How do you resolve seeming contradictions in SR?

[curiousburke]

This seems like a contradiction: if I have two spatially separated observers in one frame (A and B), and one observer in another frame (C) at the same point in space as A. As I understand it, B will have the same definition of "now" as A, and A will have the same "now" as B and C, but B will have a different "now" from C. Expressed as age, B might say they are the same age as A, and also say that A is the same age as C, but still say that they (B) are a different age from C. How does this get resolved?

 

[Hill]

and A will have the same "now" as [ ] C

No, they will not.

 

[curiousburke]

why not if A and C are at the same space and time?

 

[Hill]

why not if A and C are at the same space and time?

Because "now" is not an event but a spacelike hypersurface of all events which are simultaneous in a given frame. These hypersurfaces differ between A and C although they might intersect, for example in that space-time event you refer to.

 

[curiousburke]

So, you're saying that my misunderstanding is that participants in space-time events don't have to agree on "now"?

 

[Hill]

So, you're saying that my misunderstanding is that participants in space-time events don't have to agree on "now"?

Yes, they don't.

 

[PeterDonis]

As I understand it, B will have the same definition of "now" as A

Yes.

and A will have the same "now" as B

Yes.

and C

No. A and C have the same "here and now" at the event you describe, where A and C meet; but they do not have the same "now" because "now" is not just a single event but a set of events, and the set of events "now" for C is not the same as the set of events "now" for A.

Expressed as age

"Age" is a different thing from "now".

B might say they are the same age as A, and also say that A is the same age as C, but still say that they (B) are a different age from C.

No, this is not possible. Think of A, B, and C as all carrying clocks that read their age. At the event where A and C meet, if A and C's clocks both read the same age, that is an invariant; it doesn't matter what frame you choose. And if B, spatially separated from A and C, is at an event which is in the same set of "now" events for A and B as the event where A and C meet, says that his (B's) age--the reading on B's clock--is the same as A's age--the reading on A's clock--at the event where A and C meet, then B's age must also be the same as C's age--the reading on C's clock--at that event, because A's and C's clock readings don't depend on what frame you choose.

 

[curiousburke]

I don't understand what that means that two observers at the the same point in space-time don't agree on "now"; do there agree on "here"?

 

[PeterDonis]

I don't understand what that means that two observers at the the same point in space-time don't agree on "now"; do there agree on "here"?

As I said, they agree on "here and now": they are both at the same point in spacetime.

If they are moving relative to each other, then they do not agree on either "here" or "now" separately: both "here" and "now" are sets of events in spacetime, not single events, and they are both different sets of events for A and C. They only agree on "here and now" at the single event at which they meet.

 

[Hill]

do there agree on "here"?

They agree on "here" at that moment but generally they do not. Need an example?

 

[curiousburke]

Yes.


Yes.


No. A and C have the same "here and now" at the event you describe, where A and C meet; but they do not have the same "now" because "now" is not just a single event but a set of events, and the set of events "now" for C is not the same as the set of events "now" for A.


"Age" is a different thing from "now".


No, this is not possible. Think of A, B, and C as all carrying clocks that read their age. At the event where A and C meet, if A and C's clocks both read the same age, that is an invariant; it doesn't matter what frame you choose. And if B, spatially separated from A and C, is at an event which is in the same set of "now" events for A and B as the event where A and C meet, says that his (B's) age--the reading on B's clock--is the same as A's age--the reading on A's clock--at the event where A and C meet, then B's age must also be the same as C's age--the reading on C's clock--at that event, because A's and C's clock readings don't depend on what frame you choose.

This is exactly how I was thinking, but wouldn't the velocity difference and spatial separation mean that C and B would interpret 'now' differently?

 

[Ibix]

So, you're saying that my misunderstanding is that participants in space-time events don't have to agree on "now"?

They will agree on "now at the same place". So if you swing your left hand and your right hand and they meet and make a noise, everyone will agree that your hands were at the same place at the same time. Thus no-one hears a noise with no cause, nor sees a cause with no noise. But they will, in general, disagree that your hands started their swinging motions at the same time - because they are not at the same place there's no immediate physical consequence of whether they start to move simultaneously or not. Only whether they meet - and everyone will agree on that.

 

[curiousburke]

Okay, so I should have been explicit and said "here and now", which A and C will agree on.

I started thinking about this because of the triplet paradox posed by FloatHeadPhysics on youtube. He makes the claim that C would not see A and B as the same age. I just turned this around and asked the question how B would see their three ages. So, is FloatHead wrong, and A B and C will all see their ages being the same? In which case his resolution of the paradox using simultaneity is wrong.

 

[PeroK]

How does this get resolved?

