# Time dilation on a heavily oblong planet

Once upon a time writer Hal Clement had created a world named Mesklin. One of features he didn't detail for historical reasons is a time dilation between polar areas, where gravity is 700g and equator, where gravity was 3g. Today it is known that at smaller gravity time goes faster.

Today we for sure know that time on Moon is slightly faster than on Earth, roughly 58 microseconds per day is gained. It's already may pose a problem for network communications. That's with 1:6 gravity values ratio, not 3:700.

What would be value of similar paramter for Mesklin-like planet (or formula to estimate one) and what problems it may pose for communications?

• Very strongly related: worldbuilding.stackexchange.com/a/254568/16949 Commented Apr 8 at 12:12
• @jcaron funny that one actually hits the case I was thinking of - a megastructure. Think of a matreshka brain. One of problems with that concept is its gravity and shell theorem - each layer got different gravity levels. Plus, today sciece fiction of often intermeshes with power fantasy in odd ways, expectations may differ from "original" world. Commented Apr 8 at 21:44

### It is very, very hard to tell, but it hardly matters

Mesklin is so oblate that Newton's shell theorem stops applying: even according to Hal Clement himself that "To be perfectly frank, I don't know the exact value of the polar gravity; the planet is so oblate that the usual rule of spheres... would not even be a good approximation." Calculating the time dilation value for gravity sources near spheres is easy enough, but around Mesklin, not only is the shape not spherical, but the Lense-Thirring effect begins to kick in due to the planet's extremely high angular momentum. I work extensively in real life with the Lense-Thirring effect, and it gets complicated as the angular momentum of the central body increases.

Then again, what communications network? It's not like any human-built probe could possibly survive near the surface of the poles.

### Okay but that's not fun

Okay, fine, I'll use a Schwarzschild approximation. Assuming that the mass of Mesklin is 16 Jupiter masses (about $$3.0368\cdot10^{28}\text{ kg}$$) and has a polar radius of $$31,770\text{ km}$$, then we can use the elements of the Schwarzschild metric to calculate a time dilation factor:

$$k=\sqrt{1-\frac{2GM}{rc^2}}\approx\sqrt{1-\frac{2\cdot6.6734\cdot10^{-11}\cdot3.0368\cdot10^{28}}{3.1770\cdot10^7\cdot8.9875517874\cdot10^{16}}}=0.999999290155.$$

For an observer distant from Mesklin, they would perceive a clock on Mesklin's pole ticking through 709 fewer nanoseconds every second than their own Mesklin-distant clock. That's about 61.3 milliseconds per day.

### What does that mean for communication networks?

Oh, there are no communication networks in orbit. See, the Lense-Thirring precession I mentioned earlier changes the way orbits work around the central body; any communications satellites near Mesklin will observe a precession of their line of nodes (a critical-to-know astrodynamic value that essentially determines where on the planet the satellite passes over), and their time and space will also be screwed up. At first it might not be an issue, but over time one will observe that the satellite's orbit becomes erratic, and it might become impossible to keep it in a stable orbit.

Additionally, it's not like there's much down there to communicate with. 3 g of surface gravity makes it virtually impossible for modern-day chemical rockets to lift stuff off the planet and makes it hell to live there. If there absolutely needs to be communications networks going across the planet, they're definitely not going to be at the poles, so worrying about that shouldn't matter. Even if it did, it wouldn't mean that much: 61.3 milliseconds a day sounds bad, but for humans that kind of time dilation is so far below noticeable that nobody will even care, and for computers it just means changing a few values to synchronize the transmitters and receivers.

• I think, much later he published corrected gravity values (and other model parameters). Someone helped him with calculation or he had access to a computer. Yeah, with anything non-spherical the shell therem stops working, even for Earth it doesn't work perfectly - "shells" start appear in 80-100 km above surface - but today we have methods of integrations Newton couldn't use. Commented Apr 8 at 1:22
• for humans 60 milliseconds per day doesn't matter. For computer networks, or precise navigation it does. Albeit if they could reach Mesklin, land and even take off from it, they likely had ways to deal with issues. And yeah, whole "Mission"was because there was no comms with errouneously landed probe. Currecntly our IT have problems with Moon's 58 microseconds, they need updated protocols :P Commented Apr 8 at 1:25
• And why should 700g preclude any human probes? We can certainly build things to stand up to 700g--that's far, far below gun-rated stuff and we were putting electronics in shells in WWII. Commented Apr 8 at 2:31
• "3 g of surface gravity makes it virtually impossible for modern-day chemical rockets to lift stuff off the planet": rockets can accelerate at hundreds of gravities. It's not the surface gravity that's the problem, it's the orbital velocity you need to reach...somewhere around 100 km/s. Commented Apr 8 at 2:39
• @Syndic only in that the orbital velocity and surface gravity are both functions of mass and distance from the center. And I specifically stated that orbital velocity was the main problem. Commented Apr 10 at 11:50