B Simplest symmetrical time dilation question

[GR86]

[Mentor's note - split off from a thread about the Twin Paradox]

In the meantime let me ask the most simplified question i can here then. If the universe is void of all matter save two rockets and they are passing eachother at the speed of light like two cars on the freeway in opposing lanes, which is said to be the one moving?

My understanding is neither, or both. And what happens with time in this situation? Neither are accelerating, and each would look out a window and see the other rocket flying by at the speed of light. How does the universe determine what happens with relativity? The only answer that makes sense to me is, nothing happens, there is no time dilation differences between the two.

I believe i have heard a theory that both would see the other's time sped up in relation to their own, meaning both would have time slow relatively speaking.

 

[PeterDonis]

If the universe is void of all matter save two rockets and they are passing eachother at the speed of light like two cars on the freeway in opposing lanes, which is said to be the one moving?

Neither one is moving in any absolute sense.

More important for this discussion, the twin paradox has nothing to do with the relativity of motion; it has nothing to do with the question of which twin is "moving" and which is not. Again, you're barking up the wrong tree if you're trying to understand the twin paradox along these lines.

what happens with time in this situation?

This is also barking up the wrong tree; the twin paradox has nothing to do with "time flowing differently" for one twin vs. the other. "Time dilation" is a coordinate convention. It's not a physical thing.

Differential aging is a physical thing, as shown by the two twins, but it has to do with different path lengths through spacetime. But each twin's clock registers time along its own path through spacetime at the same rate: one second per second.

How does the universe determine what happens with relativity?

By spacetime geometry and path lengths of worldlines. These are invariants and have nothing to do with any of the concepts you are trying to use.

I believe i have heard a theory that both would see the other's time sped up in relation to their own, meaning both would have time slow relatively speaking.

You're going to need to give a reference for this. "I believe I have heard" is not a useful basis for discussion.

 

[Nugatory]

They are passing each other at the speed of light...

That's not possible of course, but the question works just as well if we say "at close to the speed of light" so we'll make this mental correction and proceed.

which is said to be the one moving?....My understanding is neither, or both.

There is no frame in which they are both at rest, but we can choose to treat either one as at rest and the other moving, or both moving (but that makes the math harder without telling us anything new so you won't often see this in the intro textbooks and I won't go into it here).

And what happens with time in this situation? Neither are accelerating, and each would look out a window and see the other rocket flying by at the speed of light. How does the universe determine what happens with relativity? The only answer that makes sense to me is, nothing happens, there is no time dilation differences between the two.

They both find that the other's clock is running slow relative to their own (and note that although this result is not what you were expecting, it does honor the symmetry of the situation).
The apparent paradox (How can clock A be slower than clock B and clock B be slower than clock A) is resolved by remembering the relativity of simultaneity. Say both clocks are set to 12:00 noon at the moment that they pass each other. One hour later A looks at their clock and sees that it reads 1:00 PM, as we expect. A uses the Lorentz transformation to calculate what B's clock reads AT THE SAME TIME that their clock reads 1:00, or equivalently they are watching B's clock through a telescope and by allowing for light travel time they can see what B's clock read when light left B AT THE SAME TIME that A's clock reads 1:00 PM. Either way, they find that B's clock reads 12:30 AT THE SAME TIME that A's clocks reads 1:00. In other words, the events "A's clock reads 1:00" and "B's clock reads 12:30" are simultaneous when we use the frame in which A is at rest.
Clearly B's clock is running slow, by a factor of two.

But this is where the relativity of simultaneity comes in. Using the frame in which B is at rest, these two events do not happen at the same time. Instead, the event "A's clock reads 12:15" is simultaneous with the event "B's clock reads 12:30" so just as clearly A's clock is running slow by a factor of two.

How confident are we that the universe really works this way? Well, the GPS system relies on this analysis many thousands of times every day with multiple satellites moving relative to one another and the surface of the earth.... And the GPS system works.

I believe i have heard a theory that both would see the other's time sped up in relation to their own, meaning both would have time slow relatively speaking.

