Time and Light speed communication

[jfy4]

Since I began studying relativity, there has been a lot of talk about treating time on equal footing with space. Then there is the space-time. I have been pondering this a little while and I came across a (what I find) curious pair of situations. They are somewhat modified versions of the twin paradox.

These will seem like quite a fuss, just to point out a puzzlement I am having, but I want to try and be clear.

Here is the first:

Consider the space-time flat (and 2D, just x and t). Let's have 4 observers (A, B, C, D) and me. Their coordinates are

$$x_A=\{x_1,t_1\}$$
$$x_B=\{x_2,t_1\}$$
$$x_C=\{x_3,t_1\}$$
$$x_D=\{x_4,t_1\}$$

and I am going to be stationary with A, C, and D. Note then that the clocks are started in sync. B heads off quickly to a distant flag pole in space, light years away and then returns to the original coordinate, x₂. Our coordinates now, as observed by me are

$$x_A=\{x_1,t_2\}$$
$$x_B=\{x_2,t_2^{\prime}\}$$
$$x_C=\{x_3,t_2\}$$
$$x_D=\{x_4,t_2\}$$

where t^'₂<<t₂.


Here is the second scenario:

There are two observers, and me, with coordinates

$$x_A=\{x_1,t_1\}$$
$$x_B=\{x_2,t_1\}$$

and I am stationary with A (for now). Then B heads off for some time (according to me) and then comes to rest relative to me. Now the coordinates are

$$x_A=\{x_1,t_1^{\prime}\}$$
$$x_B=\{x_3,t_2\}$$

where t₁^'>t₂. Now A heads off for some time (according to me) until I read that his clock reads that the same time has passed for him, as it has for B. He then comes to rest so that A and B's clocks are again in sync. The coordinates read

$$x_A=\{x_4,t_2^{\prime}\}$$
$$x_B=\{x_3,t_2^{\prime}\}$$

so now A and B are at two different (now further apart) locations but their time coordinate is the same.


Now the reason I found this interesting is that, if A and B try and communicate at the end of all their traveling with flashlights, it only depends on their relative spatial distance, not on their relative temporal "distance". That is, in the first scenario A and B are close spatially, however, they clearly are "separated" extremely far in time (likely years!).

But in the second scenario, A and B are separated quite far in terms of spatial distance, but they posses the same temporal coordinate. However, in both circumstance, the communication with light only depends on their relative spatial separation (this is nothing new).

I was just a little puzzled that we have stressed this huge change in our mind set about the relationship between space and time, but as this example shows, the exchange of information is only dependent on relative spatial location, not relative temporal "location".

Does anyone have any insight into why that is? Or, has their been research into why the the relative readings on A and B's wrist watches as observed by me do not play a role in the exchange of information?

 

[JesseM]

Since I began studying relativity, there has been a lot of talk about treating time on equal footing with space. Then there is the space-time. I have been pondering this a little while and I came across a (what I find) curious pair of situations. They are somewhat modified versions of the twin paradox.

These will seem like quite a fuss, just to point out a puzzlement I am having, but I want to try and be clear.

Here is the first:

Consider the space-time flat (and 2D, just x and t). Let's have 4 observers (A, B, C, D) and me. Their coordinates are

$$x_A=\{x_1,t_1\}$$
$$x_B=\{x_2,t_1\}$$
$$x_C=\{x_3,t_1\}$$
$$x_D=\{x_4,t_1\}$$

and I am going to be stationary with A, C, and D. Note then that the clocks are started in sync. B heads off quickly to a distant flag pole in space, light years away and then returns to the original coordinate, x₂. Our coordinates now, as observed by me are

$$x_A=\{x_1,t_2\}$$
$$x_B=\{x_2,t_2^{\prime}\}$$
$$x_C=\{x_3,t_2\}$$
$$x_D=\{x_4,t_2\}$$

where t^'₂<<t₂.

What do you mean "our coordinates now"? By definition, the coordinates of a bunch of different observers "now" in your frame should all involve the same time-coordinate. B's age will be less than the rest of you "now", but the time-coordinate you assign in your frame to an event on B's worldline is different from the B's age (i.e. B's proper time) at that event.

