B Triplets Paradox: Is There a Solution?

[PeterDonis]

I mean the point on the wordline of B that appears to be simultaneous with Aa1, from the perspective of A.

Ok.

As for the diagram, I am picturing it, it's just that it requires 2 dimensions for space

This is a huge increase in complication, because the two Lorentz boosts in question--from A's rest frame to B's rest frame, and from A's rest frame to C's rest frame--are in different directions. That means the transformation from B's rest frame to C's rest frame, or vice versa, is not a pure Lorentz boost; it's a boost plus a spatial rotation. I doubt that anyone's powers of visualization are up to handling that by intuition. Cases like this need to be worked by grinding through the math. But as @Dale has pointed out, before even trying that, you need to be sure you have a firm grasp of the simple 2D (1D space + time) case.

You can at least get a start on the 3D (2D space + time) case by looking at the two obvious edge cases, though: the case where B and C are moving in the same direction (in which case they are at rest relative to each other--easy), and the case where B and C are moving in opposite directions, in which case the relative speed between B and C can be obtained from the simple relativistic velocity addition formula.

It is all straight lines though, right?

As I understand things, you are specifying that the objects always move inertially, correct? If that is the case, then yes, all of the lines of interest will be straight lines.

 

[PeterDonis]

Depending on the angle between those trajectories, we have a varying relative velocity between B and C.
Ignoring A, we calculate γ=1.5 between B and C

Where are you getting 1.5 from? Or are you just specifying it arbitrarily?

 

[villy]

Just an arbitrary number to account for a difference in velocity.

 

[PeterDonis]

Just an arbitrary number to account for a difference in velocity.

How do you know that you can even achieve $\gamma = 1.5$ between B and C, given that $\gamma = 2$ between A and B, and between A and C?

 

[villy]

How do you know that you can even achieve $\gamma = 1.5$ between B and C, given that $\gamma = 2$ between A and B, and between A and C?

As far as I can tell it could be anything varying from 1, for an angle of 0 to a value that has to be larger than 2 for an angle of 180.

You are correct though, this has gotten quite further away from where I'm comfortable with the math.
When I said I'm visualizing it, I only meant the simple diagram from the perspective of A, with two symmetrical straight lines going out. I wasn't claiming I'm doing any transformations with it.
Just imagining the relative positions of the points based on the values (33<50).

 

[PeterDonis]

As far as I can tell it could be anything varying from 1, for an angle of 0 to a value that has to be larger than 2 for an angle of 180.

Ok. But you can calculate the value for the angle of 180, since that's the case where C and B are moving in opposite directions, so it's actually a 2D (1D space + time) case and you can apply the simple relativistic velocity addition formula to get the relative speed between B and C, and calculate their $\gamma$ from that.

 

[PeterDonis]

When I said I'm visualizing it, I only meant the simple diagram from the perspective of A, with two symmetrical straight lines going out.

If they're symmetrical, then in terms of proper time of B and C, they should both have the same time dilation factor relative to A.

Just imagining the relative positions of the points based on the values (33<50).

I don't understand where the value of 33 comes from.

 

[villy]

Can I ask this, for clarification?
There is a way to establish simultaneity between 2 points on two world lines right?
If one point is simultaneous with another (on different lines obviously) does it follow that the reverse is true?

I don't understand where the value of 33 comes from.

Choosing a time interval of 50 years for B and calculating the corresponding value for C, just γ=1.5

 

[PeterDonis]

There is a way to establish simultaneity between 2 points on two world lines right?

No. You don't "establish" simultaneity because it's not something that is given to you by physics. It's a convention that you adopt. You have to specify the simultaneity convention you're using when you specify a scenario, if you want to make any statements about simultaneity in that scenario.

If one point is simultaneous with another (on different lines obviously) does it follow that the reverse is true?

This question is unanswerable until you specify what simultaneity convention you are using.

Choosing a time interval of 50 years for B and calculating the corresponding value for C, just γ=1.5

But that makes no sense, because $\gamma = 1.5$ is a time dilation factor between B and C, but the 50 years for B is based on the time dilation factor between A and B. Those are different things and you can't mix them together and expect to get anything meaningful.

 

[villy]

Right, I can see how my phrasing is quite bad.
What I mean is this:

We can analyze the A B system, from A's perspective, ignoring C.
If we assume (arbitrarily) a value of Δt=100 years, we can calculate where B will be in space, and how much time will have passed for B (Δτ=50 y), based on the known relative speed.

We can perform a similar analysis, for the B C system, from B's perspective, ignoring A.
We also know their relative velocity, whatever it is, corresponding to γ=1.5 .
If we arbitrarily choose a Δt'=50 y for B, we'll get Δτ'=33 y for C.

Am I still making some mistake?

Edit: I understood this doesn't describe the same moment in any meaningful way.

 

[Dale]

Right, I can see how my phrasing is quite bad.
What I mean is this:

We can analyze the A B system, from A's perspective, ignoring C.
If we assume (arbitrarily) a value of Δt=100 years, we can calculate where B will be in space, and how much time will have passed for B (Δτ=50 y), based on the known relative speed.

We can perform a similar analysis, for the B C system, from B's perspective, ignoring A.
We also know their relative velocity, whatever it is, corresponding to γ=1.5 .
If we arbitrarily choose a Δt'=50 y for B, we'll get Δτ'=33 y for C.

Am I still making some mistake?

Edit: I understood this doesn't describe the same moment in any meaningful way.

