# Voyager local time dilation (caused by gravity)

Voyager I, as an example, taking account gravity
and setting aside effects of speed as cause of time dilation.

If it is very far away from earth and sun, so then there must be a difference in the spacetime curvature there in the ship compared with here in earth, It means a detectable difference between our local clocks and its onboard clocks.

Imagine a signal transmision was designed to be at 1 byte for second at local Voyager clock

Should we receive it at higher and higher rates ?

( because of solar system gravity decrease as it(Voyager) moves away and it will keep decreasing while it doesn't reach a middle point between another massive object)

Or the middle Voyager-Earth light path would compensate the effect, making the high rates generation from low curvature zones being delayed enough for us to receive it at same rate that it was generated?

There is in principle a "gravitational blueshift" for signals traveling from Voyager to us. The data rate we receive will be higher than the data rate transmitted by a factor $(1+\Delta\Phi/c^2)$, where $\Delta\Phi$ is the difference in Newtonian gravitational potential between the locations.
If I'm not mistaken, $\Delta\Phi/c^2\approx 10^{-8}$ for this sort of system, so the shift is quite small in practice. In particular, it's much smaller than than the ordinary Doppler effect due to the fact that both Voyager and Earth are moving. A motion at a speed of just 3 m/s would cause a Doppler shift as large as this gravitational shift, and both bodies are moving much faster than that.
• For the purposes of plugging into the above, taking infinity to be the zero of gravitational potential, the surface of the earth is around 53 km/s or $\Delta \Phi = 1300\text{ (km/s)}^2$ down the well. – dmckee --- ex-moderator kitten May 16 '11 at 19:39
• And that is exactly the source of the blue shift. Light falls down the well, but instead of accelerating (it cannot since $c$ is constant) it picks up energy in the form of frequency. – John Alexiou May 16 '11 at 19:44
• It's a weak-field approximation. That is, it applies only when gravity is, in some appropriate sense, "nearly Newtonian." I don't immediately remember all the details of what has to be assumed, but it's something like (1) all masses moving nonrelativistically, (2) $\Phi/c^2\ll 1$ over the region of interest. So it does work for the entire solar system, for example. So yes, if you can measure this effect, you're probing the gravitational potential. In practice it's not a great way to probe gravity, though, because the effect is small compared to other signals such as Doppler shifts. – Ted Bunn May 17 '11 at 17:03
• I'm sorry, but I don't understand your last comment. Anyway, here's what I'm saying. If the distribution of mass is complicated, then the function $\Phi({\bf r})$ is complicated, but it's still true that the time dilation factor is $1+\Delta\Phi/c^2$. The validity of this depends on the fields being weak and the matter being nonrelativistic, but not on any assumption about the geometry being simple or symmetric or anything like that. The proof of this result is found in general relativity textbooks. I like the one by Schutz for this sort of thing. – Ted Bunn May 17 '11 at 20:22