SR #23: Light Clocks

sr23-0This week I’ve focused on the relativity of time under motion, and we’ve seen that moving very fast allows “time travel” into the future. Very handy if you don’t mind the one-way trip. What’s more, a spaceship capable of such a flight is physically possible, so it’s a “time machine” we know works!

On Monday I described how fast-moving, but short-lived, muons created high in the atmosphere live long enough to reach the ground due to time dilation. That’s just one place we see Special Relativity actually working exactly as Einstein described. For another, fast-moving particles at CERN have decay times showing they, too, have slow clocks.

As we’ll see today, light’s behavior requires time appear to run slower!


(Imaginary) light clock.

Of course, that’s true of all of Special Relativity — that light moves at a constant speed to all observers is the basis of SR.[1]

But the connection light’s speed has with time dilation seems more direct here than how it affects length (or energy or mass — which I haven’t discussed). The connection with simultaneity is more direct, but the concept of lines of simultaneity is still a bit abstract.

That time appears to run slower is very concrete, and its physical basis seems almost obvious once the idea clicks.[2]

Some texts on SR more or less start with this (but it does require being clear about the speed of light’s consistency and what that implies — I really wasn’t).

What I’m talking about here is the idea of a light clock.

As a general rule, any clock works by ticking. That is, any clock works by having something that goes “back and forth” (for some useful definition of “back and forth”). A grandfather clock, for example, usually has a pendulum that swings back and forth.


Clock escapement.

If you’ve ever taken a watch or smaller clock apart, you know the gear train starts with what’s called an escapement — often a little curved, clawed bar that ticks back and forth.

Its movement creates the ticking sound clocks make as it slices the pressure of the spring into moments the watch ticks off.

Electronic clocks use a vibrating crystal (remember “quartz clocks”?) that does the ticking. These crystals vibrate at extremely high rates of speed, and are very stable in holding the rate constant. As such they make for very accurate clocks.

Cheaper clocks that plug into the mains use the alternating current (A.C.) as their ticking. In the USA, the mains “ticks” 60 times per second, which is convenient. (In other parts of the world, 50 cycles per second is common.)


Watch quartz crystal.

What if we made a clock using the most accurate, most consistent, fastest thing we know: light!

If we bounce light back and forth over a known distance, we can consider each trip across that distance a “tick” of some extremely fast pendulum (it probably just goes “tck” or even “tk”).

You’ll recall[3] that light travels about one meter in three nanoseconds. If we bounce light back and forth over a one-meter distance, then each tick is three nanoseconds. When we slice time this finely, we end up with a very accurate clock.

So far, so good.

Now let’s imagine our clock bounces light up and down vertically. For us to use the clock in our own frame of reference, it can bounce light any which way. There is no preferred orientation for us.


Diagram 1. Al’s clock.

But to illustrate what happens when motion is involved we’ll use up and down because that’s where we’ll see the effect of time dilation.

Em is our usual traveler, so we’ll haul one of these clocks onto her spaceship. She’ll fly past Al to see what happens.

Meanwhile, Al has a clock of his own so they can compare notes. Diagram 1 shows Al and his clock.

Note that this is not a time-space diagram, but a flat (schematic) drawing of Al with his light clock beam bouncing up and down.

The t indicates the time it takes for one tick of Al’s tclock.

The question for Al is what does Em’s clock look like from his point of view as she speeds past. Remember that, to Em, her own clock looks just like Al’s does to him in diagram 1. Her light clock bounces light straight up and down, just like his.

But what Al sees as Em passes by looks like this:


Diagram 2. Al watches Em (and her clock) pass by.

Al sees Em’s clock bouncing light up and down (as she does), but he also sees the light having to take a longer path because the clock is moving!

Em’s clock moves some amount (x) during each tick, so the light not only has to make the vertical distance, it has to cover some horizontal distance as well. (But this is relative only to Al! Em sees her clock normally.)


Pythagoras’ pretty Theory!

Maybe you recall the post where Pythagorean Theory entered into calculations for physical motion that wasn’t along the line of travel.

When Em threw the ball upwards (or dropped it), the slanted path — again covering both vertical and horizontal distance — required Pythagoras to tell us the total distance.

This is the same situation. From Al’s perspective, the light in Em’s clock has to travel vertical and horizontal distance, so he uses Pythagoras to determine the total length the light travels in her clock.

What matters is that the total distance in Em’s clock is always longer than in his. It has to be. Adding any amount of horizontal distance has to make the total distance greater. Basic geometry.


