A relativistic framework to establish coordinate time on the Moon and beyond
arxiv.org
arxiv.org
the CSPICE toolkit from JPL/NAIF has a bunch of routines to calculate local time, local time of arrival of events from other places (light cone, I suppose?), down to the nanosecond.
https://en.wikipedia.org/wiki/Barycentric_Coordinate_Time
https://en.wikipedia.org/wiki/Ephemeris_time
https://en.wikipedia.org/wiki/Coordinate_time
Etc
Can someone kind explain this unit for me? microseconds per day.
Does that mean that the relativistic difference is cummulative? In other words, does it add up over time?
This multiplies, so after a million days a clock on the moon will read 58 seconds behind a clock on the earth.
In an Earth-centered inertial frame, the Moon is moving faster than the Earth, but it is also at a very high altitude, and the altitude effect, which speeds up the Moon clock relative to Earth, is much larger than the speed effect, which slows it down.
(Note that the above only takes into account the effects of the Earth's gravity. The paper also takes into account the effects of the Moon's gravity, which don't change the above answer qualitatively but do add small corrections numerically, so the 58.721 microseconds per day is not the actual value the paper ends up with.)
I'm assuming this affects clocks (and things) but not time, right? Time itself is no different on the moon (it's not the future there).
I know this must be true otherwise we would be surrounded by time travellers by now. So, where does this cancels out?
My intuition says that if we have two clocks, each clock with a display and a laser pointing to each other, and we put one of them on the moon and the other on earth, someone observing it from a third equidistant point would see both lasers blinking at the same rate.
If that's true (I don't know if it is) both of your answers are kinda right, aren't they? From earth, the moon clock ticks slower compared to earth clocks, therefore it lags behind. From the moon, the moon clock ticks faster compared to earth clocks, therefore it skips ahead, and vice-versa. I am not sure though, I feel like I'm missing something.
What's the difference?
> otherwise we would be surrounded by time travellers by now
I'm not sure what your reasoning is here.
> where does this cancels out?
I don't understand the question. What is supposed to cancel out?
> someone observing it from a third equidistant point would see both lasers blinking at the same rate.
No, they wouldn't.
> From earth, the moon clock ticks slower compared to earth clocks
No, it ticks faster. That was my point.
- I get into a spacecraft, go into orbit around earth.
- Once in orbit, I start accelerating until my clock ticked faster for long enough to be 1s ahead of earth.
- Once I'm 1s ahead of earth, I capture transmissions my buddy sent to me of the stock exchange rates.
- My buddy, on the ground, has a telescope and is ready to invest or sell stocks depending on whether I stop or keep accelerating.
- My buddy now has knowledge about the future, I land and share the money we stole from the future with him.
Let me re-describe your scenario properly:
Your buddy sends you information about stock exchange rates when his clock reads 12 noon exactly. You are one light-second away from him, so his message is received by you at 12 noon + 1 second, his time. That is also 12 noon + 2 seconds, your time, but that doesn't matter; it's still information about rates when his clock said 12 noon, not rates when his clock said 12 noon + 1 second, 1 second after he sent the message.
In other words, the fact that your clock ticks faster and gets "ahead" of your buddy's does not mean you receive information from your buddy's future.
Going further than this is way above my skillset :D Thanks for the time to indulge my curiosity, I'll study more and try to develop a better intuition.
If you mean the 1 second light travel time is the same as the 1 second clock difference, you can increase the clock difference by just spending longer in orbit. For example, you could wait until the clock difference was 1 hour. Then, if your buddy sent you a light signal at 12 noon by his clock, it would arrive at 1 pm + 1 second (1 second light travel time) by your clock. But it would still be telling you stock exchange rates at 12 noon by your buddy's clock, not 1 pm by your buddy's clock.
Note that this is wrong: you don't have to accelerate to make your clock tick faster. You just have to be in orbit at a high enough altitude for the speedup due to altitude to outweigh the slowdown due to your free-fall orbital speed.
From the paper:
> Φ0 is the effective gravitational potential in the rotating frame, which is the sum of the static gravitational potential of the Earth, and a centripetal contribution
This is just so I can catch up with a specific desired dilation relative to earth.
I don't want to make my orbit higher, on the contrary, the less distance the better so communication is faster.
It does.
> therefore accelerating matters.
No, it doesn't. In a free-fall orbit, proper acceleration is zero; you are weightless. "Accelerate" would mean firing your rockets to change your orbit. You don't want or need to do that.
It is true that there is a coordinate acceleration for a body in a circular orbit, in coordinates centered on the Earth, but coordinate acceleration is irrelevant to what we are discussing.
