# Combining time dilation and length contraction

## Main Question or Discussion Point

So, I am having trouble understanding how time dilation and length contraction in special relativity merge together and describe space-time. I guess the best way to ask this question is to pose a question, which i know how to answer mathematically, but I would like to discuss some thoughts about it.

Question:
Jack boards a spaceship and travels away from Earth at a constant velocity 0.45c toward Betelgeuse. One year later on Earths clocks, Jack's twin, Jim, boards a second spaceship and follows him at a constant velocity of 0.95c in the same direction.

Would the people on earth measure the spaceship to be in the same position as the people in the spaceship or would both of the observers measure the spaceship to be at different locations? From my current knowledge I would think that the earth observers would view the spaceship to be closer than it actually appears because of the time it takes for light to travel back. Is this correct? I asked my physics professor and he said that I have to factor in Lorenz contraction at the same time and he said both observers would see the spaceship at the same position, which I found to be extremely confusing. Which would be the case. Can anyone explain how to understand time dilation and length contraction together in and mathematical or even intuitive sense?

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ghwellsjr
Gold Member
So, I am having trouble understanding how time dilation and length contraction in special relativity merge together and describe space-time. I guess the best way to ask this question is to pose a question, which i know how to answer mathematically, but I would like to discuss some thoughts about it.

Question:
Jack boards a spaceship and travels away from Earth at a constant velocity 0.45c toward Betelgeuse. One year later on Earths clocks, Jack's twin, Jim, boards a second spaceship and follows him at a constant velocity of 0.95c in the same direction.

Would the people on earth measure the spaceship to be in the same position as the people in the spaceship or would both of the observers measure the spaceship to be at different locations?
I'm not sure what you're asking here because you have two spaceships, Jack's and Jim's, but generally speaking, whenever you have different observers traveling at different speeds, they're all going to have a different measurement of where the others are.
From my current knowledge I would think that the earth observers would view the spaceship to be closer than it actually appears because of the time it takes for light to travel back. Is this correct?.
I'm confused again by this question. What is the difference between what "observers would view" and what "actually appears"?
I asked my physics professor and he said that I have to factor in Lorenz contraction at the same time and he said both observers would see the spaceship at the same position, which I found to be extremely confusing. Which would be the case. Can anyone explain how to understand time dilation and length contraction together in and mathematical or even intuitive sense?

There are different ways of talking about scenarios such as the one you described. You can discuss what each observer sees. This type of discussion is independent of any specific application of any theory. Or you can apply Special Relativity and assign your scenario to a specific Frame of Reference which will describe what is happening according to the space-time coordinates of that FoR. Your questions imply that you would like to cover both types of discussion, is that correct?

You also mentioned that you understand at least some of the mathematics involved but you don't say exactly what. That almost implies that you would like a full and complete explanation of Special Relativity which could be a very long answer and may not even be what you are asking about.

Your questions about time dilation and length contraction would lead me to suggest an answer along the lines of a light clock but I'm going to guess you are already familiar with that, correct?

So could you please be more specific with your questions and let us know how much you already know?

Yes, I know the basic math of the light clock and the derivation of gamma. I guess one of my questions is. Why do you have to have the Lorenz contraction in the first place?
Why can't you just change the variable of time and not length to make things work? What problem derives the Lorenz contraction, where does it come from?

As of my other question. Say one spaceship takes off from earth and it travels for one year in earth time. Well would the the distance that the earth observers see the ship at match the distance that the observers on the ship see? No right? because the ship has traveled farther than earth sees because of the time required for the light to travel back. Hope this clears things up.

I think i made my first part a little more clear, i hope, sorry I'm new to this and it can be hard to get my questions out at first. Thank you very much for taking the time to help me.

ghwellsjr
Gold Member
Yes, I know the basic math of the light clock and the derivation of gamma. I guess one of my questions is. Why do you have to have the Lorenz contraction in the first place?
Why can't you just change the variable of time and not length to make things work? What problem derives the Lorenz contraction, where does it come from?
I think maybe the reason why you think that time dilation will take care of everything is because you have studied a light clock in which the light beams are traversing at right angles to the direction of motion. In this case, there is no length contraction because it only applies in the direction of motion. If you studied a circular set of mirrors set up as a light clock, you would see that time dilation is not enough to solve the problem, you also need length contraction along the direction of motion. Someone else recently asked a similar question about https://www.physicsforums.com/showthread.php?t=530141". Why don't you take a look at that thread and see if it helps you with regard to this issue?
As of my other question. Say one spaceship takes off from earth and it travels for one year in earth time. Well would the the distance that the earth observers see the ship at match the distance that the observers on the ship see? No right? because the ship has traveled farther than earth sees because of the time required for the light to travel back. Hope this clears things up.

I think i made my first part a little more clear, i hope, sorry I'm new to this and it can be hard to get my questions out at first. Thank you very much for taking the time to help me.
I'm still confused as to why you presented two spaceships in your scenario running at different speeds and different times. Did you want to know how earthlings view both spaceships, and how Jack and Jim both view the earth, and how Jack and Jim view each other? And when you say that Jim follows Jack at 0.95c, did you intend for that to be his speed relative to the earth or relative to Jack? And finally, did you want to know what each observer (earthlings, Jack, and Jim) actually see of each other or did you want to see how a Frame of Reference for each one would interpret your scenario?

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