# Measurement of the velocity of a celestial body by means of (relativistic and classical) gravitational effects on clocks

Imagine a planet with the same properties as Earth, this time moving in an elliptical orbit around a black hole of a large number of solar masses. Also imagine that the surface of this planet is as massive as that of the Earth and that you can therefore experience a normal force on it as on the Earth's surface. At a speed of 30 km/s, the Earth has a radius of 6500 km and a mass of 6*10^24 kg.

Imagine that the planet is currently at the apoapsis (the farthest point) of an elliptical orbit around the black hole. Now place a pendulum clock and an atomic clock on the planet's surface and somehow get them in phase. This therefore depends on the pendulum length of the pendulum clock for the frequency that must be in step with the frequency of the atomic clock. The pendulum motion depends on the (Newtonian) field strength g on the surface of the planet, while the rate of the atomic clock depends on the gravitational time stretch on the surface according to general relativity.

Special relativity says that an object increases in mass as its speed increases. Now the planet varies in speed in its elliptical orbit around the black hole: according to Kepler, it will absolutely move faster in its orbit at the periapsis (i.e. closer to the black hole) of the ellipse than at the apoapsis (farthest from the black hole). ). According to special relativity, this change in speed should affect the mass of the planet.

Imagine that the speed increases to 30,000 km/s, then the mass of the planet will have increased to 1.26*10^25 kg. This increased mass of the planet increases the gravitational pull on the pendulum: the field strength g increases on the planet's surface (from 9.47 m/s^2 to 19.9 m/s^2) and the frequency of the pendulum clock will rise.

However, the gravitational time stretch will also increase on the surface due to the increased mass of the planet, which will cause the atomic clock to tick slower here. The pendulum clock and the atomic clock will therefore become out of phase on the faster parts of the planet's elliptical orbit around the black hole.

Is this conclusion correct? If not, I'd like to see where the error lies in the design

• I’m voting to close this question because such nonsense does not even belong to physics stackexchange. Jan 20 at 5:30
• I am going to repose the query due to obvious flawed assumptions in this one, as was pointed out by many kind contributors. Would love to hear your comments on that if you would be willing to look into that Jan 23 at 21:16

You have misunderstood relativistic mass. It's a relative phenomenon. An observer going with the object, like the clocks, does not see any increase of the mass.