Diagram 3. The length Al sees the light in Em’s clock travel depends on Em’s speed relative to him. The greatest angle (45°) is at light speed; the minimum angle (0°, straight up and down) is at zero. Depicted above (left to right) are: 0 c, 1/12 c, 1/2 c, 5/6 c and c.

But here’s the kicker!

Remember: Light always travels at the same speed to all observers!

So if a tick in Al’s clock is just the distance t, but he sees the distance in Em’s clock as t+x (for any x other than zero), then Em’s ticks have to be longer.

If the speed of light is constant (which it is), and if — from Al’s point of view — the light in Em’s clock takes longer to “tick” (which it does), then Al has to see Em’s clock as running slower.

It’s the only possible conclusion.

warp drive

Sorry, looks like there won’t be much going forth boldly and exploring strange new worlds. At least not at warp speed.

What’s interesting is that the geometry in diagram 3 leads directly the Lorentz equations we’ve been using to determine gamma. Special Relativity is a geometrical theory that only requires Pythagorean math. (General Relativity is a whole other kettle of tensors.)

On that note, we’re essentially done!

The last posts in this series will use what I’ve shown you so far to explain why FTL is almost certainly forever an impossibility — even in principle.

Suffice to say a new Einstein would have to come along and change our notions of reality just as Einstein did and as Newton did before him.

[1] It’s what happened when Einstein pondered the implications of the two rules:

  1. Physics works the same in all inertial frames of reference.
  2. Light moves at the same speed to all observers.

Put those together, require both be true, and, poof, out pops Special Relativity.

[2] In fairness, it took years for it to click with me. It’s one of the first SR concepts I encountered (way back in high school), and I wasn’t clear enough about how light behaved to make head or tail of this “light clock” business.

[3] If not see SR #4: Two Rules. Also touched on in SR #9: Light Diagrams.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

10 responses to “SR #23: Light Clocks

  • SelfAwarePatterns

    An excellent series Wyrd! Thanks again for helping me work through the twin paradox.

    • Wyrd Smythe

      Thank you! And you’re welcome.

      It’s been beneficial for me, as well. It’s often said you never understand something so well as when you have to teach it. I’ve proved that’s true many times in life. It’s been true here.

      What started as my own attempt to fully understand the Twins Paradox led to a fully exploration of SR. That led to making a crap load of diagrams mostly to see better the hand-drawn diagrams I was making on graph paper. (For fun I was using my old slide ruler for most of the math. Graph paper, pens, straight edges, and a slide ruler… blast from the past! 😀 ) I’d been intending to write a post (a post!) about SR to celebrate Einstein’s birthday, but each year the date slipped past.

      Doing this series at least gave me a chance to use those diagrams! XD

      It does have some first version roughness, but when I look back over it, it seems to hold together and have pretty good structure. I’m not embarrassed by it! (I am glad I can stop thinking about SR for a while!)

      • SelfAwarePatterns

        It was definitely well done. I especially liked the post on train and tunnerl sizes. I think that was when I started seeing my way through this.

        I’m still looking for something to do diagramming. I usually use Visio at work, but I used a Macbook Pro at home. I’ve tried Excel a few times, but it doesn’t seem well suited to anything but business graphs.

        Enjoy your SR vacation!

      • Wyrd Smythe

        It can depend a lot on what type of diagrams you need to make. Visio is pretty powerful (especially if you get into tweaking shapes and anchor points or making your own shapes). I used it a lot at work, too.

        For stats work (for me that means baseball), Excel made great graphs. I loved making data graphs in Excel. I did a lot of that at work when I supported manufacturing groups. I’ve seen it used to draw diagrams, but I agree — it seems a tedious misuse to me.

        Some diagrams (such as many of my SR diagrams) have repeating elements and a required geometrical precision that begs for a programmatic approach. I never needed that in a work context, though.

        For general work Visio may be one of the best out there (again, especially if you really get into the app), but for specific types of diagrams there are other specialized tools for them. At least I’ve seen ads… most of them seem fairly pricey.

      • Wyrd Smythe

        You had me thinking back to using Visio. For me, learning to manipulate connection points was a jump to using Visio at a new, more powerful, level. There are a variety of useful properties you can set, but mainly the ability to control how they move when the shape is resized is invaluable. If you haven’t gotten into editing connection points, that’ll open new doors for you in that app.

        Creating your own shapes to suit whatever kind of work you do a lot of is yet another level. If you find yourself going through the shape libraries a lot trying to find the right shape (and often not finding something that fits the bill), then making your own shapes is an option. For example, at one point I was doing a lot of electronics drawings, so I made transistor, resister, and other component, shapes. They turned Visio into a pretty nice to draw a schematic with the computer!