From this we can determine that the speed of B in A's frame is $\sqrt{3/4} \ c$ and the speed of C in B's frame is $\sqrt{5/9} \ c$. And if I understand your setup for this current scenario, all three coincide at the origin. And if I understood your setup it is not necessarily the case that the boost from A to B is in the same spatial direction as the boost from B to C. As such, there is no way to determine the speed of C in A's frame.

You have arbitrarily chosen an ending event on A, but that will not correspond to a unique event on B or C. You can choose events on B and C and assert that they match somehow, but it is not a physically meaningful thing.

 

[villy]

No I set C up to have the same relative speed with A as B did, but moving in a different, unspecified direction, such that we get the appropriate speed between B and C.

 

[PeterDonis]

We can analyze the A B system, from A's perspective, ignoring C.
If we assume (arbitrarily) a value of Δt=100 years, we can calculate where B will be in space, and how much time will have passed for B (Δτ=50 y), based on the known relative speed.

Yes.

We can perform a similar analysis, for the B C system, from B's perspective, ignoring A.
We also know their relative velocity, whatever it is, corresponding to γ=1.5 .
If we arbitrarily choose a Δt'=50 y for B, we'll get Δτ'=33 y for C.

Yes.

Am I still making some mistake?

No. But:

I understood this doesn't describe the same moment in any meaningful way.

Exactly. Your two analyses are both correct, and have nothing whatever to do with each other.

 

[PeterDonis]

there is no way to determine the speed of C in A's frame.

Actually, I think there is, but it's not straightforward.

 

[Dale]

No I set C up to have the same relative speed with A as B did, but moving in a different, unspecified direction, such that we get the appropriate speed between B and C.

Ah, I missed that. Yes, I think that there is a way to figure that out, but it would probably take me an hour or two using mathematica to figure it out. I don't think this will be a fruitful approach for someone just learning this material.

 

[villy]

I'm still missing something. I see no need for advanced calculations.
Looking at B, A can calculate then when A measures the passage of 100y, B will have experienced the passage of 50 y (and it's relative position in space).
Looking at C, B can similarly calculate then when B measures 50 y, C will have experienced 33 y.
How does this not refer to the same moment for B? When their clock measures 50 y.

 

[Dale]

I'm still missing something. I see no need for advanced calculations.
Looking at B, A can calculate then when A measures the passage of 100y, B will have experienced the passage of 50 y (and it's relative position in space).
Looking at C, B can similarly calculate then when B measures 50 y, C will have experienced 33 y.
How does this not refer to the same moment for B? When their clock measures 50 y.

That is the relativity of simultaneity. Simultaneity is frame variant.

You actually don't need to bring C into this at all. This happens just between A and B. When A measures the passage of $100 \mathrm{\ y}$, then at that same time in A's frame B measures $50 \mathrm{\ y}$. But when B measures the passage of $50 \mathrm{\ y}$, then at that same time in B's frame A measures $25 \mathrm{\ y}$. So one event (B measures $50 \mathrm{\ y}$) is simultaneous in A's frame with A measuring $100 \mathrm{\ y}$ and is simultaneous in B's frame with A measuring $25 \mathrm{\ y}$.

The key is that the phrase "at that same time" means different things in A's frame vs in B's frame. They do not agree on which events happen "at the same time". That is the relativity of simultaneity.

 

[PeterDonis]

I'm still missing something. I see no need for advanced calculations.

Then yes, you are definitely still missing something.

Looking at B, A can calculate then when A measures the passage of 100y, B will have experienced the passage of 50 y (and it's relative position in space).
Looking at C, B can similarly calculate then when B measures 50 y, C will have experienced 33 y.
How does this not refer to the same moment for B? When their clock measures 50 y.

Here is a hint at what you are missing: the two bolded "when"s in the above mean different things and cannot be directly compared.

 

[villy]

Yeah, I finally got what I was missing.
Out of curiosity, what was the part that required advanced calculations?

 

[PeterDonis]

what was the part that required advanced calculations?

To actually calculate the speed of C relative to A.

 

[villy]

Thanks for all the answers.

 

[Ibix]

A useful analogy for all of this is a couple of straight roads on a Euclidean plane. The roads cross and make an angle $\theta$ between them, and they have mile markers on them including one at the crossing point.

If you get to the first mile marker and look across you'll discover that you are level with the mile marker numbered $1/\cos\theta$ on the other road. This is true whichever road you are on, and it is not contradictory because people on the two roads are using different notions of "level with". They are both defining "level with" to mean "lying on a line perpendicular to my road", but since their roads are not parallel this produces notions of "level with" that are not parallel.

That's the underlying truth of time dilation. The roads diverging at angle $\theta$ become worldlines on a Minkowski plane diverging with rapidity $\psi=\mathrm{tanh}^{-1}(v/c)$, and the "level with" lines become "simultaneous with" lines, and the Euclidean perpendicular and distance becomes the Minkowski orthogonal and interval. But the insight is the same - time dilation just comes from projecting regularly spaced lines orthogonal to your own path onto a path that is not parallel.

The complication when you add another dimension is just that the lines orthogonal to your path become planes orthogonal to your path. If B wants to calculate where his planes intersect C's path, all he needs is the angle/rapidity between those paths. But you haven't directly specified that - you have specified the angle/rapidity between the paths of A and B and of A and C, and the angle between the projections of B's and C's paths onto A's spatial planes. There's enough information there to do the calculation, but some rotation/boost matrices are involved.

This is also an insight into why naive time dilation calculations don't work for accelerating observers. Their paths are curved, so a plane that is locally orthogonal to the path at one point/event inevitably cross planes orthogonal to the path at other points/events - so they end up either missing out or double counting parts of other paths. An inertial observer has a straight path and may analyse an accelerated observer's path this way, but not vice versa.