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I'm going to try my best to describe what I'm looking for, but it's a bit hard to explain.

I'm imagining a world with multiple realms and time fluctuates between them according to some sinusoidal wave and I want to figure out how to track the flow of time.

As a simple case, let's assume there are just two realms: A and B. I don't know if a regular sine wave is the right approach or not, but say realm A's time is flowing according to $\sin(2\pi x/3)$ and realm B's is flowing according to $\sin(2\pi x/5)$. These two sync up every 15 days, meaning that if you spend 15 days in either realm, then 15 days will have passed in the other realm as well. But any other point along the way, sometimes realm A will be ahead of realm B, and sometimes it'll be behind.

So my question is: is it possible to setup such a system, in which if I know that it's day x within the 15 day cycle in realm A, I can determine what day in the cycle it is in realm B? If so, what is the math behind it? And how could this be extended to more realms, each with a different sine wave?

In other words; Wanting to have a function that given a time past some epoch will give the phase angle in units of hours. Eg day 43.5 realm B is lagging realm A by 3 hours. On day 50.8 realm B is leading realm A by 10 hours.

Hopefully that all makes sense.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Apr 26, 2023 at 16:57
  • $\begingroup$ Do you understand that if the time "speed" is following a sine curve, it probably means your realm is living an infinite looping cycle (sin(t) will become negative at one point)? Is it what you intended? $\endgroup$ Commented Apr 26, 2023 at 17:43
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    $\begingroup$ "Realm A's time is flowing according to $\sin(2\pi x/3)\,$: Your confusion will most likely magically disappear once you think hard about what is $x$ in this equation. Specifically, in what way is $x$ anything other that a uniform and not fluctuating time dimension common to the two realms? P.S. $\sin(2\pi x/3)$ is bounded between −1 and +1. That's not a lot of time span. You probably want something like $t = x + 1 − \sin(2\pi x/3)$... $\endgroup$
    – AlexP
    Commented Apr 26, 2023 at 18:42
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    $\begingroup$ VTC - As written, and as per the current only answer - This is not a worldbuilding question, but a Maths question - as such, would be better asked in one of the Math-specific SEs. $\endgroup$ Commented Apr 26, 2023 at 21:55
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    $\begingroup$ @TheDemonLord So what you're telling is that if you remove all worldbuilding elements this is not anymore a worldbuilding question. Indeed, and that's in fact very common as worldbuilding relies on many real-world matters so you still have a question with everything removed. The actual issue to be solved is what is it about if you remove nothing? $\endgroup$ Commented Apr 27, 2023 at 9:01

2 Answers 2

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Use integration.

enter image description here

I took your two values on Desmos and graphed them out, adding +1 to make them easy to read. The y axis represents the speed of time, the x axis represents real time passed to an outside viewer.

I then integrated the values to calculate the total time spent (real time times rate of time movement), the area, with integrator calculator and took away -1 to make it easy to read.

This is a simple way to work out when the realms sync up. When the integrated values touch, the green and purple values, time has synced, when the red and blue values touch the realms are moving at the same speed. You can adjust the values as you wish by tweaking the equations.

To find when the times converge, do a simple calculation.

Take the difference in x coordinates (days) between a convergence point, for example, 0.938, and your chosen point, such as 2 days.

Then work out the average time speed between those two points. At 0.938 it's 1.924, and at 2, it's .134 for the first equation, and 1.588 for the second. Take the first speed value and divide by the second.

For time one, (2-0.938) * (1.924+.134)/2=1.092 for time two, (2-0.938) * (1.924+1.588)/2=1.864872. The second curve will be about a day ahead.

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  • $\begingroup$ I see what you're saying. So using this information, how would I then go from day x in realm A = day y in realm B? $\endgroup$
    – tstrickler
    Commented Apr 27, 2023 at 2:20
  • $\begingroup$ I showed how you do the maths. You'd need a different equation of course to get a notable time difference of days. $\endgroup$
    – Nepene Nep
    Commented Apr 27, 2023 at 8:23
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This is very simple. It's really no different than how relativity based time dilation works.

Put a record player on a cart. Move that cart at a fixed speed. Have each world on opposing edges of the turntable. As long as the linear speed of the cart exceeds the linear speed of the edge of the turntable, You can measure "now" in each world based on their distance from an arbitrary starting point.

If the linear speed of the edge exceeds the linear speed of the cart, then you have time travel, but otherwise it's just time dilation.

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