One resolution is to note that global simultaneity (global "now", in your terms) is not a thing. Locally, it looks like simultaneity is a thing. In a race, all the athletes ought to hear the gun at the same time. But, that's a local scenario, where any lack of absolute simultaneity is hardly measurable. If we try to define simultaneity between Earth and Mars, say, then we have the significant (20 minutes plus) time lag of light signals each way. This creates some ambiguity in when precisely "now" on Mars is. There's a 20-minute variation each way.

If we try to define now on Earth and now somewhere in the Andromeda galaxy, then there is a 2 million years ambiguity each way. So, global now is just not a thing.

Note that the clocks on the GPS satellite systems have to take this ambiguity in "now" into account. It's only a fraction of a second, but enough to through the GPS system off if you try to impose a global now on them.

 

[Ibix]

I started thinking about this because of the triplet paradox posed by FloatHeadPhysics on youtube.

Perhaps you could state the triplet paradox that you are talking about, rather than expecting us to hunt through YouTube and watch videos on subjects we already understand.

All (almost all?) "paradoxes" in relativity boil down to failing to apply the relativity of simultaneity correctly.

 

[PeterDonis]

wouldn't the velocity difference and spatial separation mean that C and B would interpret 'now' differently?

The relative velocity is what makes C and B interpret "now" differently. The spatial separation does not affect that, as should be obvious from the fact that A, who is not spatially separated from C at the event where they meet, also interprets "now" differently from C (but the same as B, who is spatially separated from A).

 

[curiousburke]

Perhaps you could state the triplet paradox that you are talking about, rather than expecting us to hunt through YouTube and watch videos on subjects we already understand.

All (almost all?) "paradoxes" in relativity boil down to failing to apply the relativity of simultaneity correctly.

Sorry, you're right. It's basically the scenario I posted with the PeterAdonis' addition of synchronizing the clocks when A and C are at the same here and now. The question is how old are B and C when C gets to B. C is traveling parallel to the line between A and B.

 

[PeterDonis]

C would not see A and B as the same age

More precisely, consider the two different sets of "now" events for A and C that both contain the event where A and C meet. The event on B's worldline that is in A's "now" set shows B's "age clock" reading the same as A's and C's "age clock" at the event where A and C meet. But the event on B's worldline that is in C's "now" set shows B's "age clock" with a different reading from A's and C's "age clock" at the event where A and C meet. This is just a simple consequence of A's and C's "now" sets being different.

Notice how being precise about exactly what is being said helps to resolve any apparent "paradox"?

 

[Dale]

So, you're saying that my misunderstanding is that participants in space-time events don't have to agree on "now"?

That is correct. They agree on “here and now” at the event where they meet, but not “now” even at that event.

 

[curiousburke]

More precisely, consider the two different sets of "now" events for A and C that both contain the event where A and C meet. The event on B's worldline that is in A's "now" set shows B's "age clock" reading the same as A's and C's "age clock" at the event where A and C meet. But the event on B's worldline that is in C's "now" set shows B's "age clock" with a different reading from A's and C's "age clock" at the event where A and C meet. This is just a simple consequence of A's and C's "now" sets being different.

Notice how being precise about exactly what is being said helps to resolve any apparent "paradox"?

I think I do. Like, I ... kinda get it. It's about looking at two different world lines for each question.

 

[curiousburke]

That is correct. They agree on “here and now” at the event where they meet, but not “now” even at that event.

"now" being the entire line, not just this event.

 

[PeterDonis]

It's about looking at two different world lines for each question.

A's and C's worldlines are different, yes. (And B's is different from both.) Each observer's worldline defines "here" for that observer.

 

[PeterDonis]

"now" being the entire line, not just this event.

Yes.

 

[curiousburke]

Thank you all for your insights. It's a difficult concept for me, and really hard to eliminate all vestiges of the global "now" assumption.

 

[Ibix]

Sorry, you're right. It's basically the scenario I posted with the PeterAdonis' addition of synchronizing the clocks when A and C are at the same here and now. The question is how old are B and C when C gets to B. C is traveling parallel to the line between A and B.

So, we have two observers, A and B, at rest relative to each other and a third travels from one to the other. The issue is, presumably, that B sees the traveller's clock tick slow and the traveller sees B's clock tick slow. Which one is older when they meet?

The relevant application of the relativity of simultaneity is to note that the traveller's frame does not see A and B's clocks as synchronised - he says B's clock is ahead of A's. Thus he is unsurprised that B's clock shows a larger value than his own - it was initially ahead, and although it is ticking slowly there has not been enough time for it to fall behind.

 

[curiousburke]

So, we have two observers, A and B, at rest relative to each other and a third travels from one to the other. The issue is, presumably, that B sees the traveller's clock tick slow and the traveller sees B's clock tick slow. Which one is older when they meet?

The relevant application of the relativity of simultaneity is to note that the traveller's frame does not see A and B's clocks as synchronised - he says B's clock is ahead of A's. Thus he is unsurprised that B's clock shows a larger value than his own - it was initially ahead, and although it is ticking slowly there has not been enough time for it to fall behind.

