Do this for all objects and you can determine the distances between the objects at any particular time. That seems pretty simple to me. But maybe I misunderstood your problem.

 

[Nugatory]

b

I think I understand how, using Lorentz transformations, I can find relative speeds of objects within different frames of reference. However, I'm struggling to transfer that understanding to distance. I'm sure that the distance as measured by observers within two different references must be different, but if I know the relative speeds, and I have a distance as measured by one observer, how do I translate that into distance as measured by the other observer?

You're ahead of the game by starting with the Lorentz transformations instead of (as so many do) the time dilation and length contraction formulas that are derived from them. But now what you're looking for is the relationship between distance between two points in one frame and distance between the same two points in another frame - and that's exactly what the length contraction formula is for. You'll find it at http://en.wikipedia.org/wiki/Length_contraction. Just imagine that the "object' they're talking about is a stick set out between two points, so the "distance" between the points is the the length of the stick.

BTW, the thing that makes distances tricky is that when we speak of the distance between two points, we mean the distance between the position of an infinitesimal object (like the end of your measuring stick) at rest at one of the points, and another infinitesimal object (the other end of the measuring stick) at rest at the other point at the same time... and of course relativity of simultaneity means that "the same time" in one frame isn't "the same time" in another.