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since the Sun generates a gravitational field it also generates gravitational time dilatation. Hence, time further from the Sun should pass quicker than in its proximity.

Can we, therefore, say that the time on Mercury is different from the time on Pluto?

Do space probes take into account this difference? For instance, was 'New Horizons' adjusted for the time dilatation during its trip to the Kuiper's belt?

And if there is a difference, does not this rise a paradox, for planets have been generated in the same geological era but then have different relative time? [this actually works also on Earth itself, for Wikipedia reports that there is a difference of 39 h between sea level and the top of the Everest].

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  • $\begingroup$ There is a difference, and I'll leave it to someone else to do the back of napkin calculations with properly formatted equations, but the difference is virtually imperceptible. $\endgroup$ – ohwilleke May 7 at 9:25
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    $\begingroup$ What paradox would there be? I can't see any. 39h is a ridiculously short time over the span of 4.6 billion years. No dating experiment is this exact. $\endgroup$ – infinitezero May 7 at 11:04
  • $\begingroup$ OK 39 h out of 4 billion years is negligible, but still, there is a difference. How much would be the difference between Mercury and Pluto then? The paradox is that, although we are in the same moment in time, the time elapsed is different. Thus the tip of the Everest is 39 h older than the rest of the planet although part of the same system. More or less like having a finger that is older than the rest of the body. $\endgroup$ – Gigiux May 7 at 12:14
  • $\begingroup$ Consider the relative magnitude of the time dilation caused by the Sun's gravity well to that caused by the planet's own gravity well. And I guess you should also consider the SR time dilation due to the planet's orbital velocity. $\endgroup$ – PM 2Ring May 7 at 14:41
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    $\begingroup$ "The paradox is that, although we are in the same moment in time, the time elapsed is different." You need to let go of the idea that "we are in the same moment in time." That's a holdover from your intuition of absolute time. $\endgroup$ – D. Halsey May 7 at 22:06
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Clocks on the surface of each planet tick at different rates. It is a small effect, amounting to a moderate number of parts per billion.

I used Mathematica to calculate the time dilation on the surface of the Sun and the surface of the planets, relative to a clock far from the solar system that is stationary relative to the Sun. (Mathematica has data on lots of physical quantities.)

I took into account both gravitational time dilation and kinematic time dilation, using the formula for the time dilation factor,

$$\sqrt{1-\frac{2\phi}{c^2}-\left(1-\frac{2\phi}{c^2}\right)^{-1}\frac{v^2}{c^2}}\approx 1-\frac{\phi}{c^2}-\frac{1}{2}\frac{v^2}{c^2}.$$

Here $\phi$ is the positive-ized Newtonian gravitional potential,

$$\phi=\sum_i\frac{G M_i}{R_i},$$

and $v$ is the orbital speed.

For the Sun, I included only the gravitational potential from itself. For each planet, I included the gravitational potential from the planet and from the Sun, but not from other planets.

For calculating the orbital speed, I approximated the planetary orbits as circular, with an orbital radius equal to the average of the semimajor and semiminor axes.

Of course, all the results are very close to 1, but a bit smaller. The following table expresses the results as how much less than 1 the time dilation factor is, in parts per billion.

$$\begin{array}{ccccc} \text{Sun} & 2122. & 2122. & 0 & 0 \\ \text{Mercury} & 38.35 & 0.1005 & 25.50 & 12.75 \\ \text{Venus} & 21.07 & 0.5972 & 13.65 & 6.823 \\ \text{Earth} & 15.50 & 0.6961 & 9.870 & 4.935 \\ \text{Mars} & 9.86 & 0.1406 & 6.478 & 3.239 \\ \text{Jupiter} & 23.01 & 20.16 & 1.897 & 0.9485 \\ \text{Saturn} & 8.80 & 7.247 & 1.0350 & 0.5175 \\ \text{Uranus} & 3.313 & 2.542 & 0.5143 & 0.2572 \\ \text{Neptune} & 3.58 & 3.089 & 0.3283 & 0.1641 \\ \end{array}$$

The first numeric column is the body's total time dilation on its surface. The other three show the breakdown into gravitational dilation due to the body's own gravity; gravitational dilation due to the gravity of other bodies (for the planets, the Sun); and kinematic dilation due to orbital motion around the Sun.

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  • $\begingroup$ Thank you. This is exactly what I have been hoping to get. But how do I read the table? That the time on the Sun's surface is 2122/3.58 times slower than on Neptune? $\endgroup$ – Gigiux May 8 at 15:50
  • $\begingroup$ No. These numbers are slowdowns in parts per billion, compared with a reference clock far outside the solar system but at rest with respect to the Sun. Suppose that clock ticks 1,000,000,000 times. Then an identical clock on the surface of the Sun will tick 1,000,000,000 - 2,122 times. And a clock on Neptune will tick 1,000,000,000 - 3.58 times. $\endgroup$ – G. Smith May 8 at 15:57
  • $\begingroup$ Put differently, the clock on the Sun runs 0.999997878 as fast as the reference clock. $\endgroup$ – G. Smith May 8 at 16:00
  • $\begingroup$ By computing these slowdown factors relative to the reference clock, you could then take their ratios to determine how much faster or slower clocks on two planets run relative to each other. $\endgroup$ – G. Smith May 8 at 16:03
  • $\begingroup$ I don't think this is right, for the reasons given in my answer. $\endgroup$ – Ben Crowell May 8 at 16:05
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Time dilation is a concept from special relativity. GR doesn't have a concept of time dilation, except in the special case of a static spacetime, which is the only case where the metric can be derived from a scalar potential. An observer on the surface of a planet is orbiting the sun, so the spacetime they're in is not static. All GR really ultimately says is that when you integrate the metric along a world-line, you get the proper time. The result is typically not describable in terms of a static time dilation.

Do space probes take into account this difference? For instance, was 'New Horizons' adjusted for the time dilatation during its trip to the Kuiper's belt?

What is actually observable is a Doppler shift, and yes, this is taken into account for space missions. A cleaner, simpler example to correctly demonstrate the concepts you have in mind is GPS. You can find careful treatments of this online.

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