# Where in the solar system could we place a clock, such that it would run the fastest (relative to a clock on Earth)?

I know that time dilation occurs between two reference frames with differing velocity, or differing positions in a gravitational field. Many examples that illustrate this point involve things like spaceships travelling near the speed of light, assumedly this allows a larger relativistic effect to be demonstrated to the reader. While these examples do well to illustrate the point, they almost all involve a situation in which the observer on Earth sees the spaceship's clock apparently run slower than their own.

So my question is: where in the solar system could we place a clock, such that an observer on Earth would see it tick the fastest? How much faster would that clock appear to tick?

Note: I realize 'the solar system' has a loosely defined boundary. For the sake of this question, lets arbitrarily define it to end at 50 AU (~Pluto's aphelion). I suspect that the sun's gravitational effects drop off well before this distance, so while a clock in the oort cloud may appear to run a little faster still, the increase will be relatively tiny.

I did some digging on this site, and this question seems to be similar, but I'm more interested in a more local response (ie, within our solar system, not universe-wide). Additionally, I'm curious as to how large this effect could be (and I do realize it will likely be quite small).

Thus far, I've found that the Earth orbits the sun at a velocity of ~30,000 m/s, so (ignoring gravitational effects), it seems that a clock placed on the sun's surface would be seen to tick faster. This image from the wiki page on time dilation seems to confirm this:

But, the sun is the largest gravitational well in the solar system! So I suspect that the fastest clock (relative to Earth) would actually exist somewhere near the edge of the solar system. Perhaps not in orbit of the sun, but constantly accelerating directly away from the sun (with the magnitude of acceleration being equivalent to the gravitational force being exerted on the clock by the sun)?

This seems reasonable to me, but then, relativistic effects are notoriously counter-intuitive, so I'd like to enlist the help of the smart people of physics.SE to make sure I'm understanding things correctly :)

• Not a physicist here, but shouldn't the observer always have the fastest clock? Or is that only if the observer is not accelerating? Commented Sep 25, 2018 at 11:21

If you want the clock to go as fast as possible then you need to avoid the things that would slow it down.

First is the kinematic time dilation. You can slow that down to 0 by placing the clock at rest in the “local” solar system’s center of momentum frame which is approximately inertial on the scale of several centuries. The clock would need to use rockets to station keep.

Second is gravitational time dilation. That cannot be reduced to zero, but it could be made small just by being at the 50 AU limit. To avoid occasional time dilation due to Pluto passing by, we could go 50 AU from the barycenter in a direction normal to the plane of the ecliptic.

Regarding the quantitative calculations, you can use the formula $$\sqrt{1-\frac{2GM}{c^2 r}}$$ for the gravitational time dilation. At 50 AU this gives 0.9999999998

• I'm not sure if this answers the question because it was after fastest relative to Earth, so stationary relative to the CoM frame of the Solar System is wrong, I think.
– user107153
Commented Sep 25, 2018 at 20:39
• The location I described is close to a constant distance from the earth. So it is hard to see what would disqualify it as an answer
– Dale
Commented Oct 21, 2020 at 12:10
• It's not enough to be at a constant distance though – consider two objects separated by $r$ both of which are spiralling around each other. However, as I said I'm not sure: I wasn't saying this isn't right, I'm just not sure it is right.
– user107153
Commented Oct 21, 2020 at 12:45
• Consider an object at rest in an inertial frame, and another object circling it under acceleration (in special relativity). These objects are at rest in some non-inertial frame but their clocks certainly do not keep the same time.
– user107153
Commented Oct 21, 2020 at 15:37
• No, they're not. But 'being a constant distance away' does not tell you what the special-relativistic difference in clock rates (which you need to minimise) is. That's my point.
– user107153
Commented Oct 22, 2020 at 13:55