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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

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    $\begingroup$ I’m voting to close this question because such nonsense does not even belong to physics stackexchange. $\endgroup$
    – Kurt G.
    Jan 20 at 5:30
  • $\begingroup$ 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 $\endgroup$ Jan 23 at 21:16

4 Answers 4

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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.

(Please refrain from commenting about the use of "relativistic mass". I already know...)

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  • $\begingroup$ Thanks, I misunderstood and made a flaw with mass increasing due to special relativity. I am going to repose the query, without special relativity. Perhaps you could look into that one as well, would love to hear your thoughts $\endgroup$ Jan 23 at 21:13
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Note also that Newtonian gravity and Kepler's laws are only approximations, which ignore relativistic effects. For dealing with relativistic effects you need the relativistic theory of gravity, namely general relativity. This will tell you that gravity is due to rest mass, not relativistic mass (actually the stress energy tensor rather than rest mass, but for most purposes this will boil down to rest mass).

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  • $\begingroup$ Thanks for the comment, I realize I made a flaw with assuming special relativity increasing gravitational mass. I am going to repose the query, perhaps you could look into that one as well. Love to hear your thoughts $\endgroup$ Jan 23 at 21:12
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All clocks are equally affected by all relativistic effects. It does not matter how each clock works. Because the atomic clock and the pendulum clock are in the same location and are moving in the same way, they will stay synchronized. They are both affected (from the point of view of another reference frame) by gravitational and motion-induced time dilation.

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  • $\begingroup$ Thank you for your comment, I disagree with your conclusion here that all clocks are identically affected by time dilation, namely due to their differing reliance on gravity. I will repose the query and get rid of the special relativistic part, which was due to my misunderstanding of 'the myth' of relativistic mass. Perhaps you could look at that query as well and share your insights. Much appreciated $\endgroup$ Jan 23 at 21:10
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The conclusion is correct: as the planet speeds up in its elliptical orbit, the increased mass affects both the gravitational field strength and the gravitational time stretch. This dual impact causes the pendulum clock and atomic clock to fall out of phase during faster orbital segments around the black hole.

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