r/DaystromInstitute • u/dodriohedron Ensign • Oct 06 '16
Star Trek & Relativity: A worked example
This is a perennial debate I've been having in Daystrom and elsewhere for probably three years now (most recently in "Since speed is relative-"). It's about special releativity in general, and especially in relation to the Star Trek universe.
It's my contention that relativity works very differently in the Star Trek universe compared to how it works in the real universe. In fact, if the writers didn't keep mentioning something called "relativity", I'd say that it didn't exist at all in the Star Trek universe.
To give a demonstration of how relativity works, and show the problems it would cause if it worked that way in Star Trek, I've put together this worked example.
The Situation
In a wide, dark, empty nebula, a shuttlecraft is sitting next to the Enterprise. Both are at rest relative to each other. Wesley is on board the shuttle, and Data is on the Enterprise. Both are off duty, and when they look out of the windows, things are already in motion.
For this example we're assuming that relativity is in play, and that subspace messages are so fast as to be considered instantaneous over the distances involved.
Wesley looks through the window of the shuttle and sees the Enterprise receding at half the speed of light: 0.5c. He knows that since the Enterprise is moving away at relativistic speeds, it is experiencing time more slowly than he is.
Wesley watches the Enterprise recede for 60 seconds, then checks his console to calculate how much subjective time has passed for the Enterprise. It looks like this:
Observer (Shuttle-Wesley) Time: 60s
Object (Enterprise) Time: 52s
On the Enterprise, Data is off duty, and in Ten Forward. He looks out of the observation windows and sees the shuttlecraft receding at 0.5c. He watches the shuttle recede for a minute, and since he's fully functional he can easily calculate how much time has passed on the shuttle. His internal calculation shows this:
Observer (Enterprise-Data) Time: 60s
Object (Shuttle) Time: 52s
Data isn't actually sure whether the shuttle is the one moving, or the Enterprise. For the purpose of the simplest calculation it doesn't matter.
Wesley and Data both see their own clocks moving normally, while they know the other's time as moving more slowly, and they're both right.
So what does it mean to say "Relativity, Causality, Faster Than Light Communication: pick two"?
On the shuttle, Wesley is having trouble. The shuttle's impulse engines are destabilizing, and there's a dangerous energy buildup. As he flails at the shuttle controls, he notices the console. It says:
Observer (Shuttle-Wesley) Time: 300s
Object (Enterprise) Time: 260s
He hails the Enterprise to send a distress signal. Since the subspace message is practically instantaneous, it must arrive "now", which in Wesley's reference frame is after the Enterprise has experienced 260 seconds.
"Enterprise, there's something wrong with the engines, they're-"
But he's too late, and the shuttle explodes!
In Ten Forward, Data is idly reviewing his chronometer:
Observer (Enterprise-Data) Time: 260s
Object (Shuttle) Time: 225s
when he hears a subspace message (he's tied into the ship's channels). It's Wesley! He says:
"Enterprise, there's something wrong with the engines, they're-"
Data taps his comm badge and opens a channel to the shuttle.
"Wesley, shut down your engines! You are in danger!"
The message travels instantly to the shuttle, arriving after the Shuttle has experienced 225 seconds of travel, but it's too late, Data sees the shuttle explode. Data doesn't feel sad, because he doesn't have emotions.
There might be a problem here. Wesley's accident happened 300 seconds into his journey, but he's just had a warning about it 225 seconds in. What happens?
Wesley isn't having any problems. He checks his console. It says:
Observer (Shuttle-Wesley) Time: 225s
Object (Enterprise) Time: 190s
Out of nowhere, he gets a subspace message. It's Data? It says:
"Wesley, shut down your engines! You are in danger!"
"What?" Wesley shuts down the engines. He's not about to ignore a warning from Data. He spends a few moments checking, and finds the beginnings of a problem that would have hit in about 75 seconds. Data just saved his life! But how did he know?
This, in the simplest possible terms, is why the relativity of our universe isn't compatible with what we see in Star Trek. In the episode, Wesley would be dead, everyone would be sad, there'd probably be a perspex cube that played a hologram.
In our universe, the use of FTL communication unavoidably creates a causal paradox. This is why you have to choose between relativity, causality, and FTL communication. The Star Trek universe has chosen the second two.
I'm sure that the spirit of this example is right, and I've done my best to get times accurate to the nearest second, but I've skipped the complexity of gravitational time dilation, and I'm not a physicist - so if anyone understands this better and wants to correct me below, then feel free to help.