Here is the second scenario:

There are two observers, and me, with coordinates

$$x_A=\{x_1,t_1\}$$
$$x_B=\{x_2,t_1\}$$

and I am stationary with A (for now). Then B heads off for some time (according to me) and then comes to rest relative to me. Now the coordinates are

$$x_A=\{x_1,t_1^{\prime}\}$$
$$x_B=\{x_3,t_2\}$$

where t₁^'>t₂. Now A heads off for some time (according to me) until I read that his clock reads that the same time has passed for him, as it has for B. He then comes to rest so that A and B's clocks are again in sync. The coordinates read

$$x_A=\{x_4,t_2^{\prime}\}$$
$$x_B=\{x_3,t_2^{\prime}\}$$

so now A and B are at two different (now further apart) locations but their time coordinate is the same.

Again I think you are talking about their proper time, not the time-coordinate of the events of them coming to rest.

Now the reason I found this interesting is that, if A and B try and communicate at the end of all their traveling with flashlights, it only depends on their relative spatial distance, not on their relative temporal "distance". That is, in the first scenario A and B are close spatially, however, they clearly are "separated" extremely far in time (likely years!).

Again you have to distinguish between the difference in time-coordinates between two events on their worldlines, and the difference in their ages at those events. If they are x light-seconds apart in their mutual rest frame, then the coordinate time between one sending a light signal and the other receiving it will always be x/c in this frame. There may be a much larger difference in their ages at these two events, but I don't see how that's fundamentally different from you making a phone call to a person much older or younger than you, you wouldn't normally refer to this difference in ages by saying there is a large "temporal distance" between you two.

 

[jfy4]

What do you mean "our coordinates now"?

By "our coordinates" I meant: where I see them, and what I read on their clocks.

There may be a much larger difference in their ages at these two events, but I don't see how that's fundamentally different from you making a phone call to a person much older or younger than you, you wouldn't normally refer to this difference in ages by saying there is a large "temporal distance" between you two.

Right.? It doesn't matter people's age when you talk on the phone or use Morse code flash-lights, but it got me thinking. Is your age not your "distance" down the time axis? To me it seems that it is. That if me and someone else sync our watches, and then one of us runs awhile and comes back, our clocks are our of sync because one of us has "covered" less time. Then there is this stress to treat time and space similarly. Well, we measure the distance between us as our relative distance between us, and our age seems to be our location on the time axis (even if it is relative). So why isn't our relative age important when communicating with someone?

Don't get me wrong, I am not accusing anything of being wrong. But two people's relative age seems to be the "distance" between them in time. Is there any insight into this "distance" that does not have to be taken into account when light travels between us?

 

[JesseM]

Right.? It doesn't matter people's age when you talk on the phone or use Morse code flash-lights, but it got me thinking. Is your age not your "distance" down the time axis? To me it seems that it is.

What do you mean by "the time axis"? The t-coordinate axis of some inertial frame? If so then no, your "distance" along this axis at any point on your worldline is just the t-coordinate of that point, nothing to do with your own age at that point.

Suppose I draw on x-y coordinate grid on some patch of land with roads on it, and then you and I depart from the same starting point along different roads with the odometers of our cars running. It the roads later cross and we meet at the crossing point, our odometers may show very different elapsed distances traveled since last we met, but our "distance along the x-axis" is just the x-coordinate of the point we're currently at, nothing to do with our odometers.

That if me and someone else sync our watches, and then one of us runs awhile and comes back, our clocks are our of sync because one of us has "covered" less time.

You can't treat "time" as a monolithic entity, one of us has taken a path with less proper time, but the amount of coordinate time between meetings is the same for both of us. Again, it's analogous to the example with the two roads, where one of us has traveled a greater distance as measured by our own odometer, but the coordinate distance between the position we departed and the position we reunited is the same for both of us.

Then there is this stress to treat time and space similarly. Well, we measure the distance between us as our relative distance between us, and our age seems to be our location on the time axis (even if it is relative).

Again, our time coordinate is our location on the time axis.

But two people's relative age seems to be the "distance" between them in time.

No, no more so than two people's relative odometer readings is the "distance" between them in space in my above example (when the two cars reunite they are right next to each other in space, but their odometers have elapsed different amounts since last they met)

 

[jfy4]

Suppose I draw on x-y coordinate grid on some patch of land with roads on it, and then you and I depart from the same starting point along different roads with the odometers of our cars running. It the roads later cross and we meet at the crossing point, our odometers may show very different elapsed distances traveled since last we met, but our "distance along the x-axis" is just the x-coordinate of the point we're currently at, nothing to do with our odometers.

An excellent analogy. That clarifies things quite nicely. Thank you.