Exactly. So, it made me think: if C sees B's clock as ahead, why doesn't B see C's clock as starting ahead or maybe behind?

 

[Ibix]

Exactly. So, it made me think: if C sees B's clock as ahead, why doesn't B see C's clock as starting ahead or maybe behind?

C sets his clock to match A and everyone will agree that they read the same as they pass. But different frames will have different views on what "at the same time as" means in the question "what time does B's clock show at the same time as C and A meet?". B says his clock shows the same time as A because it's a tautology: they synchronised their clocks by their own definition of synchronisation and therefore they show the same time. C uses a different synchronisation convention, by which A and B's clocks aren't synchronised.

Perhaps you mean to ask why different frames have different definitions of "simultaneous"?

 

[curiousburke]

C sets his clock to match A and everyone will agree that they read the same as they pass. But different frames will have different views on what "at the same time as" means in the question "what time does B's clock show at the same time as C and A meet?". B says his clock shows the same time as A because it's a tautology: they synchronised their clocks by their own definition of synchronisation and therefore they show the same time. C uses a different synchronisation convention, by which A and B's clocks aren't synchronised.

Perhaps you mean to ask why different frames have different definitions of "simultaneous"?

No, I think I understand why simultaneity is frame dependent. I just would expect symmetry between C's perspective on B's time and B's perspective on C's time.

 

[curiousburke]

No, I think I understand why simultaneity is frame dependent. I just would expect symmetry between C's perspective on B's time and B's perspective on C's time.

I'm sorry I'm rehashing: does B view them all (A,B,C) as simultaneous at synchronization but C does not? That's where I am tripped up.

 

[Dale]

Thank you all for your insights. It's a difficult concept for me, and really hard to eliminate all vestiges of the global "now" assumption.

This is indeed the single most challenging concept in SR. You are by far not alone in this challenge. Once you learn this then you will be most of the way to learning relativity.

 

[Ibix]

No, I think I understand why simultaneity is frame dependent. I just would expect symmetry between C's perspective on B's time and B's perspective on C's time.

There is symmetry in the perspectives, but this scenario is asymmetric. You've got one observer using one frame and two observers using the other. So in one frame you haven't defined a second fixed point to make measurements with.

If you add an observer, D, at rest with respect to C and as far away in their mutual rest frame as A and B are in their frame, you can ask how old A or B is when they get to C (or D, whichever they meet later). That'll give you the same answer, which is the symmetry you expect.

 

[curiousburke]

There is symmetry in the perspectives, but this scenario is asymmetric. You've got one observer using one frame and two observers using the other. So in one frame you haven't defined a second fixed point to make measurements with.

If you add an observer, D, at rest with respect to C and as far away in their mutual rest frame as A and B are in their frame, you can ask how old A or B is when they get to C (or D, whichever they meet later). That'll give you the same answer, which is the symmetry you expect.

Ironically, that's exactly what I tried to think about when I got really confused. If frame 1 is contracted wrt frame 2 and frame 2 is contracted wrt frame 1, how can A be next to C and B be next to D? That's a rhetorical question, not meant to be answered, which would take this tread down a different path.

 

[Ibix]

Ironically, that's exactly what I tried to think about when I got really confused. If frame 1 is contracted wrt frame 2 and frame 2 is contracted wrt frame 1, how can A be next to C and B be next to D? That's a rhetorical question, not meant to be answered, which would take this tread down a different path.

Look up Minkowski diagrams. I can draw one for this scenario later. I find them very helpful - essentially the answer is that when two frames say they are measuring "distance" they are measuring different lines in spacetime, and there end up being four different things being measured when you say "they measure the distance between them". There would only be a contradiction if they were measuring one or two things.

 

[Dale]

Ironically, that's exactly what I tried to think about when I got really confused. If frame 1 is contracted wrt frame 2 and frame 2 is contracted wrt frame 1, how can A be next to C and B be next to D? That's a rhetorical question, not meant to be answered, which would take this tread down a different path.

Because of things like that, a spacetime diagram can be a very helpful tool in both organizing your thoughts and communicating them to others.

 

[PeroK]

Ironically, that's exactly what I tried to think about when I got really confused. If frame 1 is contracted wrt frame 2 and frame 2 is contracted wrt frame 1, how can A be next to C and B be next to D? That's a rhetorical question, not meant to be answered, which would take this tread down a different path.

You have to be careful with such specious arguments. That means seemingly plausible, but actually wrong. (It's a good word to know!) Consider this: two ambulances are racing in opposite directions towards each other. The first ambulance hears the second's siren at a higher pitch. And, the second ambulance hears the first's at a higher pitch. If someone suggests that is a contradiction, you can examine the basis of that argument. In this case, however, you are probably convinced from the outset that there is no contradiction - even if you can't immediately see the flaw.