# 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? – kutschkem Sep 25 '18 at 11:21

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