Solving Weird Time Dilation Question w/ Spaceships

[Looeelooee1]

I put the prefix as intermediate because I wasn't really sure how complicated of a question this is, I am currently a junior in high school...

So basically if you were to have two spaceships with people the exact same age in them traveling at 99% the speed of light and they were traveling parallel to each other, but in opposite directions, then when ship A passes ship B, to ship A ship B will be traveling slower through time, and ship A will feel normal, and vice versa with ship B. So my question is, who is actually traveling slower through time? If both people were to meet each other then who would be younger? Also, from the perspective of someone on earth, they will both be the same age right?

I kinda had a hypothesis (that I know is most likely wrong) that from the perspective of ship A ship B will be younger when they meet and from the perspective of ship B ship A will be younger when they meet, and then if both of them were to meet someone on earth, the person from Earth would see them as being the same age...

Thanks in advance any help much appreciated!

 

[mathman]

To answer your question you need to know how they got together (same inertial frame). If they both stop, turn around and meet with both stopping, they will be same age. However, if one stops and turns around to meet the other, joining his frame, that person is younger.

 

[Ibix]

The problem is the relativity of simultaneity. Two people in relative motion don't agree, in general, on whether two things that didn't happen at the same place were simultaneous or not. That means that you can't just say that the two people are the same age: that would imply that their birthdays were simultaneous, and not everybody will agree on that - unless they were twins born at the same time and the same place. But then you need to worry about how they got to be far apart so that they could be moving towards each other. The details of how they did that will tell you which one is older and which one is younger, or if they are the same age. Since you didn't specify that, there isn't enough information to answer your question. If you do specify it (and I suggest assuming instantaneous accelerations and movement in straight lines only if you don't want us to start throwing around calculus), then we can talk about an answer.

It is clear that your answer is wrong, however. When the two ships meet up, they must agree on which one (if either) is older. You can see this fairly easily by putting a bomb on each spaceship rigged to blow if the other one is younger (it just looks at the other guy's clocks). If they disagree on which one is younger they must disagree on which one exploded - which is kind of crazy.

 

[m4r35n357]

I put the prefix as intermediate because I wasn't really sure how complicated of a question this is, I am currently a junior in high school...

I kinda had a hypothesis (that I know is most likely wrong) that from the perspective of ship A ship . IB will be younger when they meet and from the perspective of ship B ship A will be younger when they meet, and then if both of them were to meet someone on earth, the person from Earth would see them as being the same age...

Thanks in advance any help much appreciated!

This is definitely a "B" question, but please don't be put off by that ;)

You have clearly understood that people in relative motion experience time differently, which is a good starting point. However to really understand this situation will require some mathematics, and I am definitely not the one to give such a course!

I can give you an answer that might help with the confusion though. It all hinges on the "symmetry" of how the two people meet up again. If B goes off somewhere and comes back to A, then B will be younger when they meet (they do different things). On the other hand, if B goes off somewhere and A later goes off and meets them there, then they will be the same age (they do the same thing).

That is it in a nutshell if you leave out the mathematical details. Hope it helps.

 

[Looeelooee1]

The problem is the relativity of simultaneity. Two people in relative motion don't agree, in general, on whether two things that didn't happen at the same place were simultaneous or not. That means that you can't just say that the two people are the same age: that would imply that their birthdays were simultaneous, and not everybody will agree on that - unless they were twins born at the same time and the same place. But then you need to worry about how they got to be far apart so that they could be moving towards each other. The details of how they did that will tell you which one is older and which one is younger, or if they are the same age. Since you didn't specify that, there isn't enough information to answer your question. If you do specify it (and I suggest assuming instantaneous accelerations and movement in straight lines only if you don't want us to start throwing around calculus), then we can talk about an answer.

It is clear that your answer is wrong, however. When the two ships meet up, they must agree on which one (if either) is older. You can see this fairly easily by putting a bomb on each spaceship rigged to blow if the other one is younger (it just looks at the other guy's clocks). If they disagree on which one is younger they must disagree on which one exploded - which is kind of crazy.

So if you assume instantaneous acceleration and movement in straight lines only, and assume that they were born in the same time / place (ik that's impossible but for the sake of the question), or better yet assume they aren't people and instead marbles or something so I can ask which is experiencing time slower, then when they see each other as passing through time slower than themselves as they pass each other, then are they just seeing that from their perspective, which is wrong for some reason? and from the perspective of someone perfectly still (relative to the two ships) directly in the middle of them as they pass they are really the same age?

Thanks for the replies btw and sorry if this is a dumb question I just learned about this stuff so it's all kinda new...

 

[Ibix]

The reason the theory is called relativity is that motion is relative. You, the guys in the ships, or anyone can consider themself to be at rest, and that other people who are moving. Thus you always see your wristwatch, at rest with respect to you, ticking normally. You see the watch of anyone who is moving with respect to you ticking slowly.

