Relativity 105b: Acceleration - Bell's Spaceship Paradox and Rindler Coordinates

0:00 Intro (Bell's Spaceship Paradox)
1:49 Resolving the paradox
6:04 Maintaining Distance during Acceleration
10:34 Rindler Coordinates
13:38 Spaceship clocks tick at different rates
17:11 Proper acceleration for each ship
19:45 Spacetime interval Rindler Coordinates
21:04 Rindler Coordinate reverse transformation
22:34 Summary
24:15 Speed of light not equal to c

[dXoverdteqprogress]

Every few years when I return to this topic and have to think about it again, it breaks my brain a little. Well done!

[mot]

Thanks very much for your clear explanations and great graphics. Unlike most of the people viewing your videos, I have learned all my math and physics by myself and was not a math or physics major. But I find math and physics very interesting and challenging. I takes me a while to get this to soak into my brain but with your superb explanations it is sinking in slowly but surely. Thanks for your clear explanations.

[mariolemelin]

These videos are fantastic. I can't wait for 105c and 105d. I like to mention that you tell about 105c in 105d around 15:00. Have a wonderful Christmas and a Happy new year 2021

[zubetto]

Thank you for such a great, detailed and consistent explanation!

[deadinsider]

I'm still waiting when you will finish Special Relativity and start General Relativity, then wait until you will finish General Relativity. Really glad that you are making the videos it helps me to learn about Relativity, because books are too formal for me to understand them :D Really thank you for the videos

[dyachenkotimofey]

I watched some videos about relativity and didnt understand some things, but you helped me. THANK U SO MICH

[ivanq]

Very well explained. Just one minor thing. At 11:37, the slide states (ct~, x~) are coordinates for observer traveling with proper acceleration alpha. It doesn't much sense because (ct~, x~) are just coordinates with an associated equation of motion. As a matter of fact, as derived later in the video, (ct~, x~) along a constant x~ has a proper acceleration of alpha D/x~

[camorimd]

Thanks a lot for your vids, they are amazingly explanatory :)

[AnthonyStauffer-rw]

I've finished your Tensor Calculus series, which was EXCELLENT, and decided to watch this series straight through without taking notes. I'm a middle-aged science nut who never had the chance to study this stuff in college. Anyway, does the light cone in inertial coordinates translate to the Rindler Horizon? I was going through my mind about switching from an inertial reference frame to an accelerating one and suddenly had that question. If you accelerated from an inertial frame into a Rindler frame, would you, in essence, cross the light cone?

[William]

Thank you for this marvellous video series! I am now looking forward to a video that handles relativistic rotation and the Ehrenfest paradox. In addition to that, I am also very curious about how a rolling circle (which has exactly one zero-velocity point at any given moment) or a ball would look like in the eye of a stationary observer in SR due to length contraction.

[nityadas]

Dear Chris, thanks for uploading 105b.
Best Wishes and Regards
----Dr. N Das

[foxhound]

Another very informative video. So Einstein, the inertial observer in your scenario, would see the distance between the spaceships remain the same, since in Einstein’s frame the acceleration was simultaneous, but Eisenstein will see the string break because the string length contracts, and is no longer spanning the distance between the spaceships. An observer in the rear spaceship would see the front spaceship start accelerating first, and therefore increasing the distance, and breaking the string. Ok, my question is on not two separate spaceships, but one long spaceship with the rocket at the back (bottom). When the rocket motor fires, momentum must be transferred, thru internal stresses, from the back of the spaceship to the front (since the front has no rocket motor of its own). In effect, the front of the spaceship is being “pushed” by the back. My question is: since time at the front of the spaceship is running faster then the back of the spaceship (according to a observer n the back) are there additional “relativistic stresses” imposed on the structure? Meaning, in addition to the “non-relativistic stress” of the back of the spaceship pushing the front, is there additional load imposed by time running faster at the front, but the front not having its own dedicated source of acceleration? (The front is relying on the back pushing, transferring momentum, to it.)

[narfwhals]

This was super interesting and the implication of the Equivalence Principle give an immediate connection to why the Rindler Coordinates look very much like the Kruskal coordinates for a black hole.
It is interesting to note what the lower Rindler Horizon represents. The accelerated observer can _only_ receive signals from behind it and never send one there. Just like a white hole.
But doesn't the event horizon of a black hole also vanish for the local inertial observer?

[pinklady]

You ae Heaven-sent. Thank you for all your tutorials.

[imaginingPhysics]

My wish for this Christmas and the next year is a video either of Thomas rotation or Ehrenfest paradox.
Thank you in any case. Merry Christmas.

[ryam]

You've missed an important lesson of Bell's ship experiment, which is that in certain reference frames the LF contraction is a dynamical effect taking place in the material structure of moving bodies, and is not a consequence of mere change of coordinates from one reference frame to another. There's no necessity to prove the tearing of the string in a comoving frame. The rest frame can allow it perfectly well when taking in account dynamical contraction.

[mehtabkamal]

Hi Chris, at the end of video you said that x~ is not always equal to Lo for non-inertial frames. However, in the video you showed that it is always equal to Lo since spacetime interval in Rindler's coordinates is independent of ct. Is this possibly an error, or is it something that will be seen in later videos?

[Amit-gi]

Your mathematical rigour on SR is at its highest

[Kiscorpion]

Hi @eigenchris . Many thanks for the video. Can you please help with the following:

Let us remember the familiar example of train and passenger: Observer, standing still on the perron at the station (frame S), train (frame S') moving with constant velocity relative to S, and the passenger aboard the train moving with constant velocity relative to S'. This is one the examples used to calculate Einsteins addition formula to find the passenger's velocity relative to S.

What would be the addition formula when

1) The train is moving with constant proper acceleration relative to S, while the passenger moves with constant velocity relative to S'

2) The train is still moving with constant velocity relative to S, but the passenger is moving with constant proper acceletation relative to S'

I cannot find the answer in books.

Thanks for your help.

Best Regards

[williamtfinnegan]

Thanks for the great set of videos. This video seeems to leave a few loose ends.
1. Suppose we substitute a fixed length rod between the space ships instead of a string. It would seem as the speed of the assembly increases, the length of the rods would shrink as viewed from the starting gate.

2. If the space ships all accelerate at the same rate, does the relative speed between each of them stay at 0 or does it change?

3. The distance "D" mentioned throughout the piece is confusing. What is the reference frame for measuring D? Is D an instantaneous value or only a starting value? Is D a specific numeric value regardless of the reference frame?

4. Not clear in the piece whether the acceleration is substantial

Thanks for the great videos.
Just a newbie..