Thursday, January 28, 2010

Special Relativity: It's All Just Rotations

I took a philosophy class where the teacher asked at one point "Have any of you heard of special relativity and can tell us what it says."

Without thinking what I was doing said: "Special relativity says that we not only rotate in space, but can also make rotations in time, as time is now on the same footing of space."

The teacher than said: "What are you talking about?  I was looking for something on length contraction or time dilation."

I smacked myself for giving such a confusing answer, but I was right!

Everything "special" about special relativity boils down to Lorentz transformations.  There are 6 generators of this "Lorentz Group".  Three spacial rotations (here is a rotation in the x-y plane):

And three boosts which represent boosting to a faster or slower speed (He is the one for the t-x plane.  Equivalent to boosting faster in the x direction):
Now look at the two objects above, they look very similar don't they?  That's because boosting is analogous to a rotation in the "time-x" plane.

How I think of it:
Everything is going the speed of light. (The magnitude of your 4-velocity is always the speed of light.)  If objects move faster in space, relative to you, this is because they have rotated themselves so that their velocity vector points more in the spacial dimension and less in the time dimension.  This is why time goes by more slowly for such and object, relative to you. 

Think I'm crazy? Here is a good back up by the Wikipedia:
The norm or magnitude of the four-velocity is always exactly equal to the speed of light. Thus all objects can be thought of as moving through spacetime at the speed of light. This provides a way of understanding time-dilation: as an object like a rocket accelerates from our perspective, it moves faster through space, but slower through time in order to keep the four-velocity constant. Thus to an observer, a clock on the rocket moves slower, as do the clocks in any reference frame that is not comoving with them. Light itself provides a special case- all of its motion is through space, so it does not have any "left over" four-velocity to move through time. Therefore light, and anything else traveling at light speed, do not experience the "flow" of time.
So there you go, I stand by my claim of what we fundamentally learn from special relativity: "we don't only make rotations in space, but can also make rotations in time, as time is now on the same footing of space."


  1. Everyone's homework assignment is this:

    Find a friend, go outside, get in a position to run but before you do say "How would you like to see what it looks like to rotate in through time?" When your friend says "sure?" take off running as fast as you can.

    Don't explain it, just let him/her figure it out on their own. :)

  2. Great post Joe! Correct me if I'm wrong, but one thing to point out is that rotations through time aren't on exactly the same footings as rotations in space. This is because time comes in with an opposite sign in the metric -- equivalently that rotations in time have cosh and sinh, where rotations in space have cos and sin. This is essentially like time taking on imaginary values -- if we measure space in meters, we'd measure time in i*meters. So, we're not only rotating through time, we're rotating in the complex plane.

  3. You're right Bill. The opposite sign on the metric gives rise to the sinh/cosh versus sin/cos.

    Your description is the more technically correct one.

  4. Your explanation also makes it more clear why going faster than the speed of light is more non-trivial than science fiction would like us to believe. Trying to go faster than light is like rotating a 1-meter vector so that its projection on the x-axis is 2 meters, or trying to rotate more than 90 degrees without turning backwards. It just doesn't work. (At least not without a flux capacitor or a critical mass of dilithium crystals, of course.)

  5. Bill, that's a really interesting point!

  6. I really like this way of thinking about special relativity. As a student I got hung up in the math so that I never "got" it in a conceptual way until later in grad school. My colleague tells me that Brian Greene has a similar presentation in one of his books.

    Great stuff.

  7. In order to understand well relativity it can be good as well to study about what has been done before Einstein about it, like with Parmenides and Zeno of Elea.

  8. Wait a minute! (woah, that statement takes on significant cosmological meaning in this context)

    This reminds me of my wife. In Relief Society when the instructor talked about Newton's law of motion. My wife had to correct her and give the definitions for all three laws. The reaction of the class was much like when Marty looked out over the crowd at the 'Enchantment Under the Sea' dance after his crazy Eddy VanHalen guitar solo.

  9. Stan,

    That's really funny! I wish I could have seen that myself.

    And while I'm at it: Wow! You wife knows the definitions of all three laws! That's awesome as I bet very few people do.

  10. Stan,

    Now what your wife needs to do is, when a disagreement breaks out in Relief Society, she needs to go to the board, write down all four of Maxwell's laws in differential form then make some hand waving argument to prove her point.

    I would pay to see this.

  11. Cartesian,

    You have to love Zeno's paradox!

  12. JS

    My wife is a mathematician and likes (yes *likes*) to derive formulas during tests rather than memorize them. She's a smart cookie and just a bit Asbergerish in social settings, as many smart cookies are. =:) I'm sure she could handle Maxwell easily enough.

    btw, you went to BYU as an undergrad right? Do you remember Vencil Skarda? Mathematics dept, taught my wife and I difeqs.

  13. Stan,

    That's really awesome. (Truth is I have a hard time memorizing so usually have to re-derive things on tests as it is forced on me.)

    Yes I know Vencil Skarda though never had a class from him. It would have been interesting to have take diffeqs from him.

  14. btw, a special programming note for all. I listen to Coast to Coast am most nights. Yes I admit it. Not because I believe in all that kookyness, but I am fascinated by the people who do.

    Tonight Art Bell, as he often does, will be interviewing a legit scientist, Michio Kaku. If you're up that late give it a listen. In addition to the interesting things they will talk about, I am amazed at the callers. They seem genuinely interested in real science even though they hunt BigFoot and tell stories of their abductions.

  15. Maybe I'll have to tune in as I also am interested in what Kaku is up to these days. Kaku sometimes says things that many believe really are on the fringes.

    Nevertheless, anyone who has published 150+ peer reviewed journal articles, many of which are highly cited, isn't going to receive much criticism from me and in some sense has earned the right to speculate about crazy things. (Though it is true its been several years since he was doing this quality of research. Today he mostly just has fun speculating these days.)

  16. Joe, just a quick question. Do you remember which philosophy class that was and which professor?

    ps: That happened to me several times. I just got used to the blank stares like Marty Mcfly got. I also had philosophers insist that they understood GR even if they had no idea what a derivative was, (I'm not talking about a covariant derivative, I mean just a plain derivative). Yeah well...

  17. It was the second History pf Philosophy course. (I believe 202?) The professor was Darin Gates.

    I must say, he actually was a great professor and I highly recommend him to anyone who may find this information useful. (But I did get a "what are you talking about?" look about this.)

  18. Ha! I liked Dr. Gates as well. I think he does a good job at teaching philosophy. I never had the opportunity to talk about relativity in his class so I never got to see his reaction to it.

  19. I had a well-meaning brother in my ward try to use cosmology to explain the creation in Genesis. I kept my mouth shut but I couldn't help but smile. :)

    And in case you were wondering, the CMB apparently has something to do with God saying "Let there be light".

  20. Yes I have to smile myself Nick. But you have to give him credit for being clever.

    Actually, a similar thing happened to me. I heard some people arguing about this saying "This light had to not be physical light because you can't have photons that fill the whole universe without stars or moons and those weren't created yet."

    The mathematician in me, ie.. the one who likes to throw out counterexamples to every claim made, wanted to say (but didn't) "I can think of a physical process that could have filled the universe with photons without stars or moons..."

    Of course, knowing that such a comment would have been foolish I also just smiled.

  21. Joe,

    This guy was actually trying to argue that "let there be light" was when the universe became transparent, but it was clear he was getting that from a History channel-type TV program and really didn't understand it. The rest of the class, however, seemed very impressed by his use of SCIENCE.


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