So far this week we have Em taking a round-trip to planet Noether at half the speed of light. Upon her return she discovers that, while she’s aged 23.8 years, Al (who stayed home on Earth babysitting Theories) has aged 27. It took her well over twenty years to do it, but Em effectively traveled 3.2 years into the future.
Last time we saw that — so long as Em is in constant motion — there is symmetry between Al and Em with regard to who is moving and who isn’t. Both can claim the other is (or they are). Both views are valid. Until Em stops. Or starts, for that matter.
Today we look at Em’s “time shadow” — it’s a key to the puzzle!
Last time we watched friend Em make a six light-year trip to planet Noether while friend Al stays home on Earth working on Theories. It turns out that Al ages 27 years while Em ages only 23 (point 8). This is not due to special diet, but to Special Relativity slowing Em’s clock on account of her fast motion through space.
We also saw that once Em stops at Noether, this breaks the symmetry of the two valid points of view regarding their motion (Em and ship are moving vs Al, Earth, and space, are moving).
Today we examine the trip before that point, while it is symmetrical.
We’ve covered a great deal of ground in the last four weeks. (Writing a series of posts this long is a new experience for me! I hope you’re getting something out of it, too.) We’ve learned that motion, velocity, simultaneity, and length, are all relative to your frame of reference — motion changes your perception of these things. This week we’ll see that time is also relative — motion changes that, too!
So far we only needed a (very imaginary) train to demonstrate the effects of Special Relativity. An Earthly frame of reference was enough to illustrate how motion affects velocity, simultaneity, and length.
But when it comes to time, we’re gonna need spaceships!
Last time I focused on how it was possible for Al to see — even enclose in a tunnel — a train that appears shorter to him due to its motion. It turns out that the train Al sees is a stack of time slices of the train at different moments. As we’ve seen, lots of things look different in a moving frame.
Today I want to say a little about Em’s point of view, run some numbers, and take you through a little math (just one equation, I promise). Then, because it’s Friday (when I try to write about light), I’ll introduce you to light cones.
They’re not actually necessary, but they’re kinda cool.
Last time we explored the Simultaneous Lightning Strikes illustration of Special Relativity. In that scenario, on-the-ground observer Al sees simultaneous lightning strikes to a passing (very) high-speed train. On-the-train observer Em agrees both bolts hit the train (one front; one rear), but sees one happening first followed by the other.
The next scenario reverses the situation. This time traveler Em sees simultaneous events on the train and bystander Al sees them happening one after the other.
Today we explore: Peace Treaty (on a Train)!
For the last three weeks I’ve been laying a firm groundwork for the more interesting part of the series. Perhaps there was too much time and detail: I seem to have lost much of my audience (not that the lecture hall was packed in the first place).
I’ve long believed in the importance of basic knowledge — it’s stood me in good stead through life. But I know not everyone shares my appetite for details. For what it’s worth, the rest is the fun part, where all that groundwork goes into action.
This week, trains; next week, spaceships!
We started by exploring the idea that motion is relative. Now we see that the idea of simultaneity is relative! Events that Al sees as simultaneous in his frame of reference do not appear simultaneous to Em — she sees them happening one after another!
A frame of reference has lines of simultaneity that allow us to assign time coordinates to events in the reference frame. If Al and Em have different lines of simultaneity, then their coordinate systems differ— they assign different coordinates to an event!
Let’s explore that in a bit more detail…
Last time our friend Al used lasers and timers to create a regular grid-like map of the space and time near him. The map allowed him to assign space-time coordinates to events in his frame of reference (even if it takes time for him to see light from those events).
An important concept is the idea of simultaneity — of events in different locations happening at the same moment according to some observer (who has to wait for the event’s light to reach their eye).
So far the events weren’t moving relative to us. What if we — or the events, same thing — are moving (and moving fast)? It turns out, this changes the picture!
In the last two weeks I’ve covered relative motion as the ancients understood it (Galilean Relativity), touched on how light doesn’t follow those rules, and introduced time-space diagrams that we can use to visualize motion. I also introduced the topic of space-time events, which are simply locations in space at a given time.
In particular, I showed how our friend Al can use a laser to determine both the location and the time (relative to himself) of an event. This allows him to map his nearby space and time using a system of regular (that is, grid-like) space-time coordinates.
Today we continue with that idea.
A couple of readers have asked about the diagrams in this series of Special Relativity posts. I created them with the freeware 3D ray tracing application, POV-Ray. The diagrams are actually three-dimensional “scenes” designed to be viewed as flat pieces. If some of the “dots” look more like little spheres, that’s because they are!
I wrote some introductory posts a while ago (here, here, and here). You can read those if you want more details about the application.
For a little (optional!) Friday fun, I thought I’d share some POV-Ray images that have a bit more “dimension” to them.