# Time dilation/difference between 2 planets

Hi folks,

I'm writing a story which involves civilisations across multiple planets, and I've been trying to work out the impact of time dilation between them. I'm hoping someone can help me out here (thanks in advance!).

(I'm not sure if this exact question may be applicable to one of the 'real' science areas in the forum, but I thought I'd at least start here and a moderator can tell me if I'm wrong..)

Essentially, I'm trying to ascertain the exact difference between 2 planets, taking into account their velocity and gravity calculations. (I should also say that I'm not amazing with maths or physics - I have a 'roughly college level' understanding of physics - so please be gentle...) Anyway:

I found the calculation for time dilation due to velocities;
• T=1 / SQRT(1 - (V^2 / C^2 ))

And I found the calculation for time dilation due to gravity;
• T=1 / SQRT(1-(2*G*M / R*C^2))

I found this thread (post # 35) where some forum members have proved that you use the product of these two calculations (rather than the sum) to get the true time dilation factor (everything in those threads is *way* over my head to be honest).

So I've been using details about Earth and Jupiter (taken from Nasa planetary fact sheets) just so that I can get the calculations in order, and I've made myself a crude spreadsheet with the various calculations. I'm not 100% sure of the rules of posting links to files here, but if it's okay to post it and anyone wants to see it, let me know and I'll link it.

The values I've been working with, and the results of the calculations (incidentally, I'm basing these calcs on individual seconds) I have are as follows;
Earth: Mass = 5.9724E+24, Radius = 6371008, Velocity = 29780
Dilation from Velocity = 1.0000000049337600
Dilation from Gravity = 1.0000000006961300
Product = 1.0000000056298900​

Jupiter: Mass = 1.89819E+27, Radius = 69911000, Velocity = 13060
Dilation from Velocity = 1.0000000009488900
Dilation from Gravity = 1.0000000201624900
Product = 1.0000000211113800​

So now I have a couple of questions
• Firstly, are these calculations correct?
• Secondly, what's the correct method to compare these two?
Do I just subtract one product from the other, and if so, how do I know which order to do that subtraction? I.e. I subtracted the Jupiter Product from the Earth product, and I have the result -0.0000000154814885, showing that Jupiter experiences less time ... but I could just as easily subtract Earth's from Jupiter's. Feeling quite frustrated at being this clueless.

(I also understand that when using these values that the differences we're talking about are so ridiculously small that it would really never be a problem to any space-faring civilisation, but these are just some real figures that we can all agree on so that I can get the right calculations worked out.)

Any help or advice would be appreciated - thanks!

Time dilation would become increasingly noticable leaving our solar system.

Gravity differences and relative motion would be greater. Also you would have to travel very fast if you want to go anywhere interesting.

That said, some technologies like GPS rely on accounting for time dilation. You would notice if they didn't.

Is it an important feature of your storyline?

mfb
Mentor
Do I just subtract one product from the other, and if so, how do I know which order to do that subtraction? I.e. I subtracted the Jupiter Product from the Earth product, and I have the result -0.0000000154814885, showing that Jupiter experiences less time ... but I could just as easily subtract Earth's from Jupiter's. Feeling quite frustrated at being this clueless.
Divide them - but as they are so close to 1, subtracting them gives the same result with any reasonable precision.

You'll have to take the gravitational field of sun into account as well. You can add the gravitational potentials of sun and the planets.

KiltedNinja
Is it an important feature of your storyline?
Hi Andrew, at the moment, yes it is - although it really depends on whether the figures I eventually use will have any real impact with time dilation. I had a particular idea of how it could be interesting to use it in my story, and I can't really get past the idea until it's solved - not that I'm obsessive or anything..

Divide them - but as they are so close to 1, subtracting them gives the same result with any reasonable precision.

You'll have to take the gravitational field of sun into account as well. You can add the gravitational potentials of sun and the planets.

Hi Mfb, thanks for the reply. It didn't occur to me to account for the sun's gravity - although it did occur to use the sun's velocity - so thanks for that!

With regard to including the gravity potential of the sun, just so that I understand you correctly:
I work out the gravity-time-dilation of the sun, and the gravity-time-dilation of the earth, and add them together.
Do I do the exact same with velocities? As in, do I work out the velocity-time-dilation of the sun and add that to the velocity-time-dilation of the Earth? Or do I just add the two velocities together and work out the velocity-time-dilation using that value?

Regarding the division (between the 2 planets [or systems]), please forgive my ignorance in what I'm fairly sure sounds simple, but what is the order of the division? I'm assuming that the larger product of the two is the system in which time passes more slowly - so I would expect to divide the larger by the smaller, and that value would be factor of how much more time passes in the larger system. Is that correct? (just using my figures above, the value I get for Jupiter is 1.00000001548149)

Thanks!

mfb
Mentor
I work out the gravity-time-dilation of the sun, and the gravity-time-dilation of the earth, and add them together.
Don't add time dilation, add the gravitational potentials, so - GM/R - Gm/r where M,m are the masses and R,r are the distances.
Do I do the exact same with velocities? As in, do I work out the velocity-time-dilation of the sun and add that to the velocity-time-dilation of the Earth?
Which velocities do you want to add? The rotation of the planets is a tiny effect, and depends on the current position of the observer. The motion of the sun is just one velocity. Again, here you would add velocities, not time dilation.

