Gravitational Time Dilation

Calculate planet speed of Neptune and Mercury relative sun orbit.  Two people live on Mercury.  One stays on Mercury and one moves to Neptune for 40 years then returns to mercury.  Did the person on Mercury age the same as the person on Neptune during the 40 year interval?  Did he age more, less, or equal? Explain.


$R(Neptune) =4 500 000 000 km =4.5*10^{12} m$  radius of Neptune orbit

$T(Neptune) =164.79 years = 60190 days =5.2*10^9 seconds$

$V (Neptune) = \omega*R = (2*\pi/T)*R = 5437.4 m/s =5.43 km/s$

$R(Mercury) = 57 909 100 km =5.79*10^{10} m$  radius of Mercury orbit

$T(Mercury) =87.97 days = 7.6*10^6 seconds$

$V(Mercury) =(2*\pi/T)*R =47867 m/s =47.87 km/s$

If you take one men from Mercury to Neptune, the orbital speeds of both planets are small comparing to the speed of light (300 000 km/s). This means the the RELATIVISTIC time dilation WITH SPEED will be small enough and will not count.  $(\sqrt{1-v^2/c^2} =1)$

What counts however is the GRAVITATIONAL time dilation (explained also by relativity) that will be significantly different for each person. The DEFINITION of gravitational time dilation is the following: clocks far from massive bodies run faster and clocks near massive bodies run slower.

In a few words time is passing slower for a person in a bigger gravitational potential. because the gravity of Neptune is much bigger that the gravitation of Mercury, than the time will pass slower for the person on Neptune that for the person on Mercury.  The person on mercury will age faster than the person on Neptune.

( Since the gravitational attraction of sun on both planets is small enough as compared to the gravitational attraction of the planet itself, the above observation is correct. If you live at higher altitude on earth, time is passing slower for you than for people on plain).