      • SelfAwarePatterns

        I did create my own shapes in Visio for a grad school database design class. The professor wanted us to use a specific ER notation. She was okay with us using the built in Visio one, but warned we would have to use the designated one on the test. Wanting to insure that I knew her notation, I made Visio shapes to match it.

        I’ve played on an off with connection points over the years. They didn’t always do what I want them to. I rarely create anything more elaborate than org charts these days, so I’m sure there’s a lot of Visio functionality I’m not tapping.

      • Wyrd Smythe

        Yeah, all the MS Office products are just chock full of features most of us never touch. (Heh, I don’t blame your teacher; I never liked the Visio database diagrams. 🙂 )

        Come down to it, Visio is a pretty good general diagram app. To find something better, I think you’d have to look into an app that someone wrote to do especially the types of diagrams you have in mind. Electronic schematic apps, for example, have a lot of intelligence about routing wires and other things helpful for that sort of thing.

        Hey, for no other reason than mentioning Microsoft and having recalled this to a buddy recently, many years ago I worked with a Unix-based group, and got to know a lot of anti-Microsoft Unix-is-the-best types. I’ve never had the reflexive hatred of MS that many of my peers do; in fact, I think they make some pretty amazing products.

        My “Unix Challenge” to my Unix-lovin’ buddies was this: I’ve got a production database (SQL Server) with current sales data. I can whip up an Excel spreadsheet with a live link to that data so that opening the spreadsheet updates the data directly from the database. In that Excel, I can make some really nice charts (I mean really nice) based on that live data. Then I can create a PowerPoint that links to my charts. I can give that to a manager to show at a meeting. When he does, he’s showing nice charts with at this minute live data simply by opening the PowerPoint. And there’s nothing he can do with a PowerPoint that tells him anything about the data or database structure let alone give him any means of modifying any of it (so it’s very secure). And the data will be current live data every time he shows the presentation. And the icing on the cake: Assuming I know what data and charts are desired, I can whip this up in a matter of a half-hour or less.

        I know this to all be true because I used to do it all the time.

        So… tell me my fine Unix friend… I love Unix as much as any one — very cool O/S — but how would you accomplish that in Unix? (No one ever came up with an even halfway decent kludge. 😀 )

      • SelfAwarePatterns

        Personally, I’m not intensely pro or anti-Microsoft. They’re simply a fact of life in IT. I sometimes get impatient with people who get ideological about technology. I use Microsoft, Apple, and Linux systems. All have their place.

        I do budget work and write up funding proposals with Office, and wouldn’t think of using anything else, particularly since many of the people I have to work with also use it. And a lot of our development work happens with .Net.

        At the same time, we run open source systems that were clearly written to run on Linux, on Linux servers, and other systems that were clearly written to run on Windows server, on Windows servers. Often we can make those systems work on other than their preferred system, but it almost always comes with pointless difficulties.

        At home, I do love my Macbook Pro, but I keep a Windows virtual machine handy 🙂

      • Wyrd Smythe

        All true. We were both an MS and an Apple shop for a long time (Macs mostly in the labs — scientists seemed to love Apple), but somewhere in the 1990s they stopped supporting Apple. (Ironically I’d transferred into a new group explicitly to be their Apple support guy. By the time I wrapped up my projects and transferred, there were no more Macs to support! That group also supported CAD-CAM on Sun “pizza” boxes running Unix, which is how I first got into Unix.)

  • Wyrd Smythe

    A couple of late notes. I had planned an “extra” post to explore some details I glossed over, but there didn’t turn out to be as much extra to say as I’d originally thought. So you get this comment:

    “As a general rule, any clock works by ticking.”

    One exception to “clocks tick” comes when using a controlled process to measure time. For example, the burning of a candle is fairly predictable given a regular candle size, and such candles were used as time pieces long ago. (Kind of tough to carry around in your pocket or on your wrist, though.)

    “Now let’s imagine our clock bounces light up and down vertically.”

    We would observe the same effect if light bounced into and out of the flat diagram — that is, in the dimension missing from the 2D diagram. The requirement is that light be moving perpendicular to the direction of travel.

    If the light bounces back and forth along the direction of travel, we have the situation described in Peace Treaty Train. We can see the time dilation effect there as well, but it’s really brought out when the beam moves perpendicularly.

    “[Em’s] light clock bounces light straight up and down, just like his.”

    And, of course, Em sees the same “time slowing” effect as Al’s clock passes as he sees in her clock. As usual, relativity means they both observe these effects in the other party (and do not observe these effects happening to them).

And what do you think?

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