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.
I had demonstrated how any form of FTL was impossible given two frames of reference in relative motion — how it violated causality. That was one of the goals of the series, to explain the quip:
- Special Relativity
- Causality
- FTL
Pick two.
Yet it still seemed reasonable to me that two frames that matched, that weren’t in relative motion, could communicate FTL without any violation. (I mean, why not?)
That might work out,… if only there were no other frames of reference.
I was so used to scenarios involving my two subjects, Al (Einstein) and Em (Noether), I never extended the putative FTL radio scenario to the presence of others — which makes causality violation possible again.
Fortunately, some other guy on the internet set me straight.
(A science fiction blog comment section of a post about FTL in science fiction. I mentioned ansibles might be possible in matched frames. A reply helped me see my error. Proof, once again, that good science fiction is seriously educational! Many of my first science lessons came from it.)
§
I really should have known better.
“No FTL!” means “No FTL, no exceptions!”
Einstein, so far, is batting one-thousand.
§
Here’s why my FTL radio, even in matched frames, can’t work (without violating causality).
Remember spacetime diagrams? It’s back to spacetime diagrams. (See SR #9 if you need a refresher.)
Diagram 1. Frame of reference for Alex and Blair. Drew and Chris pass at 0.5c. Drew passes Alex (A), and Chris passes Blair (B). [click for big]
In this frame, Alex and Blair are motionless in the frame (their vertical world lines in the diagram) and, we imagine, separated in space by many light years.
They are motionless with respect to each other, so we imagine they have an ansible that allows them to communicate instantly. The diagram shows this as a fat purple line.
Drew and Chris are in separate spaceships, both moving at 0.5 c (half the speed of light). They are in frame 2, motionless with respect to each other, and they have an ansible that allows them to communicate instantly in that frame.
At points A, Drew passes Alex and sends a (regular radio) message about a crew mishap (a twisted ankle) that occurred at point X along Drew’s world line (left dark green line).
Alex uses the ansible to send an FTL message to Blair at point B. Note that points A and B are simultaneous to Alex and Blair. (All horizontal time slices are simultaneous to them in this diagram.)
At that moment, Chris happens to passing Blair, which allows Blair to send a (regular radio) message to Chris at point B.
For Chris and Drew, points C and D are simultaneous (the light green lines are their lines of simultaneity). That means Chris can use the ansible to send an FTL message to Drew that warns about the crew mishap before it happens.
Note that point X for Drew is simultaneous with point x for Chris, so Chris could only learn about the mishap from Drew (via ansible) after that point.
The moment Drew passes Alex is simultaneous with Chris’ point m, while the point Chris passes Blair is simultaneous with Drew’s point n. From their point of view, Chris passes Blair three green time slices before Drew passes Alex.
From Alex and Blair’s point of view, the passage of Drew and Chris is simultaneous.
Isn’t relativity strange!
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For symmetry, here’s what things look like in frame 2, where Drew and Chris can consider they are the motionless ones while Alex and Blair are in motion relative to them:
Diagram 2. Frame of reference for Drew and Chris. Alex and Blair pass at 0.5c. Alex passes Drew (A); Blair passes Chris (B). [click for big]
The problem in both diagrams is the purple line going back in time!
In this case, according to Drew and Chris, the message Alex sends Blair goes back in time. According to Alex and Blair, it’s the message Chris sends Drew that goes back in time.
Either way, there’s a causality violation!
§
So.
“No FTL!” means “No FTL, no exceptions!”
No warp drive. No sub-space radio. No ansibles.
§
Diagram 3. Drew passes Alex.
As sort of picky details about the exact timing (because the diagrams above make the passing events look compressed):
More realistically (see diagram 3, right), Drew passes Alex at some point along both their timelines (the purple dot).
The only requirement is that Drew passes Alex after the crew mishap (at point X), so information about it can be passed to Alex.
The fat cyan lines represent non-FTL information about the accident moving, first, with Drew and then with Alex to point A (where Alex uses the ansible to send the information to Blair).
