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https://www.physicsforums.com/threads/does-relativistic-mass-create-gravity.68454/page-2

https://www.physicsforums.com/threads/does-relativistic-mass-induce-gravity.413594/

https://www.physicsforums.com/threads/relativistic-mass-and-gravity.626128/

https://www.physicsforums.com/threads/relativistic-mass-and-gravity.675637/

So, in lieu of observational evidence, most posters in these threads come up with thought experiments to try to gain insight on the manner. I’m going to do the same here with a fresh thought experiment I didn’t see in the other threads and see if anyone can, say, pick it apart some, here goes:

Earth has a twin planet named “Bearth,” with exactly the same radius and mass in kgs. One day Bearth comes zooming through the solar system and passes Earth at 0.87c (relative to an x coordinate which lines up parallel to the equators of both planets), giving Bearth a relativistic gamma factor of 2 relative to Earth in the x direction. On the planet Bearth there are two twin sisters, Alice and Beth. Both of these girls are 6 feet tall (being twins, of course).

Alice and Beth are playing a game, the game is for both girls to hold their arms straight out, drop a baseball, and see which one hits the ground first. They are not sure whose ball will, because Alice is standing on the north pole of Bearth and Beth is standing on the equator. Low and behold, though, when they try the experiment/game, they find the balls hit the ground at roughly the same time, and the acceleration of each ball is 10 m/s^2.

Now, while this going on, Bob on planet Earth is watching Alice and Beth play this game through his powerful telescope and wants to play too. Bob is also 6 feet tall and is standing parallel to Alice on the north pole of the Earth. Looking through his telescope, he sees when Alice and Beth drop their baseballs and then proceeds to drop his own at the exact same instant. What does Bob notice?

From what I can tell, Bob notices that, while he sees his ball fall to the ground at time t, he sees Alice’s ball fall to the ground at time 2t due to the time dilation gamma factor of 2 described above. Is this not right? In that case, wouldn’t this appear to Bob as if the force of gravity on planet Bearth is

*less than*that of Earth, rather than greater due some putative “relativistic mass.” As far as Beth is concerned, it would seem as though Bob would witness his and her ball hitting the ground at the same time because, while Beth’s ball takes twice as long to hit the ground, she is only half the size as Bob at 3 feet tall due to her length contraction in the x-direction of movement.

So, at least according to this thought experiment, the relativistic velocity of Bearth would seem to have the effect of making gravity appear to be

*weaker*on Bearth than Earth relative to Bob’s perspective. Is this wrong? If so, please explain why and what theoretical and observational work illuminates this issue.