Naked Science Forum
General Science => Question of the Week => Topic started by: thedoc on 18/10/2016 16:02:59

If you where to live on jupiter for 50 years and then return to earth what would the time difference be back on earth due to Jupiter's increased gravity?
Asked by Troy
Find out more on our podcast page (http://www.thenakedscientists.com/HTML/podcasts/nakedscientists/show/1001491/)

We answered this question on the show...
Kerstin Goepfrich put this question to Dr Stuart Higgins from Imperial College (https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2F%2Fforum%2Fcopies%2FRTEmagicC_633pxGreat_Red_Spot_From_Voyager_1_04.jpg.jpg&hash=dec1527cd4a72d3818ed91bfd169ec48)London...
Stuart  Okay, let's go to Jupiter…
Take the next left and proceed for four hundred and eightyfour million miles.
Stuart  Let’s speed that up and pretend we could just teleport ourselves directly to Jupiter’s gas surface… Jupiter is much bigger than the Earth; it has about three hundred and twenty times the mass  it’s huge. Assuming we could live on Jupiter for fifty years and somehow survive the lack of oxygen, giant storms, and the fact that there’s no real solid land to stand on, then the strong gravitational field of Jupiter’s greater mass would have some peculiar effects.
Crucially, compared to a clock on the Earth, a clock on Jupiter would tick more slowly. According to Einstein’s theory of relativity, time passes slower in a gravitational field. This is known as gravitational time dilation.
Kerstin  Being near a massive object makes time move more slowly. So on Jupiter, the big boy of our solar system, how much time would we save? On Facebook Aik suggested a hundred years. One point two three four five minutes was put forward by Martin. Stuart please don’t leave me hanging any longer...
Stuart  Well, it’s not much. For each second on Jupiter your wristwatch would be running roughly twenty nanoseconds slower than a clock left back on Earth. Over fifty years you’d end up being about thirtyone seconds younger compared to if you’d stayed on Earth. And yes, for general relativity fans, this is very much a simplification, not taking into account loads of other factors but it gives us an idea of the incredibly small size of this effect.
Kerstin  So a lot of effort for thirtyone seconds! But why do we even care about Einstein and all this time dilation?
Stuart  It turns out that even though the differences are tiny, they’re big enough to cause huge problems with the global positioning system, GPS. The constellation of GPS satellites that orbit the Earth each have their own clock on board, which broadcasts its signal to receivers, such as your smartphone. By comparing how long it’s taken the signal to reach you from different satellites, the system can work out where you are.
But there’s a problem, the clocks in space are further away from the Earth than your receiver is; they’re in a weaker gravitational field. That means after taking into account other effects, time runs faster for the satellites and while the differences are tiny, over a single day it can build up so that your GPS receiver is wrong by a whopping ten kilometres.
Oh dear, I appear to be lost.
Luckily, the satellites have Einstein’s equations built in so they account for gravitational time dilation.

I assume you will be on the same day on the planet... That means you will only spend part of 'Jupiter's day'.

A couple of points I'd like to add to this thread.
The comparison between clock on Earth and clock on Jupiter so far is a bit incomplete as it only takes into account the difference in the respective Planet's gravitational fields. In other words, if all else is equal a clock on Jupiter would run some 20 nanoseconds slower per sec than the clock on Earth. However, all other things are not equal. Jupiter orbits further from the Sun than the Earth does. This means that not only is the Earth deeper in the Sun's gravitational field, it has a greater orbital speed. These two factor both add up to cause a clock sharing an orbit with the Earth around the Sun to run slower than one sharing Jupiter's orbit.
When you factor this in, it turns out that the clock on Jupiter runs only ~7 nanoseconds per sec slower than the Earth clock when you do a clock to clock comparison.
We do this same combination of gravitational time dilation and time dilation due to speed when dealing with satellites orbiting the Earth and comparing their clock rates to ones on the surface of the Earth. For lower satellites, the speed factor dominates and the satellite clocks run slow, for higher satellites, the position in the field dominates and the clocks run fast. The dividing line is at an altitude which is equal to half the radius of the Earth.
GPS satellites orbit below this altitude, so they run slower than a clock on the surface.
One last point, Someone mentioned that clocks at high altitude run fast because they are in a weaker gravity field. This, I fear, is misleading as people can take this to mean that it is the local strength of gravity that is the factor that determines gravitational time dilation. This is not the case, It is the difference in gravitational potential.
To illustrate the difference, let's use two different planets to compare clocks, Earth and Uranus. We will use the " all other things being equal" approach and compare clocks just due to the the planets' gravitational fields.
Doing this, we find that that a clock, according to Gravitational time dilation, will run slower on Uranus than it does on the Earth, However, if we compare surface gravity of the two clocks, Earth's surface gravity is the greater of the two. A clock on the surface of the Earth will feel a greater gravitational force, but the clock on Uranus will run slower by virtue of being in a deeper gravity well.

What interesting points. I'd not thought about the relative speeds of the two orbiting bodies. Thank you.