Relativity 105d: Acceleration - Twin Paradox and Proper Time Along Curves (Rindler Metric)

Previous videos on the calculating the lengths of curves in Tensor Calculus series:

0:00 Intro
1:43 Proper Time review
4:09 Proper Time for non-inertial worldline
6:20 Twin Paradox Calculation
8:11 Reversed Triangle Inequality
12:03 Twin Paradox Resolved
14:39 Proper Time along Curves
18:04 Example calculation (inertial coordinates)
22:27 Example calculation (Rindler coordinates)
27:04 Proper Time Curves Conclusion
30:01 Summary

[eigenchris]

Error at 25:00, the components of ex-tilde should be cosh and then sinh, but I have written them as sinh and then cosh by accident.

[eigenchris]

Re-uploaded from a couple days ago because someone caught a math error.

[azeds]

I dont know if someone told you this but man you are made to explain math
The tensor series i downloaded it . Its a tresor . Finally i wish from you yhat you publish a post recommending books for all math . Thanks

[lrlrch]

Thank you for all of these videos. I know that a lot of people must say this, but you deserve more subscribers. Your videos inspired me to study more physics and maths :)

[oleg-avdeev]

Thank you so much for these videos! I found very enlightening the visual demonstration of how the zig-zag non-inertial frame of reference fails to "flatten" under Lorenz Transform.  

I think that this is the very source of confusion that people have with the Twin's Paradox—Galilean Transform is capable of straightening these kinks, and it is our default intuition for frames of reference, so we fail to imagine that this is not a fundamental property of any Relativity, but very specifically a consequence of Galilean Relativity assumptions.

I also find it cute how Galilean spacetime sits in the middle of space and Minkowsky spacetime triangle inequalities, with ||Sb1 + Sb2|| being always equal to ||Sb1|| + ||Sb2||, and how this relates to Complex, Hypercomplex and Dual numbers. Makes one wonder what a "Complex"-based (completely un-physical) relativity could look like!

[ProfeARios]

One of the best explanations I've watched about the Twin Paradox. Thank you so much for sharing. Regards from the Republic of Panama!

[patriciacosson]

Very nice video as usual félicitation for your great job

[narfwhals]

You don't _need_ acceleration to resolve the twin paradox. If you treat Bob as two different inertial frames you get the same result. The physical explanation is then that the receding and the approaching frames do not see equal halves of Alice's worldline. There is a "simultaneity gap" between the two frames.

[Archigamers]

Thank for ur work, you are really clear, I understand a lot more of relativity compare to any other media.
(Jean Pierre Petit is also really good, but he is french)

[partakerofbread]

Now see your explanation on the two definitions of proper time tau. 😊

It's not an easy topic to explain, but I think you are doing a great job!

It would help if you could explain how this is a leading into general relativity, but otherwise your explanations are very thorough.

JS

[ericbischoff]

27:00 lambda final -> lambda initial
As always, thanks for the great video

[Mysoi]

Thank you! now I have a clear understanding of the spacetime interval. It is a 4-position vector in spacetime that measures the distance between two events. It is distinct from proper time, just like how the integral of dR/dlambda dlambda gives us the arc length L, not R.

[vitrums]

31:59 wow, it took some time to make it click for me. But I finally figured, that proper time measures time of events on a world line. So by looking at a space-time diagram even if we notice a segmented world line turning different ways throughout its development in time, then it still measures spatial deviation = 0 along the whole way. All we have to do is to imagine ourselves in rest along such world line and we'll start to see how it's the whole world that starts to shift left and right instead. And the cool part about this thought experiment is that we essentially become non-inertial Bob observing inertial Alice. So even though it's Alice who's in motion not us, we still measure less time than her.

[narfwhals]

I think an interesting thing to do with this video would be combining the two topics you discuss. What does the twin paradox look like with a twin who is constantly accelerating towards earth? (aka gets thrown up and falls back down) In that case the acceleration _has_ to be the cause of the age difference since there is no inertial frame to consider. This could also be a nice tie in to General Relativity later.

[pedrolopa]

these videos are great! I just wish you went into a bit more detail in the twins paradox, especially because we've learned that sr can handle acceleration in flat spacetime. How to calculate alice's proper time from bob's perspective during his acceleration? Thank you

[dantoro]

If I understand correctly, the twin paradox cannot be solved without involving an acceleration. If Alice and Bob have different speeds relative to each other, even if they are both in a frame inertial, they will never agree on their respective proper times unless they go through the Lorentz transformation. And to compare the two clocks side by side, one of the two must accelerate or decelerate to reach the other and thus pass through a non-inertial frame.
However, if Alice and Bob have constant velocities and are converging on each other, at some point they will cross paths and their proper times will coincide without any acceleration. How do we represent this on the space-time diagram?

[slobodannedic]

Related to 20:00+, it should be noted that one cannot evaluate length of a ‘line’ which is not straight or circular unless one has such parameterization to ensure that the each an ver increment is of the same length, that is the tangential velocity is constant; good indication of this is that the surface of an ellipse cannot be integrated to ab times pi - by integrating r^2 times delta-angle, unless the angle’s time-parameter dependence is not determined by the Kepler’s Equation. That ver well might be the problem in still unsolved closed form expression for the perimeter of an ellipse … !?!

[WildGamez]

Very interesting that frames of reference are mathematical...whether minkowski or rindler

[freydrik]

Velocity is not completely relative… it is limited by c. So it seems that nature can sort things out in terms of breaking the symmetry, knowing who is moving faster, as on such absolute velocity scale.

One of the major successes of SR is the explanation of the muons detected on Earth and explained by fast moving muons time dilation or proper time increase. But this could be interpreted in the reference frame of the muon as if the Earth went into it at very fast speed, near c… not affecting the muon’s proper time, but instead dilating the time of Earth (and of the entire universe actually).

[A-_--]

​ @eigenchris
For some reason my replies aren't appearing (thanks youtube) so I'll post it here instead. A topological 3-torus can still be flat, but yes, a similar situation could arise if the universe has (roughly) constant positive curvature. Maybe you missunderstood my post, but I wasn't talking about any of the twins being in the vicinity of a massive body per se (well, I guess the twin on eath is), but rather the large scale structure of the universe. If the universe were a flat 3-torus, you could walk in a straight line and return to the place you started at.