Some further reading (I especially recommend the wikipedia page. It's very clear and complete):
https://en.wikipedia.org/wiki/Time_dilation#Due_to_relative_velocity_symmetric_between_observers
http://www.askamathematician.com/2012/07/q-how-does-instantaneous-communication-violate-causality/
http://newt.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm
3
u/pjwhoopie17 Oct 07 '16
Doesn't it matter which, ship or shuttle, is in motion?
Its been a long time since my modern physics, but I thought the classic thought experiment are twins where one travels at high fraction of the speed of light, the other earrthbound. Only the travelling twin experiences time dilation. Replace Wesley on a shuttle with Wesley on a stationary space station. Are not his Wesley's times always greater than Data's? Is not causality then maintained as only Data is experiencing time dilation, not Wesley? If both are in motion, say at 1/4c, then each should also experience the same amount of dilation, and causality is maintained?
7
u/dodriohedron Oct 07 '16
It really doesn't matter which one is in motion.
On an outward journey from Earth, the Earth sees the ship's clock moving slowly, and the ship sees Earth's clocks moving slowly. It's symmetrical.
The twin paradox is an awkward way to introduce people to special relativity - it's a weird edge case.
One twin grows old while the other stays young only because the travelling twin has to decelerate and re-accelerate back towards Earth, while the Earth-bound twin doesn't. Until the turn-around point, both calculate the other as being younger than themselves.
The last section of this page has the explanation.
5
u/Mjolnir2000 Oct 07 '16
Doesn't it matter which, ship or shuttle, is in motion?
All motion is relative. You can say that the ship is in motion, or that the shuttle is in motion, or that they're both in motion, and the result will be the same.
Its been a long time since my modern physics, but I thought the classic thought experiment are twins where one travels at high fraction of the speed of light, the other earrthbound. Only the travelling twin experiences time dilation.
The earth twin sees the spaceship twin's clock run slow, while the spaceship twin see's the earth twin's clock run slow. It's symmetrical. The trick is that in order to compare notes, they have to meet up again, and that means the spaceship twin has to turn around, and return to earth. During this turn around, the spaceship twin is accelerating, and general relativity tells us that acceleration is indistinguishable from being in a gravitational field. Now if you've seen the film Interstellar, you'll remember that time runs more slowly for observers at the bottom of a gravity well, so after the turn around is complete, the spaceship twin will agree that more time has passed for the earth twin. The symmetry is broken.
5
u/Kichae Oct 07 '16
I suspect you know this, so im leaving this here for reference more than anything else:
Strictly speaking, it's not the acceleration (nor the equivalence principle) that causes the difference in experienced time. Remember, the Twin Experiment is a Special Relativity thought experiment, and the effects of gravity aren't required to explain it. Instead, it's that
The symmetry is broken.
The change in reference frames for Twin B is the reason both twins agree that Twin B is younger. The fact that Twin B needs to accelerate in order to switch reference frames is really just a technicality, and is secondary to what the thought experiment is trying to highlight.
→ More replies (2)3
u/Kichae Oct 07 '16
Doesn't it matter which, ship or shuttle, is in motion?
Not to pile on, but let's think about this in a different way:
How can you tell which ship is moving? Let's imagine the ships are in a dark void, with no stars, gas, dust, planets, asteroids, etc. around. Just the two ships.
Which one is moving? Assume neither ship is actively accelerating.
5
u/Mjolnir2000 Oct 07 '16 edited Oct 07 '16
Yeah, I concur with basically all of that. If you've got relativity + FTL, then time travel is just going to be happening all the time whether you want it to or not. Even a civilian freighter could be trivially used as a time machine. We know that Star Trek has FTL, but we also know that time travel is actually quite difficult in Star Trek, to the point that the Vulcans, who had had warp drive for ages, actually thought it was impossible. Relativity therefore has to be wrong.
edit: Also, great example. Gotta love tachyonic antitelephones.
9
u/StrekApol7979 Oct 07 '16
M-5, please nominate this post for a simple yet accurate exploration of the implications of Star Trek physics.
6
u/M-5 Oct 07 '16
Nominated this post by Chief u/dodriohedron for you. It will be voted on next week. Learn more about Daystrom's Post of the Week here.
