# Feedback Requested: Special Relativity Simplified by Phil

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• Philip Dhingra
In summary, the conversation discusses a draft of a Medium article simplifying Special Relativity using a deck of cards analogy. While one person finds the analogy to be good, another points out some inaccuracies in the physics, such as the use of Euclidean geometry in Minkowski space and the misunderstanding of time dilation. The conversation also highlights the importance of considering the relativity of simultaneity and the difference between the twin paradox and time dilation.
It looks pretty good. I like the analogy with the deck of cards. I did have one minor quibble: you said, "every particle in your body is moving forward at the speed of light." Is that what you intended? I don't think that statement is quite accurate. Otherwise, a perfect article.

Unfortunately, you are butchering the physics. The geometry in Minkowski space, which is what you are trying to reproduce, is not based on the Pythagorean theorem, but on a modified version with a minus sign, namely the spacetime interval ##\Delta s^2 = c^2 \Delta t^2 - \Delta x^2##. Furthermore, the direction of the slices of simultaneity in the different frames differ so the card representing simultaneity in one frame will not represent simultaneity in another frame. Your card analogy works only for Galilean spacetime. In SR, the simultaneities of the moving observer would contain a line from each of the cards of the stationary observer.
I did have one minor quibble: you said, "every particle in your body is moving forward at the speed of light." Is that what you intended? I don't think that statement is quite accurate.
It is accurate taken as the statement that the 4-velocity is normalised to the speed of light.

Philip Dhingra
The partiçles in your body do not move at the speed of light. Period.

The partiçles in your body do not move at the speed of light. Period.
You clearly are not familiar with the 4-vector formalism. All 4-velocities are normalised to the speed of light and this gives you the spacetime interval per unit proper time. Of course this is clearly different from an object having 3-speed being the speed of light.

The statement you are arguing against is about the time-speed, not the 3-speed. This is not saying anything else than that ##c\Delta t/\Delta\tau = c##, ie, relating the change in coordinate time to the change in proper time for a stationary observer.

Otherwise, a perfect article.
is clearly false as applying Euclidean geometry to spacetime is wrong and misleading.

Philip Dhingra
Towards the beginning you say
According to special relativity, time slows down noticeably if you move really fast. And this isn’t just theoretical. GPS satellites that move away or towards each other will have their clocks out of sync, enough to cause navigation errors. Or if you were to send a watch on an airplane that flies around the world, when it returns, it would be slightly behind the watch on your arm.
That "time slows down if you move fast" bit is not exactly wrong, but it obscures the distinction between two very different phenomena and it is commonly misunderstood (it's the source of what is probably the single most frequent misunderstanding we see in this forum).

Instead of force-feeding you another way of phrasing it, I'll try asking you questions that are raised and not clearly answered by your article.
1) Symmetry of time dilation: If you are moving, I will correctly say that your clock is running slower than mine and time is passing more slowly for you. However, you can just as correctly take the position that you are at rest while I am moving away from you, so in fact my clock is running slower than yours. How do we reconcile these two equally correct but apparently mutually contradictory claims? (As an aside, this problem is somewhat baked into your metaphor of the cards because it suggests that there is a physical difference between the two decks - one is stacked straight and one is slid sideways).
2) You are right about the airplane watch coming back slightly behind your wristwatch - all observers regardless of their state of motion have to agree about what the two watches say when they are side by side, and it turns out that the airplane watch will be behind.. But because of time dilation, someone on the airplane will correctly find that at every moment of their journey your wristwatch (moving relative to them) is slower than the clock on the plane. So how can the airplane watch end up behind?

The resolutions will be found in two things that your article doesn't really touch on: First, the difference between the twin paradox and time dilation; and second, the relativity of simultaneity. @Orodruin's point about the spacetime geometry not being Euclidean is also important - that and not the relative motion is why the airplane watch is the slow one in this case.

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Philip Dhingra
The partiçles in your body do not move at the speed of light. Period.
Orodruin said:
You clearly are not familiar with the 4-vector formalism.
Stuff like this is reason why it's really hard to get the physics right without using the math. The squared norm of the four-velocity is indeed ##c^2## (after we've sidestepped the sign convention distraction) and I expect that you both agree about that. Whether it's a good idea to attach the English word "move" to that statement is a different question altogether.

Nugatory said:
Whether it's a good idea to attach the English word "move" to that statement is a different question altogether.
I would say that it is not, but I was trying to be generous in my interpretation ... My main point stands that the post tries to apply Euclidean geometry instead of Lorentzian geometry and therefore is doomed to failure.

