The Voyager spacecrafts are moving with an approximate constant speed of 16 to 17 km/s. As per Einstein's theory of relativity, time passes more slowly for them compared to us. Then, is it possible that these spacecrafts have a longer nominal lifetime than we actually assumed?
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You can approximate the situation by treating both the Earth and Voyager as "test-bodies" moving in a spherically symmetric gravitational field of the Sun. This symmetry is described by the so called Schwarzschild solution. The Earth actually travels at about 30 km per second in this approximation where the "speed of the Sun" is zero while the Voyager is travelling at about 17 km per second.
As a simple approximation we can ignore the fact that for an observer at infinity the velocity of light in the gravitational field of the Sun will vary slightly with radial distance as well as with direction and you can write:
$$\frac{d\tau}{dt}=\frac{1}{\sqrt{1-\frac{2GM}{rc^2}-\frac{v^2}{c^2}}} $$
This expression says that you, measuring time in $\tau$, experiences less time the faster you go and the deeper in a gravitational field you are as compared to a distant observer at rest at infinity measuring time in $t$.
Voyager 1 is about 145 times as far away from the Sun then the Earth and its velocity is 17/30 times the velocity of the Earth. Both these facts contribute to time going faster for Voyager 1 then for us on Earth.
Voyager is actually carrying radioisotope thermoelectric generators to power it and those will not function as long as they would have on the Earth becasuse time goes faster for the Voyager.
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