> This is just so I can catch up with a specific desired dilation relative to earth.
As I said, you do that by staying in an appropriate free-fall orbit for a long enough time. You don't need or want to fire your rockets.
> I don't want to make my orbit higher, on the contrary, the less distance the better so communication is faster.
But the lower your orbit, the less your clock speeds up relative to Earth clocks. And if your orbit is low enough, your clock will actually run slow compared to Earth clocks (because the altitude effect no longer outweighs the effect of your orbital speed). For example, clocks on the ISS run slow compared to Earth clocks.
That's almost what I meant! Spacecraft, acceleration to pick up speed (not to go higher), stock-exchange cheaters.
I don't get the "don't need" or "don't want". It is part of my scenario. I also don't want a twin falling into a black hole, but it is a thought experiment that helps put things in perspective, specially for layman like me.
I always heard of the scenario of the twin at the speed of light that remains younger. I am introducing an element of communication (originally, clocks with laser beams then stock exchange rates) into that scenario and trying to understand what the offsets mean.
This is all above my paygrade, I know, so don't worry! I know I'm far from getting it and I don't want to bother :)
That would mean you would no longer be in a free-fall orbit, you would be moving faster than free-fall orbit speed, and your clock would run slower. Depending on how much you sped up, you might even end up having your clock run slower than Earth clocks.
> stock-exchange cheaters
Nothing you can possibly do with your rocket will enable you to cheat on the stock exchange. No matter what you do, you can't have your buddy's light signals contain information from his future.
> I don't get the "don't need" or "don't want".
You don't need or want to fire rockets to speed up if your objective is to have your clock run as fast as possible relative to Earth's at a given altitude. Indeed, if you really want your clock to run as fast as possible relative to Earth's clock at a given altitude, you should use your rocket to "hover" motionless (meaning zero speed relative to Earth's center of mass) at that altitude, not to speed up relative to free-fall orbital speed.
I'm not trying to come up with an experiment that enables stock-exchange cheating. I'm trying to come up with a thought experiment that highlights the effect of speed on time dilation, with the purpose of understanding what the accumulation of "microseconds per day" means, and in the spirit of the paper posted I want to put an element of information/communication there (clocks with laser beams, stock exchange, doesn't matter what it is, for this purpose they're equivalent).
> Depending on how much you sped up, you might even end up having your clock run slower than Earth clocks.
So there is at least a component of the "microseconds per day" offset formula that contributes to a slowing down compared to what is being orbited, is that correct?
If I got this right, speed matters but it is negligible compared to other factors for objects like the moon or a human made satellite. That's OK. Like I said, I don't want to actually cheat in the stock exchange, I want to understand that effect.
I'm not sure what you mean by "compared to what is being orbited". We are comparing a clock in a free-fall orbit to a clock on Earth.
> speed matters but it is negligible compared to other factors for objects like the moon or a human made satellite
For the Moon, yes, the speed effect is small compared to the altitude effect.
For human made satellites, it isn't. For example, as I have already said, clocks on the ISS run slow compared to Earth clocks--because in low Earth orbit, the slowdown due to speed is greater than the speedup due to altitude.
Your answer seems to indicate that yes. So, in principle, I should be able to come up with a thought experiment that highlights what that speed means.
Would this hypothetical faster moon, billions of years orbiting earth at a different speed, look geologically younger than our regular speed moon? (genuine don't know, but that's the kind of coloquial illustrative example I'm looking for).
Meaning, if you attached rockets to the Moon to speed it up while not changing its orbital radius? In that case the speedup relative to Earth clocks would be less, yes.
> in principle, I should be able to come up with a thought experiment that highlights what that speed means
I gave you one upthread, in the subthread where the movie Interstellar was being discussed.
> Would this hypothetical faster moon, billions of years orbiting earth at a different speed, look geologically younger than our regular speed moon?
Meaning, if you compared the two after billions of years of Earth clock time had elapsed? Yes.
Let's try this scenario:
You are in your spaceship in free-fall orbit around the Earth at the altitude of the Moon. Your buddy remains on Earth, and every day, when you in your ship pass directly overhead, he sends you a light signal with a time stamp. More precisely, he will do this every 24 hours and 50 minutes by his clock, since that's how long it takes from one overhead passage of the Moon to the next. (We'll ignore all the variations in the Moon's orbit and assume your ship is in a circular orbit at the appropriate altitude and in the right plane to pass directly over your buddy each day.)
Let's say that on day 0, you pass over your buddy at 12 noon precisely by his clock. He sends you his signal at that instant, and you receive it 1.25 seconds later, the light travel time to the Moon. And let's say you've adjusted your clock so it reads exactly 12 noon + 1.25 seconds when you receive that signal.