What they actually see, as in watching each other through telescopes and regardless of which one if either we're considering to be at rest is: both see the other's clocks running fast as they approach and slow as they move apart. This is just the Doppler effect, well understood in classical physics. Relativistic time dilation only comes into play when they find that the observed speedup/slowdown does not match the classical result; there's a bit less blueshift when approaching and a bit more redshift when separating. They calculate the discrepancy and they both find that the other clock is running slow relative to their own - that's time dilation.

 

[Ibix]

which is said to be the one moving?

Either or both, depending on your choice of rest frame. Clocks of moving rockets will tick slowly as measured in the frame you choose. This turns out not to be paradoxical because the two frames don't agree on what "all of space at one instant" means. It's a similar situation to two cars driving at the same speed along two straight roads that are at an angle to one another - both will say the other is falling behind because they have different notions of what "keeping up" means.

Note the critical difference between this scenario and the twin paradox. The spaceships do not meet up again, so there's no unambiguous way for them to compare clocks.

I stringly recommend learning about Minkowski diagrams for understanding this.

 

[pervect]

[Mentor's note - split off from a thread about the Twin Paradox]

In the meantime let me ask the most simplified question i can here then. If the universe is void of all matter save two rockets and they are passing eachother at the speed of light like two cars on the freeway in opposing lanes, which is said to be the one moving?

My understanding is neither, or both. And what happens with time in this situation? Neither are accelerating, and each would look out a window and see the other rocket flying by at the speed of light. How does the universe determine what happens with relativity? The only answer that makes sense to me is, nothing happens, there is no time dilation differences between the two.

I believe i have heard a theory that both would see the other's time sped up in relation to their own, meaning both would have time slow relatively speaking.

I have a strong suspicion that you are viewing time dilation as a relationship between the flow of time as experienced by an observer, and some underlying absolute time.

If this in fact how you are thinking, the issue is in your assumptions that there is some underlying absolute time to compare to.

"Time dilation" doesn't make any sense unless one is comparing two different notions of time. In the standard presentation, one of these two notions of time is called "proper time", and that's the sort of time measured by a clock. It's independent of any observer or frame of reference - it's just the sort of time that a clock measures between two events.

There is another notion of time that must be distinguished from proper time. This notion of time serves a different purpose. It's basic purpose is to assign labels or dates to events, so we know when they happened. This is the process of assigning time coordinates to events, and the sort of time used is called coordinate time. A real-word example of coordinate time is UTC, "Coordinate universal time". See for instance https://en.wikipedia.org/wiki/Coordinated_Universal_Time. UTC is a coordinate time.

This is a somewhat involved process, involving a fair number of conventions that are used so that standardized equations work. Note that in special relativity, every frame of reference has a different notion of coordinate time, as a result of the "relativity of simultaneity" as demonstrated by "Einstein's train" thought experiment. I can provide references on these two catch phrases (relativity of simultaneity and Einstein's train), and will upon request. The reason I'm not volunteering is my low success rate in past conversations of this type, so I feel that pointing out the resource is sufficient. It's up to the reader to be curious enough to want to do the research. My goal is to motivate thyem to do that resarch for themselves.

Anyway, time dilation can be regarded as the relationship between proper time and coordinate time, and of the two, the proper time is the one that has physical significance independent of the observer. The coordinate time is highly observer dependent, it depends both on an observer and some standardized convetions, while proper time is the sort of time talked about by the SI standard on the second.

I'll give a passing mention to the third aspect of time, causality, which is separate from the notion of proper time and coordinate time.

 

[KingGambit]

In the meantime let me ask the most simplified question i can here then. If the universe is void of all matter save two rockets and they are passing eachother at the speed of light like two cars on the freeway in opposing lanes, which is said to be the one moving?

According to whom? As long as both are not accelerating, both will see themselves stationary and the other is the one who is moving.

My understanding is neither, or both. And what happens with time in this situation?

