How to calculate the time on planet B for an event that happened on planet A?

Question

We have had a number of "universal time" questions on this site — basically the various OPs have been looking for a way to establish a single time reference for the galaxy. I believe that's impractical, possibly not even useful, especially since it should be possible to calculate the time on planet B for an event that occurred on planet A. A solution for one planetary pair would be the stepping stone for a solution between any two planetary pairs — and thereafter the foundation for the programmatic concept (not the SI concept) of a "universal time."

Conditions

• Here on planet Earth, the Great Sovereign Emperor of the Known Universe, Malodorous a la Flatuleum de la Kineticist, was assassinated by Quaternary Admiral of the Galactic Fleet, Alistair "Big Dog" Alcott. "Big Dog" wants the empire to know about the change in fiefdom as quickly as possible.

• Now, the Fleet has access to FTL, wormholes, bent space, super-Einsteinian physics, Clarkean magic, really fast ships... but let's ignore that for the purpose of this question. Information simply cannot travel faster than the speed of light.

• But, whatever voodoo magic is used to transmit the information, it's receivable by every planet. How this happens is irrelevant.

• For the purpose of this question, the receiving planet happens to be a thoroughly terraformed Kepler-1047(c), known to its residents as Plerte (but was probably named "Planeta de la Muerte" by its original colonists...). Orbital period: 3.2 Earth days (and if that's not movin' the mail, nothing is), so that's "one year" on Plerte. We don't know it's rotational period, but let's claim for the sake of a lot of fun that it's 40 Earth hours and it's spinning backward during it's orbit compared to the spin of Earth in its orbit. Oh, and Kepler-1047 is 2,665.4 light years from Sol.

• To help everybody out, your answer can use decimal time.

• The message is sent at 2:27PM (GMT+12), September 19, 2020. In decimal time, that's 0.718965325567 of the Year of Our Glorious Emperor "Big Dog," 2020 (for convenience, let's say 2020.718965325567). You, the receiver on Plerte now know where it came from (Earth) and when it was sent — Earth time.

• One last thing: the converted time must be accurate to the "minute" (in decimal time, of course). So, 12 decimal points of accuracy.

Question: Assuming I've given you enough information... What equation would permit the local government on Plerte to convert the event date and time to Plerte local time?

It's worth noting that at 2,665.4 light years it should be true that the two planets have changed position relative to one another by more than just one terrestrial minute — which is why I don't think this is a matter of simply counting backwards by 2,665.4 years. Especially since a "light year" is in reference to Earth.... heh, heh, heh....

Oh... and don't forget that hard-science tag....

Answer 1

I agree with the other commentators - this question is impossible to answer with the hard science tag because of the FTL. There's also a tonne of missing data which I've asked for in the comments but it isn't known. But, I can give you an overview of how this can be calculated once the true values are known, and assuming that both general and special relativity apply as observed in our universe in the relevant cases, and that your FTL drives only affect a special corner case that doesn't come into play and modify these results.

Any periodic signal (eg a ticking clock) from planet A will not be periodic when observed on planet B! 1 EarthSecond will not equal X PlerteSeconds at a constant rate.

This is due to Special Relativity. Sol and Keplar 1047 are travelling around the milky way at orbital speeds around ~300km/s, and from either Suns frame of reference, the other sun will be accelerating, in a non linear way. A perfectly accurate equation would require a general solution to the N-body problem for the entire Milky Way. This is an unsolved mathematical problem.

However for the purposes of time zone conversion within a galaxy we're not without hope. I asked in the physics stackexchange for some help, and the solution is to define a coordinate time as the standard, rather than measure time. This is how we've defined International Atomic Time vs UTC and such definitions scattered all over the galaxy can allow us to solve the problem.

The chain of clock definitions:

Your message is sent in GMT+12. (The Great Sovereign Emperor was killed in Nauru? Was he trying to sneak into Australia? That seems unlikely).

$$ GMT = GMT_{+12} - 0.0013689535 $$

Now we're in GMT. We need to go from GMT to UTC. These are very close, but there's an error of about 0.9 seconds

$$ UTC = GMT + GMT_{error}\\ GMT_{error} \approxeq 2.85198647 \cdot 10^{-8} $$

Now, we need to get from UTC to TAI (International Atomic Time - French acronym). This is a weighted average of 400 atomic clocks over earth, and removes leap seconds.