Then, 24 hours and 50 minutes later, you pass over your buddy again and he sends you his next signal at 12:50 pm by his clock. You receive it at 12:50:01.250058 pm by your clock, i.e., 1.25 seconds of light travel time + 58 microseconds that your clock gained. But, as I've said, the information in the light signal is still the information from 12:50 pm by your buddy's clock, not 12:50:000058.
24 hours and 50 minutes after that, you pass over your buddy again and he sends you another signal, this time at 1:40 pm by his clock. You receive it at 1:40:01.250116 by your clock, since your clock has now gained 116 microseconds. But, again, the information in the signal is still from 1:40 pm by your buddy's clock, not from 1:40:000116.
And so on, for as long as you want to run the experiment.
In other words, the light signals your buddy sends you are "markers" in spacetime: they govern what information your buddy can send you. The difference in your clock rates is not in "how fast you go into the future"--your clock moving faster doesn't mean you get your buddy's light signals "in advance". All it means is that there are more ticks of your clock--58 microseconds more--in between each "marker" light signal. Or, to put it in the geometric language that is common in General Relativity, the length of your worldline between two successive "markers" is 58 microseconds longer than the length of your buddy's worldline between those same two "markers". But the causal structure of spacetime--what information you and your buddy can exchange--doesn't depend on your clock rates, it only depends on the "markers", the light signals, between you.
Thanks, this helps a lot! I think I have just one more question :D
I can send him a signal too, right?: I can stop accelerating, or beam him back at the speed of light.
When I receive the signal at 12:50:01.250058 I stop accelerating. 1.25 seconds later at 12:50:02.500000 (his clock) my buddy sees that I sent him a response by stopping my acceleration. Or does he see me stopping when his clock is at 12:50:02.500058?
If he sees it at 12:50:02.500000, no one gets information in advance, but I gained 58 microseconds that I could use to run a computer program for a little longer, as an example, and make a better statistical prediction than he could do on the ground. Or do I lose the ability to use those 58 microseconds in some way?
Yes. Rather than stop accelerating (you weren't in the first place, you were in a free fall orbit in my scenario), you can just send a light signal back, which is a lot simpler--and uses a lot less rocket fuel. :-)
Suppose you receive your buddy's signal at 12:50:01.250058 and immediately send one back. Your buddy will receive that signal at 12:50:02.5000000 by his clock. No 58 microsecond gain, since his clock didn't gain that time, only yours did.
> gained 58 microseconds that I could use to run a computer program for a little longer, as an example
Yes. The Interstellar scenario discussed upthread gives a much more extreme example (and in my response I gave a similar scenario but involving speed instead of using the gravity of a supermassive rotating black hole to get the time dilation effect).
> the twin at the speed of light that remains younger
Neither of these are actual scenarios in relativity. I'm not sure where you are getting them from but your information appears to be garbled.
There is a so-called "twin paradox" in relativity (not actually a paradox so the name is a misnomer), where two twins who take different trips (in the original scenario, one stays at home and one travels out to a distant star and back again at high speed) can end up with one younger than the other when they meet up again. But neither twin can travel at the speed of light; that's impossible for an ordinary object like a person. And neither twin can fall into a black hole, because if they did they could never come back out to meet up with the other twin.
If I modify that scene a little bit, it serves to showcase my doubts.
4 astronauts are on orbit around the water planet, but Brand is sick. Cooper, Brand and Doyle go down so Romily can spend 20 years researching a cure.
After an hour, Cooper, the sick Brand and Doyle go back to orbit and reunite with Romily. For Romily, 20 years passed and he completed the sythesis for the cure. Romily has "stolen" some information about the future by "walking it" in a relatively different pace, and it was able to share that information with someone that "walked it" in a slower pace.
Now, this is for gravity. This was possible because of Gargantua. But gravity is not the only component of the "microseconds per day" formula, there's seems to be more.
So I'm interested perhaps in an illustrative scenario where the speed is highlighted (whatever the outcome is, doesn't matter), not the gravity, and there is an attempt at information exchange (don't care if can be successful or not, as long as the example actually explains it).
That would be something like the standard twin paradox. You could set that up so that, for example, Cooper, Brand, and Doyle go out in their spaceship at very high speed and don't come back until 20 years have passed on Earth, where Romily has spent that time finding a cure. With the right numbers for the speed of the spaceship and the distance it travels out and back, it could be arranged that only an hour would pass for those aboard the ship. The downside of this is that the rocket power you would need to boost the ship to the required speed, and then turn it around and come back and decelerate to land on Earth again, would be huge, way, way beyond what would normally be considered reasonable.