So, yes, both twins say the other is aging more slowly when they cross. But that doesn't tell you who is younger - that depends on how they have aged over all the time since the twins split up, not just their aging rate at the instant they meet. That, in turn, depends on how they have moved. Did they both set out at the same speed relative to the Earth (note that you should always specify who measures a speed, distance or time), turn around and come back at the same speed? Or did one wait for a while, stationary with respect to the Earth before heading back, while the other carried on a bit further? Or something else?

In the first case they end up the same age, but a third twin (a triplet) who stayed on Earth would end up older. In the second case the stay-at-home triplet is oldest; the one who stopped is the middle, and the one who traveled furthest is the youngest. This is actually fairly easy to calculate if everybody moves at constant speeds. It's easiest to treat the Earth as stationary. You write down how far (as measured by the Earth) each triplet travels in each phase of their movement (call this $\Delta x$) and how long (as measured by Earth clocks) it took (call this $\Delta t$). Then you calculate $c\Delta\tau=\sqrt{(c\Delta t)^2-(\Delta x)^2}$, and $\Delta\tau$ is the amount the triplet has aged on that leg of their journey.

So, for example, say that one triplet sets out at 0.8c, travels for a year and comes back at 0.8c (all measurements made with instruments at rest with respect to the Earth). On the outbound leg, $\Delta t=1$ year and $\Delta x=0.8$ light years. That makes $c\Delta\tau=\sqrt{(c)^2-(0.8)^2}$. I cleverly chose units where c=1 light year per year, so that gives us $\Delta \tau=0.6$. You can do the same calculation for the return leg, so the total elapsed time according to the triplet in the ship is 1.2 years, compared to the 2 years measured by the stay at home triplet.

Hope that makes sense.

 

[Looeelooee1]

The reason the theory is called relativity is that motion is relative. You, the guys in the ships, or anyone can consider themself to be at rest, and that other people who are moving. Thus you always see your wristwatch, at rest with respect to you, ticking normally. You see the watch of anyone who is moving with respect to you ticking slowly.

So, yes, both twins say the other is aging more slowly when they cross. But that doesn't tell you who is younger - that depends on how they have aged over all the time since the twins split up, not just their aging rate at the instant they meet. That, in turn, depends on how they have moved. Did they both set out at the same speed relative to the Earth (note that you should always specify who measures a speed, distance or time), turn around and come back at the same speed? Or did one wait for a while, stationary with respect to the Earth before heading back, while the other carried on a bit further? Or something else?

In the first case they end up the same age, but a third twin (a triplet) who stayed on Earth would end up older. In the second case the stay-at-home triplet is oldest; the one who stopped is the middle, and the one who traveled furthest is the youngest. This is actually fairly easy to calculate if everybody moves at constant speeds. It's easiest to treat the Earth as stationary. You write down how far (as measured by the Earth) each triplet travels in each phase of their movement (call this $\Delta x$) and how long (as measured by Earth clocks) it took (call this $\Delta t$). Then you calculate $c\Delta\tau=\sqrt{(c\Delta t)^2-(\Delta x)^2}$, and $\Delta\tau$ is the amount the triplet has aged on that leg of their journey.

So, for example, say that one triplet sets out at 0.8c, travels for a year and comes back at 0.8c (all measurements made with instruments at rest with respect to the Earth). On the outbound leg, $\Delta t=1$ year and $\Delta x=0.8$ light years. That makes $c\Delta\tau=\sqrt{(c)^2-(0.8)^2}$. I cleverly chose units where c=1 light year per year, so that gives us $\Delta \tau=0.6$. You can do the same calculation for the return leg, so the total elapsed time according to the triplet in the ship is 1.2 years, compared to the 2 years measured by the stay at home triplet.

Hope that makes sense.

ok thank you so much for that response it made a lot of sense and cleared it up for me!

but I also have the question of, at that moment where the two twins (or objects) cross each other's path, both going the same speed (relative to earth), who is traveling slower through time at that moment, or, how can both see the other as traveling slower through time, if only one of them can be correct. I know that if they were to both return to Earth with the exact same path they would be the same age, but at that point where they cross they see each other as different ages, or at different points in time... Is this because time itself rotates and therefore although they see each other as traveling through time slower they are really not? So basically do both objects have different relative times but the same "real" time (from the perspective of someone on earth)? Once again thanks for all the help!

 

[Ibix]

There is no "real" time. You get a lot of flexibility in how to define "time" in relativity - basically there are a set of different directions through spacetime that are called "timelike" and all bodies with mass follow these. But bodies in motion with respect to each other are following different timelike paths. That's a complicated way of saying that both twins are traveling at the same rate through time (one second per second), but have chosen different definitions of what "time" is.

