Speaking of Special Relativity, back when I wrote the SR series, one topic I left along the wayside was the concept of the spacetime interval. It wasn’t necessary for the goals of the series, and there’s only so much one can fit in. (And back then, the diagrams I wanted to make would have been a challenge with the tool I was using.)
But now that we’re basking in the warm, friendly glow of March Mathness and reflecting on Special Relativity anyway, it seems like a good time to loop back and catch up on the spacetime interval, because it’s an important concept in SR.
It concerns what is invariant to all observers when both time and space measurements depend on relative motion.
Earlier, in the March Mathness post, I mentioned Albert Einstein was born on March 14th. That’s also Pi Day, which deserved its own pi post (about pizza pi), so old Al had to wait for me to address a topic I’ve needed to address for several months.
To wit: Some guy was wrong on the internet.
That guy was me.
Back in 2015 (also celebrating Einstein’s birthday), I wrote a series of posts exploring Special Relativity. Near the end of the series, writing about FTL radio, I said (assuming an “ansible” existed) I wasn’t convinced it violated causality if the frames of reference were matched.
Over the last five weeks I’ve tried to explain and explore Einstein’s Theory of Special Relativity. We’ve seen that motion, velocity, simultaneity, length, and even time, are all relative to your frame of reference and that motion changes the perceptions of those things for observers outside your frame.
All along I’ve teased the idea that the things I’m showing you demonstrate how the dream of faster-than-light (FTL) travel is (almost certainly) impossible. Despite a lot of science fiction, there probably isn’t any warp drive in our futures.
Now it’s (finally) time to find out just exactly why that is.
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’s Too Long Train illustration demonstrates that length is relative. Observers moving at different rates measure the length of an object differently. The faster something moves in your frame of reference, the more its length contracts along the direction of motion.
In previous weeks we saw that motion, speed, and simultaneity, are relative; now we see that length is also relative. Next week I’ll talk about the relativity of time. Today I want to dig a little deeper into the length contraction part of Special Relativity.
It’ll be a factor when we get to the spaceships!
The last two train examples (Lightning Strikes and Treaty Train) focused on how simultaneity is relative to motion. Our final train example focuses on how length is relative to motion. The faster something goes relative to you, the more it appears foreshortened along its direction of travel.
This example involves a train that, if it stopped halfway through, is too long for a tunnel — it would stick out both ends. But motion contracts length, so if the train goes fast enough, it becomes short enough to fit entirely inside the tunnel.
And it’s not an illusion; the train really does fit inside!
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!
The main topic this week was how simultaneity is relative to your frame of reference. How there are (virtual) lines of simultaneity where all points on some line — at all distances from you — share the same moment in time. For any instant you pick, that instant — that snapshot — includes all points in your space.
A line of simultaneity freezes the relative positions of objects at a given moment — which enables distance measurements. Simple example: When their watches both read 12 noon, Al and Em were 30 miles apart. A more mathematical example uses x, y, & z (& t), but it amounts to the same thing: a coordinate system.
The gotcha is that simultaneity and coordinate systems are relative when motion is involved!
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…