I could have made the main diagrams like this to make it clear that it need not be the case that Drew passes Alex who instantly communicates to Blair. Presumably Alex needs to absorb the message and take steps, so there’s some delay.
Diagram 4. Chris passes Blair.
Likewise, on the other end, it’s not the case that Blair gets the message and instantly passes it to Chris.
Alex “needed a moment” to absorb and pass the message on to Blair. We’ll grant Blair the same moment.
That Chris passes Blair a little after, and that Drew passes Alex a little before, what is shown on the main diagrams is just a detail.
Full disclosure, it’s a detail that makes the diagrams harder to draw, not because of the details, but because it requires more separation between Alex and Blair for the scenario to work.
Special Relativity is geometric, and distance matters almost as much as relative speed. Walking past someone is fast enough to make simultaneity in the Andromeda galaxy completely different for the two of you.
§
Making this experiment work, where Drew tells Alex, who tells Blair, who tells Chris, who tells past Drew to beware, does require specific elements of distance and timing.
But that’s not the point.
The point is that, if FTL radio were possible, the experiment would work.
And it would violate causality.
§
So,…
- Special Relativity
- Causality
- FTL (of any kind!)
Pick two.
Stay casual, my friends!
Logos con carne 
March 15th, 2019 at 7:45 pm
As an aside to the more mathematically inclined, it seems the skewed frame of reference in spacetime diagrams can be described as the linear transformation, L, as:
Where c is the speed of light and v is velocity.
Thus, at a velocity of zero, we get:
Which is the identity — no change.
At 0.5c, we have:
Which does exactly describe the diagram’s skew.
Finally, at c, we have:
Which correctly flattens both world line and simultaneity to the light line.
Interestingly (or is it), for a velocity greater than c, say 2c, we’d get:
Which would appear to reverse the roles of space and time!
I’d like to work out if the matrix can help determine Lorentz shifts or does it just help with the spacetime diagram?
Also, I wonder if the full 4D spacetime is like this:
Does it describe how things shift in a 3D moving frame? Is the time vector (last column) really the same as the distance vectors (other columns)?
March 16th, 2019 at 8:40 am
Special relativity is a drag. It seems to ensure that we’ll never explore more than a miniscule portion of the observable universe. And increases the probability that we will never encounter intelligent aliens.
Of course, given the appalling amounts of energy necessary to accelerate to any substantial fraction of the speed of light, this may be, to a large extent, sour grapes.
March 16th, 2019 at 10:53 am
Yeah, it’s kind of a bitter irony that the universe is so big and so inaccessible. Species may be able to explore and expand, but individuals are far more limited (unless we discover some kind of near immortality). As it stands now, it seems only our robots will go very “far” (not even out of our own back yard, yet).
There is also that colonizing a galaxy is a million-year project. Can any acquisitive, exploring species hope to last that long? The dinos lasted millions of years, but they were just lizards. Our intelligence seems to be both enabling and disabling.
March 16th, 2019 at 1:29 pm
Even with immortality, the fact that reaching other galaxies takes millions of years, and other galactic superclusters billions, combined with the ongoing expansion of the universe relentlessly moving more and more galaxies beyond our cosmic horizon, means we’ll only ever be able to access about 3% of the observable universe. (And we’ll never be able to hear back from those who go out that far.)
I think it is possible for us as a species to last millions of years. Lots of species do. But I’m fairly skeptical that biological humans will colonize the galaxy, or at least any substantial portion of it. Keeping human bodies alive across those times and distances has a lot of obstacles. If mind copying is possible, we might be able to be transmitted to interstellar destinations. If not, then the stars likely belong to our AI progeny.
March 18th, 2019 at 1:21 pm
It does seem a shame. Such a vast universe.
I’m still not 100% convinced it’s all even real (99.99%, but not 100%). For instance, if this were a virtual reality, maybe only the solar system is actually modeled; the rest is just backdrop.
Fanciful, yes, but the more I think about it the more preposterous any form of existence seems. (I may be just channeling my own bad week(s) here. My mood isn’t sparkling right now.)