The reason a supermassive black hole was used instead in Interstellar was to allow Cooper, Brand, and Doyle to be able to travel at much more reasonable speeds using much more reasonable rockets, and take advantage of the spacetime geometry of the hole to get the extreme time dilation effect. The reason a rapidly rotating hole was used (aside from the fact that we believe most supermassive holes in our universe are rapidly rotating in this way) was that it allows free-fall orbits to exist very close to the hole's horizon that have very large time dilation factors like 1 hour to 20 years, so that a planet in a free-fall orbit could plausibly exist in that region. For a non-rotating hole that is not possible; to get that close to the horizon and get the time dilation factor you would need, you would have to use rockets to "hover" and the rocket power involved would be of the same order of unreasonableness, if not more so, as what would be needed for the "twin paradox" trip I described above.
This is false. Time itself is in fact progressing at a different rate in both frames, and a second in one is not the same as a second in the other.
But, time still "works the same way" for both reference frames in the sense that its not like the clocks in one appear to move in "slow motion". An observer on Earth or the Moon still sees clocks ticking at one second per second, and objects appear to move and react in the ordinary physical ways at the same rate you expect them to. But this is because the observer is also an object in that reference frame, so their perception of everything is in the same "slow motion" as the objects they observe.
Now, if an observer on the Moon was to watch an observer on Earth with a telescope or vice versa, then they would indeed see "the video playing at a slightly wrong speed". The effect is relative between observers, not local to an observer. That's why its called relativity.
The 3rd party seeing blinks at the same rate was just a guess, I'm happy to learn that this guess is false.
Otherwise, as you have been told, your intuition was trully false, and spacetime behaves in some unintuitive ways. Whats weird about it is that it seem to behave differently depending on where you are observing something from, thats cruical. Its all about point of view, literally. Wherever you are (earth, moon, space, near of blackhole), time and space will work normaly for you because you are there. But if you are looking how things are happening somewhere else, it suddenly starts looking very weird.
You said that you would be surrounded by time travellers, and you indeed literally are. The boring answer is we are all tkme travelers, but the interesting part of it that if we accelerate differently (one going up the tower while other waits down), their times literally start moving different amounts, and when they meet, one is younger than another. The reason we dont notice is because we keep moving at very, very, very low fractions of speed of light where time dilatation is easy to ignore (must be mathematically compensated for GPS sattelites to even work, tho), but that doesnt mean its not ever-presently there.
Jack Campbell's Lost Fleet series (starting with Dauntless) has a lot of time involved, due to military fleets/flotillas maneuvers at light-hour distances:
Fleet A jumps into a system. Fleet B is 6 light hours distant across the system. B won't see light from A's arrival for 6 hours. A won't see B's reaction for 12 hours, less any distance covered. Ships can engage in combat at up to 0.2c relative velocity, at a maximum distance of about 1 light second.
Fleet A accelerates to 0.1c, Fleet B does the same and they approach. Time to contact is 30 hours, and the books discuss how difficult it is for humans to rest and focus on other things during that time. Our instinctive fight/flight/fawn/freeze reaction doesn't translate well to waiting 30 hours!
Fleet A and B approach, but because the ships take a few seconds to pivot/roll/yaw to maneuver, so there's a critical time period where A or B can change position and the opposing fleet won't have time to react before the fleets clash. This is used to great effect by the skilled fleet commander of the Good Guys.
The lost fleet books also discuss maneuvering fleet formations when everyone is at light-minute or multiple light second distances. There's lots of orders like "at time 15, Echo One pivot formation down 30 degrees and starboard 15 degrees". Learning to bring all your firepower to bear on the enemy at a single moment, while denying them the same opportunity, is a very difficult skill to learn in these books.
Aaand I've written way too much. And probably not done the fantastic combat in the lost fleet books any justice at all.
Yeah, I deeply hope someone can write it one day. I've tried thinking about how you'd model and calculate such behaviour, but it's mostly beyond me.
Thank you for the compliment :)
It is a heavy game. https://youtu.be/eqn7F97M8XA
I'll also mention Dread Empire's Fall by Walter Jon Williams - https://en.wikipedia.org/wiki/Dread_Empire%27s_Fall
You again deal with long communication distances and most fleet actions are done with missiles. This then has the challenge of "how do you control the missiles / select the targets at such distances (no AIs)" and there is a class of pilots that fly a 'pinnace' which is a small, one person ship that is capable of doing high acceleration to keep up with the missiles.
During peace times, the pinnace pilots tend to be more about prestige and racing.
