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
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.)It’s diagram 1, so let’s call this the point of view from reference frame 1.
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!
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:The events, all the labeled points, are the same as described above, but shown from the perspective of frame 2, with Drew and Chris.
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!
“No FTL!” means “No FTL, no exceptions!”
No warp drive. No sub-space radio. No ansibles.
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.
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.
- Special Relativity
- FTL (of any kind!)
Stay casual, my friends!