“I think it is possible for us as a species to last millions of years.”
I think it was a Frederik Pohl story where the aliens tell humanity that no “hierarchical” species ever survives their own nature; they all self-destruct. Given our history and basic nature, I’d be very surprised we last a million years. The species that have tend to be static.
But who knows. I’m no better at predicting the future than anyone else!
March 18th, 2019 at 5:23 pm
Sounds like your March is similar to mine. Hope things get better for both of us sometime soon.
I’ve noticed that the aliens always give advice that correlates with the author’s philosophy. Charles Stross’ aliens told the humans that they wouldn’t consider accepting humanity into their federation until we were all communists.
Interestingly, humans have both hierarchical and egalitarian instincts. We’re kind of a cross between chimpanzees, who are actually more hierarchical than we are, and bonobos, who in addition to having lots of sex, are also very egalitarian. In humans, the egalitarian instinct won out during our hunter-gatherer days, but when we started farming and needed to organize, hierarchies came to dominate.
Given our instincts were honed during the long ages of the hunter-gatherer period, no one, except those at the top of the hierarchies, are really happy about them. A lot of the post-axial age social movements have been about putting limits on the privileges and abuses of those high in the hierarchy, or promulgating philosophies that attempt to replace them with something else.
Unfortunately, no one has yet figured out a way to have civilization without them. Although technology seems able to further constrain abuses and increase transparency to mitigate some of their worst effects.
March 18th, 2019 at 6:50 pm
My March wasn’t as bad as yours sounded! Just winter getting old and the aches and pains of aging.
Of course the aliens speak for the author! That goes back at least to The Day the Earth Stood Still. Speaking of which (funny how thoughts lead to other thoughts), I’ve always remembered how JMS used to say that, ‘When the whole story is told, many of you will likely agree with the Shadows.’ He was right. I did. Screw the Vorlons!
Part of his genius was having all sorts of aliens with all sorts of points of view. I can’t help but think his Vorlons and Shadows both reflect his ideas on life. We often have mixed feelings about the big ticket stuff (if we’re fair).
It’s a bit like what you’re saying with hierarchical structure versus egalitarian structure. Lot of mixed feelings there! It does seem an organizational structure is (or can be) very effective at getting stuff done. There’s that old saying about “too many chefs.”
But power corrupts, and, as you say, we’re left to try to moderate it (which becomes an arms race).
There was a good lesson to be learned in the Occupy Wall Street movement. I knew it would end up as it did: It turned into a big party slash opportunity to sell stuff. These things always do if they go on long enough. Part of their failure came from trying to have a leaderless structure. That works as well as it always does.
There’s a reason the military is so focused on chain of command. In times of war, effectiveness is everything.
March 18th, 2019 at 7:55 pm
On the Vorlons and Shadows, I actually agreed with Sheridan’s eventual position: a pox on both their houses. Leave us alone! Of course, they might argue that we were only in a position to reject them after millenia of their cultivation.
When remembering the aliens of B5, I always think of the Drazi, a race that seemed to be the token stupid race. They were the ones that had the pointless civil war between people wearing the different colored sashes.
I actually lost track of the Occupy movement, and so didn’t really notice their commercial stage. I’m generally liberal, but not to an impractical extent. Most of the Occupy spokespeople struck me as naive.
March 18th, 2019 at 9:31 pm
Oh, sure, younger races gotta take over, older races gotta go. It’s that between the Vorlon nanny mode and the Shadow chaotic mode, I tend to favor the latter.
“Most of the Occupy spokespeople struck me as naive.”
That, too, mos def.
What I’ve seen over and over with movements is that they lose focus on their goals and get caught up in the process. It becomes about the show and not about the goal (which often is nebulous and without real plan to accomplish). It’s almost like another tribe people can belong to.
I’m not big on movements…
January 24th, 2023 at 2:35 pm
[…] Last week we celebrated Albert Einstein’s birthday (he turned 140). Now we need another cake so we can celebrate the other March major mathematician’s birthday — Emmy Noether turns 137 today. […]