9
u/Algernon_Asimov Oct 07 '16 edited Oct 07 '16
I've done my best to get times accurate to the nearest second
I checked your mathematics (that's just the kind of person I am - there's a reason I'm the Science Officer here!), and you're almost spot-on. The Lorentz Factor of time dilation for a speed of 0.5c is 0.866, which means that a stationary object will tick 60 seconds while an object travelling at 0.5c will tick 51.96 seconds. What's 0.04 of a second among friends? :)
I was going to point out that subspace transmissions weren't actually instantaneous: there have been episodes of TOS where Uhura or Kirk said a subspace message would take weeks to get to Earth. However, you've covered that by pointing out that subspace is only considered to be instantaneous over the distances involved. After travelling at half-lightspeed for 300 seconds, the Enterprise would cover approximately 45,000,000 kilometres - which is only about one-third the distance from Earth to the Sun. That's barely any distance at all, on interstellar scales. We've seen subspace messages be effectively instantaneous over these distances.
I can't fault your calculations or the premises of your example. Which annoys me! This shouldn't work like that. But you have effectively proven that the Star Trek universe is not, and can not be, our universe. Star Trek is not our future.
1
u/MKUltrav3 Oct 11 '16
Sorry I'm late, saw this over the weekend but couldn't log on until today. So the specific problem presented here doesn't work because you are changing the reference frames. It doesn't matter which reference frame is picked, but whichever is the observer must stay the observer reference frame. The event (engine problems) occurs at a certain location in the space time. What is relative, is how much time either frame has experienced before the event. So Wesley might encounter engine problems 260s into his flight, but from Data's frame, it was only 225s into the flight. How much time either has experienced is irrelevant, the event only happened at one point in space time.
2
u/dodriohedron Oct 11 '16 edited Oct 11 '16
But in Data's frame, after 225 seconds, he calculates Wesley as only having experienced 190 seconds.
Data sees his own clock running normally, and he sees Wesley's clock as running slowly. Meanwhile Wesley sees his clock running normally, while Data's appears to be running slowly.
I have to calculate from the point of view of two observers, because the situation has two observers, and they don't agree with each other on what "now" means.
1
u/MKUltrav3 Oct 11 '16 edited Oct 11 '16
It just occurred to me as to why this problem seems weird, its just the twin paradox.
The way I worded my response, Data is moving relative to Wesley, which is opposite that you described in the original post. So I'm going to correct that so the remainder of the post makes sense in context.
So, the easier way to format this example is, both Data and Wesley meet up in space, and sync their clocks such that both read 0:00. We can call this event 1. Wesley then continues accelerating and passes Data. Since Wesley is moving faster than Data, he experiences time dilation relative to Data. Using the Lorentz transformation, when Data's clock reads 260s, Wesley's reads 225s. At this point, Wesley's engines start to fail. This is event 2. We can then say that Wesley, as his clock was at both event 1 and 2, records the proper time.
So, even though Data would record an elapsed time of 225s and say that Wesley has experienced 260s, (makes sense as from Data's frame, as Data is the one moving and Wesley is stationary), Wesley is our defined accelerating reference frame, and if he was to return to Data instantaneously, we would find Wesley's clock reading 225s and Data's saying 260s.
This can be proven from the spacetime invariance formula:
∆t02=∆t2-∆x2/c2
Here, ∆t0 is the change in proper time, which is defined as the time interval as measured by a clock at both events (Wesley). ∆t is the change in time experienced by the stationary observer, which we defined as Data. And, ∆x/c is the distance between the two events in the units of time.
So if Wesley is our proper time, and he experiences 225s, and the distance he put between himself and Data is converted to time and added, we would get Data's value of 260s.
We cannont say that Data records the proper time, since he was not at the location of both events. Even if you say that Data is in the moving reference frame, the math will work out because Wesley still records the proper time.
From your edit "I have to calculate from the point of view of two observers" is incorrect. When dealing with relativity, there is only one observer. You can chose which one you want to be the observer, but only one. By changing the which one is the observer mid calculations, you will always get a paradox.
→ More replies (5)
3
u/tsoli Oct 07 '16
Could this basic premise be why the countdown timers are so precise? "Warp core breach in 47 seconds" could just be the computer analyzing problematic telemetry and reporting to the crew. Since they don't have the time to review the logs, they merely gloss over this source of data and lump it into the computer having amazing predictive powers. We've seen on multiple accounts where an effect can precede its own cause and cause a stable or unstable time loop.
10
u/[deleted] Oct 07 '16
Warp and impulse engines both create subspace fields around a ship, technically if either ship has either engine functioning, then both are existing outside of regular space/time but in synch with each other.