# Calculating Distances in Special Relativity - Bob's Question

• bobbles22
In summary, the conversation discusses using Lorentz transformations to find relative speeds of objects in different frames of reference and the struggle to understand how this applies to distance. The length contraction formula is mentioned as a way to convert distances between frames of reference, and the concept of relativity of simultaneity is also brought up as a potential complicating factor.
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.

Bob

bobbles22 said:
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.

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.

bobbles22 said:
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.

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.

b
bobbles22 said:
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.

Hello Bob,

Your question is a common one when it comes to understanding distances in special relativity. The key concept to keep in mind is that distances are not absolute in special relativity, but rather they are relative to the observer's frame of reference. This means that the distance between two objects will appear different to observers in different frames of reference.

To calculate the distance as measured by one observer to the other, you will need to use the Lorentz transformation equations. These equations take into account the relative speed between the two frames of reference and allow you to calculate the distance as measured by one observer to the other. The equation for calculating distance in special relativity is:

L = γL0

Where L is the distance as measured by one observer, L0 is the distance as measured by the other observer, and γ is the Lorentz factor, which is dependent on the relative speed between the two frames of reference. You can find the value of γ using the equation:

γ = 1/√(1-(v/c)^2)

Where v is the relative speed between the two frames of reference and c is the speed of light. Once you have the value of γ, you can plug it into the first equation to calculate the distance as measured by one observer to the other.

I hope this helps clarify how to calculate distances in special relativity. It may seem complicated at first, but with practice, you will become more comfortable with using the equations and understanding the concept of relative distances in special relativity.

Best,

## 1. How is distance calculated in special relativity?

In special relativity, the distance between two objects is calculated using the Lorentz transformation, which takes into account the relative velocities of the objects.

## 2. What is the difference between distance and proper distance in special relativity?

Distance in special relativity is the distance measured by an observer in a specific frame of reference, while proper distance is the distance measured by an observer in the rest frame of an object.

## 3. Can distance be measured in different frames of reference in special relativity?

Yes, the distance between two objects can be measured differently in different frames of reference due to the effects of time dilation and length contraction.

## 4. How does time dilation affect distance calculations in special relativity?

Time dilation, which is the slowing down of time for objects moving at high velocities, affects distance calculations in special relativity by making it appear that the distance between two objects is shorter than it would be at rest.

## 5. Is it possible for distance to be negative in special relativity?

No, distance cannot be negative in special relativity. The Lorentz transformation only allows for positive distances, and negative distances would violate the principle of causality in physics.

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