Regarding the division (between the 2 planets [or systems]), please forgive my ignorance in what I'm fairly sure sounds simple, but what is the order of the division?
Depends on which view you want to take. As an example with unrealistic values, imagine you get a factor 6 for Jupiter and 2 for Earth, both relative to an observer at infinite distance where the sun does not move. For every minute of the distant observer, the clocks on Earth go forwards 30 seconds, while the clocks on Jupiter go forwards by 10 seconds. Looking from Earth at clocks on Jupiter, you'll see (on average over many years) the clocks there going forward by 1 second for 3 seconds on Earth. Seen from Jupiter, you'll see the clocks "there" (on Earth) going forwards 1 second every 1/3 second on Jupiter.

Note that your actual view will be different - the time dilation from the relative motion appears symmetric, and you also have to take into account the Doppler effect, the Shapiro delay, the curved path a light signal will take, and various other effects. The calculation via the observer far away only works as long-term average.

KiltedNinja
Jonathan Scott
Gold Member
Note that the feasibility of space travel is strongly related to the planetary surface potential and hence to the time dilation. The potential in dimensionless units (e.g. Gm/rc^2) is an indicator of the fraction by which the energy of an object has to be increased to reach escape velocity. At the surface of the earth, this is less than one part in a billion (10^9), yet we have difficulty getting into space. Even if the potential were a million times lower (making it a factor of a million times more difficult to escape), the average time dilation would still only be around one part in a thousand.

KiltedNinja
Note that the feasibility of space travel...

Hi Jonathan - thanks for the reply. I have to admit, I think what you have said is outwith the scope of what I'm trying to figure out at the moment - at least in terms of the current state of my story. Having said that, it may come into play depending on whether or not I can get these calculations figured out!

Which velocities do you want to add? The rotation of the planets is a tiny effect, and depends on the current position of the observer. The motion of the sun is just one velocity. Again, here you would add velocities, not time dilation.
Okay, so it seems to me that if I want to get the true velocity of a planet moving through space, it should be it's orbital velocity plus the velocity of it's star, as it's star travels around the galaxy. So the calculation I've come up with is this (where V1 and V2 are the respective velocities of the star and planet):
= 1 / SQRT ( (1 - ( ( V1 + V2 ) ^ 2 / c2 ) ) )

Is this what you meant by 'here you would add velocities'?

Don't add time dilation, add the gravitational potentials, so - GM/R - Gm/r where M,m are the masses and R,r are the distances.
Allright, so I've come up with this (apologies for the poor readability):
= 1 / SQRT ( 1 - ( ( 2*(GM1+ GM2 ) ) / ( ( R1+R2 ) * C2 ) ) )

Is this what you meant? So this is the square root of 1 minus (2 times (GM1 + GM2), divided by (R1+R2) * C2)

Thanks for your time (and patience) in trying to help me understand this!

mfb
Mentor
There is no such thing as "true velocity". Velocities are always a relative thing: a velocity relative to something else (e.g. relative to the star). If both planets are in the same system, the motion of the star system in the galaxy does not matter at all.
Is this what you meant?
No, and it is different from what I wrote.

No, and it is different from what I wrote.

Don't add time dilation, add the gravitational potentials, so - GM/R - Gm/r where M,m are the masses and R,r are the distances.

Forgive me for being confused, but you said "add the gravitational potentials" and then wrote "GM/R - Gm/r". I know this might seem simple to you but I'm not grasping it properly.

Are my calculations even close to the right way to figure this out?

mfb
Mentor
-GM/R - Gm/r

A/B + C/D is not the same as (A+B)/(C+D) (this is a independent of the sign).

Well I think you've officially lost me.. as I briefly mentioned, I'm not entirely great with maths - so I'm struggling to understand what appears to be simple. My research time is limited due to family commitments, therefore I'm unable to spend the time to study the prerequisite material to understand this.

I'll be honest, I was hoping that someone could provide me with the info I need so that I can continue with my story with a clear mind, rather than become obsessed with a potential plot hole. It looks like this isn't to be!

Nevertheless, I thank you for your time - it has been much appreciated.

mfb
Mentor
Where is a potential plot hole if uncorrected clocks run a few microseconds slower or faster per day?

Your original formula, with just one source of gravitational potential:
$$T=\frac 1 {\sqrt{1-\frac{2GM}{Rc^2}}}$$
$$T=\frac 1 {\sqrt{1-\frac{2GM}{Rc^2}-\frac{2Gm}{rc^2}}}$$