The current formula is

$$ IAT = UTC + 0.00000117248 $$

Now we need to remove gravitational time dilation from the mass of Earth, so we need to convert to BCT (Barycentric Coordinate Time). This is a time on a hypothetical clock at the barycentre of the Earth Moon system. It's hypothetical as that's about 1500km under the surface. BCT has a scaling factor that applies since 1977 Jan 1st, as each year is 490 milliseconds too short.

$$ BCT = (IAT - 1977) * (1 + 1.550505 × 10^{−8}) + 1977 $$

We can't use GCT (Geocentric Coordinate Time) as then the mass of the moon will cause gravitation time shifting.

Now, this is the end of existing published physics, everything beyond here I'm making up.

We need to define:

• Sol System Barycentric Coordinate Time SSBCT
  • The time recorded on a hypothetical clock in the Solar Systems BaryCentre (which may be inside or just outside the sun depending on the planets orbits.).
  • No-one has defined this yet.
• Milky way Barycentric Coordinate Time MWBCT
  • The time recorded on a hypothetical clock in the Milky ways barycentre
  • We only know the distance accurate to a few thousand light years, so there is no way we can define this clock now.
  • This conversion is non linear, as we need to take into account special relativity of the motion of Sol.
  • Sol probably doesn't keep a constant velocity in the milky way, so this will have trig and calculus to determine the speed at the exact moment in orbit.
• Keplar1047 Barycentric Coordinate Time K1047BCT
  • The time recorded on a hypothetical clock in the Keplar1047 barycentre. Which is probably also in the sun, but we don't know.
  • This conversion is non linear, as we also need to take into account special relativity from the motion of Keplar 1047.
• Plerte Barycentric Coordinate Time PBCT
  • The time recorded on a hypothetical clock in the centre of the system of Plerte and its moon(s, if any).
• Plerte UTC - PTUC. Getting back to a standard of time on the planets surface thats kinda useful.
  • This will correct for general relativity.
• Plerte Local time at your city on the surface or whatever. The answer.

To work out the time, your inhabitants of plerte will need to do some maths, first thing:

• Forget about the time the message arrived in plerte local time. Over the 2600 years the galaxy has change so much that the path the message followed has changed length a non-trivial amount.
• Convert the time in the messsage from GMT+12, to GMT, to UTC, to IAT, to BCT, to SSBCT, and then finally to the galactic central time. MWBCT.
• Convert that time from MWBCT, to K1047BCT, to PBCT, to PUTC, to Local time.

And then marvel at the peice of trivia you now know. It was ~304224 PlerteYears ago, and you now know it to the Plerteminute, congratulations. This is equivalent in Earth culture to me knowing the exact time to the minute that Assurbanipal succeeds his father Esarhaddon as king of Assyria in 669 BC. (I'm very popular at parties).

Answer 2

Your world has FTL ships, therefore in your world relativity works differently from how it works in our world as science currently understands it, therefore no hard-science answer can be given. This is the only possible answer to your question as it currently stands.

There is an answer, within relativity theory, to the question of how to precisely determine simultaneity within a given reference frame. You can find the details of this here, on Wikipedia, for example, and it works not only if the planets are moving relative to each other, but also if they are accelerating. The primary reference for that would most likely be Einstein's 1905 paper, "Zur Elektrodynamik bewegter Körper", of which you can find an English translation here. (Of course, in relativity the notion of simultaneity depends on the reference frame, but in your case that's ok, because you're only interested in the reference frame of planet B.)

However, in a world in which ships can travel faster than light without causing time paradoxes, there must be some kind of universal preferred reference frame, or some other modification to relativity, in order to make that possible. Consequently, in such a world, it's unlikely that people would calculate simultaneity the same way we do. They would instead use whatever made-up physics exists in your world to make FTL possible.

Answer 3

You can't come up with "an" equation, because you haven't provided sufficient information to cover everything: what's the relative velocity between Kepler-1047 and Sol? How deep in the gravity well of the star is the planet? What's the planet's mass?

That said, in principle, the solution is relatively easy, if requiring multiple calculations. If you know the relative velocities between Sol and Kepler-1047, you can calculate the relativistic effects caused by that. Assume the message is transmitted at light speed, so that's a fixed velocity which both would measure the same which makes that part easy. You can calculate the time dilation effects caused by the gravity well of each star, and each planet, and therefore you can determine that if an event happened at time X on Earth, taking all that into account, when word reaches Kepler-1047(c) they can calculate back to determine it would have happened at time Y in their system.

Oh, and you'd need to take into account where each is in the galaxy's gravitational well, and so on.