B Understanding Special Relativity: 2nd Year SR Course Explained

[starstruck_]

Hey everyone!

So, I just finished my 2nd year intro to SR course,
we spent most of the term on Taylor expansions for relativistic corrections vs concepts. I’ve watched a crash course video and a documentary on SR and I do somewhat understand the thought experiment involved with respect to simultaneity and such (the train and the lightning strikes) but when it comes down to more formal stuff like what’s in the Feynman lectures ( and applying those concepts to a scenario) it makes no sense. I have a bit of a inking for how I should approach a certain SR scenario but I just can’t seem to break it down.
It seems so unintuitive. I have more questions than answers.

There was an example a grad student gave us with a magician and 2 knives. The magician is traveling very fast as the knives go up and then down and the magician comes out alive.

This is my understanding of it

According to the postulates (I’m assuming ?) the same events happen in both reference frames. The magician must be alive in both reference frames (of the magician and an outside observer) and the events - knife 1 going up and then down and knife 2 going up and then down must happen in both reference frames too.

This is where my understanding diwndles. The way the diagram was set up for this scenario was similar to the train thing with the lightning strikes so that’s kind of how I tried to approach my explanation .

To the outside observer, both knives would go up and then down at the same time. For the magician however, there should be a lag I think. I just can’t understand why. I tried explaining it the same way that train scenario is explained but it’s not a complete answer.

I’m so confused. I know this has to do with length contraction and time dilation? But I just can’t figure out how to apply those concepts to this scenario. Any explanation will be appreciated!
(Please just assume I don’t know anything about special relativity).

 

[Ibix]

This experiment is usually called the "ladder and barn" or "rod and barn" paradox. The resolution involves understanding that two things that happen at the same time according to one frame of reference do not happen at the same time in other reference frames. You can see this from the time Lorentz transform: $t'=\gamma(t-vx/c^2)$. If two events happen at the same time $t$ at different $x$ then the other frame measures different $t'$.

That's what's happening in this case. The knives always brush the magician's hair and feet. In any frame where he's not exactly the right length to fit exactly between the blades, they do not fall simultaneously.

I strongly recommend looking up Minkowski diagrams. They are just displacement-time graphs, but you can represent more than one frame at a time and develop intuition about how the transforms work.

 

[PeroK]

There was an example a grad student gave us with a magician and 2 knives. The magician is traveling very fast as the knives go up and then down and the magician comes out alive.

According to the postulates (I’m assuming ?) the same events happen in both reference frames. The magician must be alive in both reference frames (of the magician and an outside observer) and the events - knife 1 going up and then down and knife 2 going up and then down must happen in both reference frames too.

This is where my understanding diwndles. The way the diagram was set up for this scenario was similar to the train thing with the lightning strikes so that’s kind of how I tried to approach my explanation .

To the outside observer, both knives would go up and then down at the same time. For the magician however, there should be a lag I think. I just can’t understand why. I tried explaining it the same way that train scenario is explained but it’s not a complete answer.

I’m so confused. I know this has to do with length contraction and time dilation? But I just can’t figure out how to apply those concepts to this scenario. Any explanation will be appreciated!
(Please just assume I don’t know anything about special relativity).

First, an event either takes place or it doesn't. It doesn't take place "in a reference frame". Reference frames have nothing to do with whether a magician lives or dies or whether any event takes place or not.

A reference frame gives coordinates (time and space) to an event.

Note also that if two events coincide in time and space in one reference frame, then they coincide in time and space in all reference frames. For example, a knife reaching a point just as a magician's chest reaches the same point. In fact, you can say that if two events coincide in time and space then they are in fact the same event.

The magician's trick, as I think you see, is easy to explain in the frame in which the apparatus is at rest. The magician is length contracted and fits between the knives.

What that means is that the magician definitely survives. You need only analyse the probem in one frame to find out what happens. Whether the magician lives or dies is not frame dependent.

Or, if the trick goes wrong and the magician gets knived, then again that is not frame dependent. And, if you analyse the problem in any frame, you must come to the same conclusion that he gets knived.