According to whom?

and each would look out a window and see the other rocket flying by at the speed of light.

Yes, that's true. According to relativity velocity addition formula. But in this case you'll need a third observer E.
So this observer, E, see that A is moving to the 'east' at the speed of light very close to the speed of light, and B is moving to the 'west' at the speed of light very close to the speed of light.
So B will see that E is moving to the 'east' very close to the speed of light, and A is moving
A = (99.99% + 99.99%) / (1 + 99.99% ²) = (calculating...) 99.99999950% speed of light.

How does the universe determine what happens with relativity?

I don't know. God? But that's the law of nature.

The only answer that makes sense to me is, nothing happens, there is no time dilation differences between the two.

Fortunately (unfortunately?) yes, there is!
Both will see the others time slowing down.

Imagine Alice turn on her torch toward a mirror (blue). Alice will see herself stationary.

And imagine what Bob ( who is stationary according to Bob), sees Alice's (who is moving according to Bog) light?

The path of light from Bob's view takes longer than what Alice views. So Bob will see Alice's time slowing down.

Imagination is more important than knowledge - Albert Einstein

I believe i have heard a theory that both would see the other's time sped up in relation to their own, meaning both would have time slow relatively speaking.

Both would see other's time slowed down.
Both would see each respective time past normally, one second every second.
Even if each of them is near a black hole, one second every second.

 

[Nugatory]

Both would see other's time slowed down.

Careful…. Although you may be hitting an ambiguity in the meaning of the word “see”. They will both calculate, based on the arrival time of light from the other ship, that the other’s clock is running slow. What they will see, as in watching the other through a telescope, is the other’s time being sped up as they approach and slowing as the move apart, due to the Doppler effect. The time dilation effect shows up when they realize that the speedup while approaching is a bit less and the slowdown while moving apart a bit greater than the non-relativitistic calculation predicts.

 

[KingGambit]

Careful…. Although you may be hitting an ambiguity in the meaning of the word “see”. They will both calculate,... based on the arrival time of light

Ahh, yes. I agree. They can't see it slowed down or faster. It's a calculation based on the doppler effect, relativistic doppler effect I might say.

...is the other’s time being sped up as they approach and slowing as the move apart,...

Yess, I also agree. The OP states that the ship passing each other, each will see the other clock sped up, again to see the "real" clock speed must be calculated according to doppler effect. And slowed down after moving appart.

Say each other approaching at 0.3c. or 0.6 c in each frame. Lorentz Factor = 1.25

So in phase 1, approaching, each ship will calculate see the other clock sped up by 200%.
1 second for the rest one, 2 second for the approaching ship.

Phase 2, after passing and moving appart.
1 second for the rest one, 0.5 second for the traveling away one.

 

[jbriggs444]

Say each other approaching at 0.3c. or 0.6 c in each frame. Lorentz Factor = 1.25

Be careful here. It seems that you are suggesting that each is approaching at 0.3c in the mid-point frame. In that frame their closure rate would be 0.6c. But a closure rate is not the same thing as a relative velocity.

Their relative velocity (velocity of one in the rest frame of the other) would be given by the relativistic velocity addition formula:$$\frac{0.3c + 0.3c}{1 + 0.3^2} = \frac{0.6c}{1.09} \approx 0.55 c$$for a Lorentz factor of about 1.20.

A "closure rate" as I use the term is the rate at which the vector separation between two objects changes as assessed in some chosen third party frame of reference relative to which both objects are moving. Closure rates between pairs of material objects are limited to $2 \text{c}$.

[Some sources use the term "closure rate" to refer to the rate at which scalar separation (distance) is changing rather than the rate at which vector separation (displacement) is changing. But as long as our objects are on a collision course, this distinction need not bother us]

 

[KingGambit]

Their relative velocity (velocity of one in the rest frame of the other) would be given by the relativistic velocity addition formula:$$\frac{0.3c + 0.3c}{1 + 0.3^2} = \frac{0.6c}{1.09} \approx 0.55 c$$for a Lorentz factor of about 1.20.