There was also an instance of moving a jump point thingy that _really_ messed up a fleet action (since the fleet, traveling at high speed to the edge of the system missed the "go here to jump to the next system" and instead had to turn around and go back ... which represented a lot of time and acceleration (not all species in the universe don't have the same tolerance for acceleration).
And then I got sneakier and would send a false fleet to attack the wrong planet in turn X, and the real fleet would show up one or two turns later on the opposite side of the victim’s empire, just after they had committed every spare ship in their navy to a jump in the wrong direction, tying up their logistics until after the fight was over and I was rebuilding planetary defenses, preparing for the counterattack.
> Take the Traders' method of timekeeping. The frame corrections were incredibly complex - and down at the very bottom of it was a little program that ran a counter. Second by second, the Qeng Ho counted from the instant that a human had first set foot on Old Earth's moon. But if you looked at it still more closely ... the starting instant was actually about fifteen million seconds later, the 0-second of one of Humankind's first computer operating systems.
From the TNG era onward the various Writer's Bibles took an approach of 10,000 days per season to make the math easy if a script was given a Stardate in the right season and plenty of room for all the scripts in a season to have their own Stardates. If the Stardate day is a 24 hour Earth day equivalent then seasons were expected to take roughly 3 earth years. Or was that a sign that the Stardate day was about a third shorter than an Earth day and "five-year mission" did still just refer to Earth standard years?
There are all kinds of fan theories. Some weird things in canon. Some weirder things in the books and other bits of Beta and Gamma "canon".
(I did a bit too deep of a dive into Stardates recently as an offshoot of getting the idea to try to display a Stardate-like calendar and date stamps in the configurable calendar system of the Fantasy Grounds TTRPG hosting application.)
It basically turns the "range" of communication into relative velocity, as relative time dilation shifts transmission frequency beyond processability. An intermediary with lower relative velocity to both can act as MITM.
It made me realise that we currently live in a time that can never be accurately referenced outside the planet.
So no software can ever truly define 2024-07-11T00:00 on Pluto.
It also makes me think that the Mass Relais in Mass Effects could actually also be atomic clocks forming this galactic time grid of reference, so in-game lore seems less implausible.
I've not yet seen a full proposed standard for such a thing, but it's also not far from how GPS itself works and how GPS forms its standardized time, which is synced to Earth based atomic clocks but includes more relativistic effects. Most cellphones actually use GPS time rather than "atomic time" today, because they need GPS services in general and also because cell towers use a variant of GPS time in their communications with cell devices for complicated three dimensional dances like tower to tower hand-offs.
It's kind of neat how even "Earth time" at its most accurate involves relative effects and triangulation in some of our most used and trusted devices. (Even NTP sync of non-cellphones uses IP pings as an approximate for triangulation to try to reduce error in syncing a laptop or desktop to nearby atomic clock sources. Though at least today in most internet usage, the Internet Protocol is less impacted by relativistic effects.)
While the GPS satellites internal clocks might operate on whatever timescale, it is important to note that the time signal that is broadcast through GPS is strictly fixed offset from TAI without any additional relativistic shifts. And TAI is defined to use the geoid as reference frame.
That is to say that if you have a clock synchronized to GPS signal then it should tick exactly at same rate as TAI and UTC.
It's an incredible mathematical achievement that the byproduct of triangulation of your position on Earth also syncs your clock extremely close to our best and strictest clocks. Some of that math is relativistic. Certainly not on the scale (velocity/time/distance) that most people think of when they hear relativistic but partly because something as incredible as GPS satellites orbiting in space is no longer science fiction but a mundane part of our day to day lives and devices. There are satellites broadcasting clock signals from space 24/7! With relativistic math our phones figure out where we are down to foot, nearly (sometimes more) and figure out a precise time for where we are! It's incredible and it is a wonderful thing to take advantage of. Relativistic time math happens every day now (sometimes a lot of times in a day) in our lives, and we don't need to think about it.
Typo?
> This understanding also underpins precise navigation in cislunar space and on celestial bodies' surfaces, thus playing a pivotal role in ensuring the interoperability of various position, navigation, and timing (PNT) systems spanning from Earth to the Moon and to the farthest regions of the inner solar system.
Keep in mind that there is difference between clock measuring seconds, and converting to days and times that requires choosing a calendar. The ship could use origin calendar, destination calendar to make people familiar, or some universal calendar. The problem with destination is that it won't sync properly until arrival. Or maybe they skew the calendar on the last leg.
Also, don't need high relative velocity to have a difference. Considering relativistic space travel may be impossible; the only may be proposed warp drives.