A very close analogy is two cars traveling at 30mph along straight roads that are at an angle $\theta$ to each other. If the driver of car A looks out of the window at the car B, car B will be falling behind because its speed in the direction of A's road is only $30\cos\theta$. But the driver of car B can see the exact same thing - A falling behind because its speed in B's forward direction is $30\cos\theta$. Which car is going faster forward? Neither - they just have different definitions of "forward".

One other thing to think about. Moving "faster through space" means that you are covering more distance in the same time. "Faster through time", then, would mean covering more time in the same time. It doesn't really make sense.

 

[Nugatory]

but I also have the question of, at that moment where the two twins (or objects) cross each other's path, both going the same speed (relative to earth), who is traveling slower through time at that moment, or, how can both see the other as traveling slower through time, if only one of them can be correct.

They're both right. The key to time dilation is the relativity of simultaneity.

Suppose that at the moment they pass each other they both set their wristwatches to 1200 noon. Then at some later time A finds that at the same time his watch read 1240 B's watch read 1220. He does this by watching B through a telescope, waiting until he sees B's watch reading 1220, and correcting for the light travel time - if B is five light minutes away and A sees B's watch reading 1220 at 1245 according to A's watch, he knows that B's watch read 1220 at the same time that A's watch read 1240. So A correctly concludes that B's time is dilated by a factor of two: 40 minutes of his time is 20 minutes of B's time.

However - and this is where relativity of simultaneity comes in - although the two events "A's watch reads 1240" and "B's watch reads 1220" are simultaneous according to A, they are not simultaneous according to B. Instead, B finds that "B's watch reads 1220" happens at the same time as "A's watch reads 1210" so he correctly concludes that A is the one whose time is dilated by a factor of two.

I know that if they were to both return to Earth with thes exact same path they would be the same age, but at that point where they cross they see each other as different ages, or at different points in time... Is this because time itself rotates and therefore although they see each other as traveling through time slower they are really not? So basically do both objects have different relative times but the same "real" time (from the perspective of someone on earth)?

Relativity of simultaneity has another consequence as well: it makes no sense to compare the ages of the twins except when they're both together at the same place. We're asking for the age of A and the age of B at the same time - and relativity of simultaneity means that if they aren't colocated, different observers will come up with different answers.

However, we can start the twins together, send them on different paths through spacetime, then bring them back together and compare how much each one has aged (and how much time has passed for each, as measured by their wristwatches). Neither time dilation nor relativity of simultaneity is involved here; they've followed different paths through spacetime between the same two points, the paths have different lengths, and the length of a path through spacetime is measured by the anount of time experienced along that path.

All observers will agree about the amount each twin ages, because that's determined by the length of the path through spacetime that twin follows - and the length is what it is, whether you're following the path yourself or just watching someone else following it. To better understand how this works (and how little it has to do with time dilation) you should take a look at http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

 

[Mister T]

but I also have the question of, at that moment where the two twins (or objects) cross each other's path, both going the same speed (relative to earth), who is traveling slower through time at that moment,

I'm not sure what the phrase "travelling through time" means to you. To me it has no meaning. We don't have to travel for our clocks to measure the passage of time. There are three clocks involved in your scenario, one on Earth and one aboard each ship. Anyone who happens to be at rest relative to one of these clocks will observe that the other two are running slow compared to the one that's at rest.

or, how can both see the other as traveling slower through time, if only one of them can be correct.

All three of them are correct. In order to understand how this makes sense, you have to delve deeper into it than you have so far. The others who have responded are trying to help you do that, but it's a really hard thing to grasp. You will probably have to study it step by step, confronting apparent paradoxes at each step along the way. There are some good books to guide you, but unfortunately there are a lot of bad ones, too. And then there's the question of whether or not you want to take the geometric approach. Some authors do, and some don't.

I know that if they were to both return to Earth with the exact same path they would be the same age,

First of all, were they the same age when they crossed paths? That's an event. To make another comparison like this you need another event where they are both together at the same place. Since they were traveling in opposite directions at the first event, they can't possibly take the same paths to get to the second event. At least one of them will have to change directions. If only one of them does, then he will have aged less. If they both do, in a symmetrical way, then they would both age the same amount.

 

[Nugatory]

At least one of them will have to change directions. If only one of them does, then he will have aged less. If they both do, in a symmetrical way, then they would both age the same amount.

... And if they both accelerate/change directions, but not in a symmetrical way, they'll both age less than someone who didn't accelerate/change directions. Because of the asymmetry it won't be by the same amount so one will end up more aged than the other and we'll have t do some calculation to find which one and by how much. This is not a problem you'll find in most introductory texts, but we have many threads here about such asymmetrical variants of the twin paradox.

(@Mister T knows this, of course. This post is for OP and anyone else who is following the thread).