Ahh.., oh my God. You're right. My carelessness.
I was mislead by the OP question
If the universe is void of all matter save two rockets and they are passing each other at the speed of light like...
And in my mind, the both are traveling, and that seems to suggest that there is another observer in the middle.
A 0.3c to the east, B 0.3c to the west, according the middle man. And of course B will not see A is traveling at 0.3c+0.3c. It's about, like you said, 0.55c.

And I definetely use this formula above.

Yes, that's true. According to relativity velocity addition formula. But in this case you'll need a third observer E.
So this observer, E, see that A is moving to the 'east' at the speed of light very close to the speed of light, and B is moving to the 'west' at the speed of light very close to the speed of light.
So B will see that E is moving to the 'east' very close to the speed of light, and A is moving
A = (99.99% + 99.99%) / (1 + 99.99% 2) = (calculating...) 99.99999950% speed of light.

 

[KingGambit]

[Mentor's note - split off from a thread about the Twin Paradox]

In the meantime let me ask the most simplified question i can here then. If the universe is void of all matter save two rockets and they are passing each other at the speed of light like two cars on the freeway in opposing lanes, which is said to be the one moving?

What if we change the statement.
They are passing each other at 0.6c, what would the middle observer see calculate each respective speed?

$$0.6 = \frac {2v} {1-\sqrt{v^2}}$$
$$v\approx 0.333$$

 

[KingGambit]

What if we change the statement.
They are passing each other at 0.6c, what would the middle observer see calculate each respective speed?

$$0.6 = \frac {2v} {1-\sqrt{v^2}}$$
$$v\approx 0.333$$

Sorry,
V = 1/3

 

[Ibix]

What if we change the statement.
They are passing each other at 0.6c,

The important thing to state is who is measuring the 0.6c. Do you mean each ship measures the other doing 0.6c? If so, then you have a rogue square root and a wrong sign in the numerator of this:

$$0.6 = \frac {2v} {1-\sqrt{v^2}}$$

It should be $$0.6=\frac{2v}{1+v^2}$$which does indeed solve to $v=1/3$.

 

[robphy]

Their relative velocity (velocity of one in the rest frame of the other) would be given by the relativistic velocity addition formula:
$$\frac{0.3c + 0.3c}{1 + 0.3^2} = \frac{0.6c}{1.09} \approx 0.55 c$$
for a Lorentz factor of about 1.20.

It should be
$$0.6=\frac{2v}{1+v^2}$$
which does indeed solve to v=1/3.

It might be worth mentioning the related geometric interpretation as analogues of double-angles and half-angles:

$$\tanh\left(2\operatorname{arctanh}\left(0.3\right)\right)=\frac{60}{109}=0.550458715596$$
$$\cosh\left(2\operatorname{arctanh}\left(0.3\right)\right)=\frac{109}{91}=1.1978021978$$

and

$$\tanh\left(\frac{1}{2}\operatorname{arctanh}\left(0.6\right)\right)=\frac{1}{3}=0.333333$$

 

[KingGambit]

The important thing to state is who is measuring the 0.6c. Do you mean each ship measures the other doing 0.6c? If so, then you have a rogue square root and a wrong sign in the numerator of this:

It should be $$0.6=\frac{2v}{1+v^2}$$which does indeed solve to $v=1/3$.

Oops, I'm so sorry. This is the first time I use BB Code,
Yes, it's 1+sqrt(v1 * v2), but in the midst of it, I typo negative instead of positive.
My apologize.

Edit: LaTex.

 

[DrGreg]

Yes, it's 1+sqrt(v1 * v2)

In this case there aren't any square roots at all. It's just $1+v_1 v_2$.

 

[KingGambit]

In this case there aren't any square roots at all. It's just $1+v_1 v_2$.

Yes @DrGreg, I'm really sorry for my carelessness. I'm new in SR (much less GR), so I don't really memorize Lorentz Factor, Gamma, Beta, velocity addition by heart. And I answer in a hurry.