Mutual time dilation seems to be self-contradictory?

[victorhugo]

The first thing I want to say: time dilation doesn't make any logical sense. If one's time is going slower than the other, how could they both see each other's time going slow? if not, then there's a way to tell who's moving as one would see the other in fast motion.

 

[Orodruin]

The first thing I want to say: time dilation doesn't make any logical sense. If one's time is going slower than the other, how could they both see each other's time going slow?

You are failing to account for the relativity of simultaneity, a common mistake. I suggest reading my insight on the geometrical interpretation of time dilation and the twin paradox.

 

[victorhugo]

I suggest reading my insight on the geometrical interpretation of time dilation and the twin paradox.

Where can I find that?

 

[Orodruin]

Where can I find that?

A Geometrical View of Time Dilation and the Twin Paradox

 

[stevendaryl]

The first thing I want to say: time dilation doesn't make any logical sense. If one's time is going slower than the other, how could they both see each other's time going slow? if not, then there's a way to tell who's moving as one would see the other in fast motion.

I like the following geometric analogy: Suppose you have a system of highways that cross each other. On each highway, there are lines painted across the pavement at regular intervals (say, one line every 10 meters). Now, as shown in the picture, suppose that two highways cross each other at an angle $\theta$. Let's number the lines on each highway, using the number $N$ for the horizontal highway, and $N'$ for the diagonal highway. Set the origins of $N$ and $N'$ so that where the roads intersect, $N = N' = 0$.

Now, you're traveling down the horizontal highway, and you pass line number $N$. You look straight to your left (perpendicular to your highway), and see what the corresponding line number, $N'$ is, for the other highway. A little bit of geometry would tell you that:

$N' \approx N/cos(\theta)$

So some traveling down the horizontal highway would tell you that $N'/N > 1$. The "rate" of increase for the line numbers on the diagonal highway is greater than that of the horizontal highway.

But consider the point of view of someone traveling on the diagonal highway. He passes line number $N'$ on his highway, and he looks straight to his right (perpendicular to his highway) to see which line number, $N'$ corresponds. Geometry again would tell you that:

$N \approx N'/cos(\theta)$

So he concludes that the relative rate $N/N' > 1$, so $N'/N < 1$

So one traveler thinks that $N'/N < 1$, and the other traveler thinks that $N'/N > 1$. How can they both be right?

The answer is that the two are using incompatible methods for associating a value of $N$ with the corresponding value of $N'$. The horizontal traveler associates $N$ with a greater value of $N'$, while the diagonal traveler associates $N$ with a smaller value of $N'$. The key is that which road lines correspond is relative to the observer.

The same thing happens in relativity with time dilation. You have Bob and Alice traveling at a relative speed that is high enough that the time dilation factor is 2. They pass each other at time $12:00$, according to both their watches. Then consider the following 3 events:

1. Bob's watch shows time $12:30$
2. Bob's watch shows time $2:00$
3. Alice's watch shows time $1:00$

In order to figure out whose watch is running slow, you have to figure out which events correspond (are simultaneous). Bob thinks that event 2 corresponds to event 3. So he believes that Alice's watch is running slow, since at time $2:00$ (according to his watch), her watch only shows time $1:00$. But Alice thinks that event 1 corresponds to event 3. So she believes that Bob's watch is running slow, since at time $1:00$ (according to her watch), his watch only shows time $12:30$. They can never resolve the question of whose watch is "really" running slower, because they can't agree on which events are simultaneous.

 

[victorhugo]

I like the following geometric analogy: Suppose you have a system of highways that cross each other. On each highway, there are lines painted across the pavement at regular intervals (say, one line every 10 meters). Now, as shown in the picture, suppose that two highways cross each other at an angle $\theta$. Let's number the lines on each highway, using the number $N$ for the horizontal highway, and $N'$ for the diagonal highway. Set the origins of $N$ and $N'$ so that where the roads intersect, $N = N' = 0$.

View attachment 107022

Now, you're traveling down the horizontal highway, and you pass line number $N$. You look straight to your left (perpendicular to your highway), and see what the corresponding line number, $N'$ is, for the other highway. A little bit of geometry would tell you that:

$N' \approx N/cos(\theta)$

So some traveling down the horizontal highway would tell you that $N'/N > 1$. The "rate" of increase for the line numbers on the diagonal highway is greater than that of the horizontal highway.

But consider the point of view of someone traveling on the diagonal highway. He passes line number $N'$ on his highway, and he looks straight to his right (perpendicular to his highway) to see which line number, $N'$ corresponds. Geometry again would tell you that:

$N \approx N'/cos(\theta)$

So he concludes that the relative rate $N/N' > 1$, so $N'/N < 1$

So one traveler thinks that $N'/N < 1$, and the other traveler thinks that $N'/N > 1$. How can they both be right?

The answer is that the two are using incompatible methods for associating a value of $N$ with the corresponding value of $N'$. The horizontal traveler associates $N$ with a greater value of $N'$, while the diagonal traveler associates $N$ with a smaller value of $N'$. The key is that which road lines correspond is relative to the observer.

The same thing happens in relativity with time dilation. You have Bob and Alice traveling at a relative speed that is high enough that the time dilation factor is 2. They pass each other at time $12:00$, according to both their watches. Then consider the following 3 events:

1. Bob's watch shows time $12:30$
2. Bob's watch shows time $2:00$
3. Alice's watch shows time $1:00$

In order to figure out whose watch is running slow, you have to figure out which events correspond (are simultaneous). Bob thinks that event 2 corresponds to event 3. So he believes that Alice's watch is running slow, since at time $2:00$ (according to his watch), her watch only shows time $1:00$. But Alice thinks that event 1 corresponds to event 3. So she believes that Bob's watch is running slow, since at time $1:00$ (according to her watch), his watch only shows time $12:30$. They can never resolve the question of whose watch is "really" running slower, because they can't agree on which events are simultaneous.

Thank you so much! :)

 

[robphy]

Here's an interactive geometrical way to see it (from a recent thread asking a similar question)
www.physicsforums.com/threads/how-can-time-dilation-be-the-same-for-both-observers.886874/page-2#post-5578058