The difficult aspect of this is, of course, to describe a sequence of events in the magician's frame fo reference. Or, perhaps more precisely, in a reference frame in which the magician is at rest. The worrying thing from the magician's point of view is, of course, that owing to length contraction, the two knives are very close together, as the distance between them is length contracted.

You can either, as @Ibix suggests, hit the problem with the Lorentz Transformation. Or, you could draw a spacetime diagram. Or, you could think it through using relativity of simultaneity (that's the crucial one) and length contraction and time dilation.

Note that a lot of people seem only to pick up on time dilation and length contraction and forget about the relativity of simultaneity. For example:

I’m so confused. I know this has to do with length contraction and time dilation?

This seems to be a general problem, so my advice is to think in threes: time dilation, length contraction and RoS.

A useful mnemonic for RoS is the "leading clocks lag" rule. In this case, in the magician's frame, a timer attached to the knife closer to the magician lags behind a timer attached to the further clock. The sequence of events in the magician's frame has the further knife fall first, then some time later, the near knife. Does that give the magician time to get past the second knife before it falls?

 

[starstruck_]

First, an event either takes place or it doesn't. It doesn't take place "in a reference frame". Reference frames have nothing to do with whether a magician lives or dies or whether any event takes place or not.

A reference frame gives coordinates (time and space) to an event.

Note also that if two events coincide in time and space in one reference frame, then they coincide in time and space in all reference frames. For example, a knife reaching a point just as a magician's chest reaches the same point. In fact, you can say that if two events coincide in time and space then they are in fact the same event.

The magician's trick, as I think you see, is easy to explain in the frame in which the apparatus is at rest. The magician is length contracted and fits between the knives.

What that means is that the magician definitely survives. You need only analyse the probem in one frame to find out what happens. Whether the magician lives or dies is not frame dependent.

Or, if the trick goes wrong and the magician gets knived, then again that is not frame dependent. And, if you analyse the problem in any frame, you must come to the same conclusion that he gets knived.

The difficult aspect of this is, of course, to describe a sequence of events in the magician's frame fo reference. Or, perhaps more precisely, in a reference frame in which the magician is at rest. The worrying thing from the magician's point of view is, of course, that owing to length contraction, the two knives are very close together, as the distance between them is length contracted.

You can either, as @Ibix suggests, hit the problem with the Lorentz Transformation. Or, you could draw a spacetime diagram. Or, you could think it through using relativity of simultaneity (that's the crucial one) and length contraction and time dilation.

Note that a lot of people seem only to pick up on time dilation and length contraction and forget about the relativity of simultaneity. For example:
This seems to be a general problem, so my advice is to think in threes: time dilation, length contraction and RoS.

A useful mnemonic for RoS is the "leading clocks lag" rule. In this case, in the magician's frame, a timer attached to the knife closer to the magician lags behind a timer attached to the further clock. The sequence of events in the magician's frame has the further knife fall first, then some time later, the near knife. Does that give the magician time to get past the second knife before it falls?

Ahhhhh okay, I know how to do the lorentz transformations. I can understand it with the transformations but not conceptually for some reason. Like instead of doing the transformations, if I had to reason it out, I wouldn’t be able to.

About the RoS, I brought that up as my initial way of approaching the scenario but that was apparently only part of the answer.

I don’t think it would give the second knife time to fall. If there is lag falling down between the two knives, there’s a lag going up. So by the time he can actually get between the two knives, the second one should have already fallen (?)

 

[PeroK]

Ahhhhh okay, I know how to do the lorentz transformations. I can understand it with the transformations but not conceptually for some reason. Like instead of doing the transformations, if I had to reason it out, I wouldn’t be able to.

About the RoS, I brought that up as my initial way of approaching the scenario but that was apparently only part of the answer.

I don’t think it would give the second knife time to fall. If there is lag falling down between the two knives, there’s a lag going up. So by the time he can actually get between the two knives, the second one should have already fallen (?)

You could always post this problem in the homework section.

In general, you are often given data as measured in one reference frame and want to transform that data to a second reference frame. Lorentz does it all, but if you do it by hand, then you need to contract lengths, dilate the time and desynchronise the clocks from the original frame.

Another useful exercise is to derive the Lorentz Transformation from contraction, dilation and simultaneity.