Philip Dhingra
Orodruin said:
I would say that it is not, but I was trying to be generous in my interpretation ... My main point stands that the post tries to apply Euclidean geometry instead of Lorentzian geometry and therefore is doomed to failure.
Further, the normalization of 4 velocity is entirely a convention. For example, Bergmann, in his classic 1942 book, normalizes to 1 even using units where c is not 1 ( he puts powers of c in the SR metric, rather than using diag (1,1,1,-1) ).

To me, the logical content of the normalization is that we are taking the rate of change of timelike interval to proper time, I.e. that clocks tick one second per second. The c is just an optional conversion factor due to the common practice of writing the invariant interval in units of distance even when it is timelike.

Philip Dhingra and Nugatory
Very good feedback. Thanks for this. I'll dig more into the math pointers you suggested, and get back if you're still following this post.

berkeman
@Orodruin pointed out that it is not otherwise a perfect article. I was just reading it from the point of view of a person totally unfamiliar with relativity (for example, as I was as an adolescent). I found the statements generally interesting from that point of view.

Philip Dhingra
I was just reading it from the point of view of a person totally unfamiliar with relativity (for example, as I was as an adolescent). I found the statements generally interesting from that point of view.

But however interesting you found them, they misstate the physics in a number of ways, as @Orodruin and @Nugatory have pointed out. And the OP was looking for feedback on how well he captured the physics, not on how interesting the article appeared to lay people. Do you really want an interesting article that tells you things that are wrong?

Philip Dhingra said:
Very good feedback. Thanks for this. I'll dig more into the math pointers you suggested, and get back if you're still following this post.

The main issue for me is that you have missed the fundamental symmetry of time dilation. Until you have fully grasped that yourself you are not really in a position to explain time dilation to others.

PeroK said:
The main issue for me is that you have missed the fundamental symmetry of time dilation

I think relativity of simultaneity is also being missed. In the "stack of cards" analogy, relativity of simultaneity means one observer's stack of cards is tilted relative to the other's. Without that there is no way to build a consistent model that properly captures time dilation (or length contraction).

Thanks for the feedback, especially @Orodruin's. I made a mistake with rotating world lines using Pythagorean's theorem, which as some of you have pointed out, is incorrect in Minkowski space. I've changed that section to show the rotation of 4-dimensional velocity vectors. Correct me if I'm wrong, but I believe that those rotations https://www.amnh.org/learn/pd/physical_science/week3/time_dilation.html.

I've also added two footnotes. One is to explain why I'm sidestepping a discussion of time dilation symmetry. I am using the Twin paradox just because it makes the spooky nature of our universe palpable. The other footnote is to clarify the idea that everything is "moving forward" at the speed of light.

Philip Dhingra said:
Correct me if I'm wrong, but I believe that those rotations https://www.amnh.org/learn/pd/physical_science/week3/time_dilation.html.
They do not. As stated before, the geometry of Minkowski space is not Euclidean. The right triangle shown in your linked page is a right triangle in space in a particular reference frame.

Philip Dhingra
Orodruin said:
They do not. As stated before, the geometry of Minkowski space is not Euclidean. The right triangle shown in your linked page is a right triangle in space in a particular reference frame.
*facepalm*, you're right, I'm still doing the same thing, rotating lines in Minkowski space.

PAllen said:
Further, the normalization of 4 velocity is entirely a convention. For example, Bergmann, in his classic 1942 book, normalizes to 1 even using units where c is not 1 ( he puts powers of c in the SR metric, rather than using diag (1,1,1,-1) ).

To me, the logical content of the normalization is that we are taking the rate of change of timelike interval to proper time, I.e. that clocks tick one second per second. The c is just an optional conversion factor due to the common practice of writing the invariant interval in units of distance even when it is timelike.

I agree (it would be hard to argue agains this), it is purely a convention. It should however be pointed out that the practice of writing the invariant spactime interval in units of distance has the advantage of having a spatial part of the 4-velocity vector that approaches the 3-velocity in the Newtonian limit. Of course you could solve this, as mentioned, by slamming powers of c into the metric components - that is just a coordinate choice. (One that I would say obscures the inherent spacetime symmetries, but an allowed one nonetheless.)

Philip Dhingra said:
*facepalm*, you're right, I'm still doing the same thing, rotating lines in Minkowski space.
You might benefit from reading my PF Insight on the geometry of time dilation and the twin paradox. This is the geometry that you would need to capture through any analogy that is to give a reasonable explanation of how spacetime works.

Edit: Also note that your explanation would imply that time runs faster for a moving clock, not slower.

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Philip Dhingra said: