This is an original research work of Manmohan Dash, published here, 3rd November 2011, 8:10 am.
What are the effect of Sun’s gravity, on time dilation, of atomic clocks, on GPS satellite and on on time dilation on clocks on earth.
GPS Satellite; revolves every 12 hrs around earth. Above ~26000 kms. Keeps time on earth driven by atomic clocks. Atomic clocks would lose time due to energy loss in gravitational field, its called as Gravitational time dilation — which would be blue-shift if time or energy is gained, in recession from the trap, and red-shift or losing energy, if entering into the trap ! So the idea is how much time is lost due to Sun’s presence or say Jupiter’s presence?
Apart from Sun, the other planets in the solar system, can exert similarly understood time dilation effect, although there will be no symmetry in such situations, but for a 24 hrs period, that concerns the GPS satellite, because thats –twice, their time period, for one single rotation around earth, this fact of non-symmetry may not matter.
The masses of other planets, in the solar system, vis a vis time dilation effects, will be far smaller eg compared to that of Sun’s mass, hence their effects as well, considering, their distances also play a role. Although we can take only the massive and closer planets, or make a table of these parameter of such planets.
The cumulative effects of time dilation will be more complicated over the year, but remember; the satellites are moving along with the earth, so fractional and absolute time dilation effects do change, but these effects correspond to the time of the year and the exact configuration of the solar system at that time of the year, not to the essence of the calculation.
That is, a few more parameters, can again be studied, such as different solar distances etc. Since the effects are supposed to be much smaller, one can start with a simple classical calculation, for distance and mass, as known over the year.
Actually one may also have to add earth’s radii of ~ 6000 kms, at the beginning of the calculation, below, which will go both ways, above and below, Sun-Earth distance, without this offset. One needs to, from a rigorous satellite trajectory, just integrate the potential energy, over a 12 hrs, multiply by 2, and see how it compares with the effect from sun on earth .
In the calculation below: there is an extra zero, on sun-distance — which was figured out, by dividing by speed of light, one must always get 8 mins.
I was thinking about the effect of the gravitational field of Sun on time dilation for atomic clocks present in it’s field.
The atomic clocks, that are placed in the GPS satellites, make an orbit around earth every 12 hrs. That means half of the time, the clocks on the satellite, orbit on top of earth, wrt Sun’s distance, and the other half below it. But this difference is 52, 000 kms, although Sun’s distance is 144, 000 x (10^3) kms, from earth-center, to which we have to add, Sun’s radii of 695 x (10^3) kms.
So, we have the clocks, from Sun’s center, either (144, 695 x 10^3 – 26 x 10^3) kms = 144, 669 x 10^3 kms or 144, 721 x 10^3 kms on top and below of earth. In second significant digit, this is 1.45 x 10^8 kms, both ways, so it should not add any significant effect, to the satellite clocks, even over that distance.
The effect, that Sun’s field adds, to times of clocks, is directly proportional, to the Sun’s potential energy, at these distances — and gravitational potential energy of Sun is inversely proportional to distance.
So, the effect is really really small, since it will be of order 6.94 x 10^-11 — one needs to see, what will be speed of light, c = 1 or its value in m/s, or we can see this, in terms of inverse of all three distances, which will give us 6.9123 x 10^-11 to 6.9098 x 10^-11.
So, we can see the difference is really really small — 0.0025 x 10^-11, alarm?
Now lets compare with earth distance from Sun of 1.44 x 10^11 — center – center, effect is 6.9444 x 10^-11.
Time dilation effect, on clocks on earth, for which we can adjust, another calculation, for baseline distance of 733 kms, but this will be really insignificant, compared to 26, 000 kms, what one has to do is, also, introduce the 6000 kms earth radii and subtract or add from Solar-center and then introduce the 733 kms, which will be altered, every 24 hrs instead of satellite time-period of 12 hrs.
I suggest we do the detail calculations, since 6.9444 – 6.9123 = 0.0321 or 6.9444 – 6.9098 = 0.0346.
I have greatly simplified this now, the only thing that remains is college level calculations.
Symmetry of the situation is preserved, in case of satellite revolving in 12 hrs, and in-case of earth revolving in 24 hrs, so we can calculate, per the respective period, and later multiply if needed, a factor of 2, for the satellite clocks.
For other planetary bodies, exerting gravitational-potential-energy, to effect a time dilation, as per general theory of relativity, we need to also see, that, these planetary bodies masses are far smaller than solar mass.
The various masses in our solar system, but none of that compares to the mass of sun. All the moons are smaller than earth. Earth is smaller than eg Jupiter and Saturn, but also Sun is massive and closer compared to Jupiter and Uranus and Saturn.
All in all we need only to calculate the effect of Sun towards time dilation of GPS atomic clocks or earth based atomic clocks and since this effect is significantly negligible, it does not make any wisdom in proceeding further to calculate the effect of other massive bodies in our solar system. If Lord Sun ain’t happy what other lords can take his place either.