Elementary Physics Disciples, please take a few minutes to evaluate the following short story. (Now also note the Time Twin Paradox at the end) Gravity=Centripetal Acceleration.
Suppose Anoman and Znoman (Identical Twins) are in deep space living on opposite sides of a hollow planet with the same radius and mass of Earth, but mass is concentrated in a very thin crust (Note; nothing else exists in the universe as they can see). They of course experience a normal g force just like on Earth. Now suppose each of them cuts a small diameter worm-hole and look inside, and Znoman sees Anoman on the other side. Znoman gently lowers himself into the hole and experiences no g force at all. He is weightless. So he kicks off the rim of his hole and sails effortlessly through the center of the planet to Anoman on the other side. Once there, Anoman helps Znoman a little to stand up and Znoman experiences 1 g again standing there with Anoman.
Anoman has been playing with rocket engines and demonstrates one to Znoman. It is mounted level with the surface which is tangent. Anoman turns it on and immediately they feel a slight force in the direction of the rocket exhaust, but not so much as they can still stand comfortably and talk. They talk and talk for days and have absolutely no idea their planet is rotating from the effects of the rocket, since there are no other bodies in the universe to relate. They began noticing during the lengthy conversation that they did not feel as heavy, and after several days began to levitate. Znoman said, “Turn off that rocket engine, it must be doing something”. They have no idea what has happened. They just float around on the surface, like magic carpets. They can carry a whole tree effortlessly for an entire season of firewood. Well, Znoman says, “I really need to be getting back to my side of the planet”.
But, BIG problem! When Znoman starts to enter the opening, and go back inside the hollow planet, he feels his weight again pushing him back to the surface. Finally, he manages to get inside and try kicking off the rim. He only travels a couple of feet and then pushed back. He could walk around the inside crust, but that is, half the circumference of the planet, approximately 12,500 miles while experiencing 1 g! So, he comes back up out of the hole and then floats around the outer surface by pulling on tree limbs to get back to his side of the planet.
They ponder for days, weeks and years and cannot explain that force of 1g pushing them out of the hole and back out to the 0 g home surface. From the story you can see that from their limited dimensions and perspective, they can in no way determine that they are rotating.
Now, please reduce the hollow planet to a very thin hollow ring say 1 meter wide, with mass and radius of Earth while doing the same exercises, with the (through inside center) visit, and rocket engine, and keeping the crawl holes on opposite ends of the ring’s diameter, giving a plane two-dimensional universe. Observations will be the same.
Now, suppose their clones just magically came aboard the 2 dimensional universe. Anoman and Znoman got old and checked out, and taught them nothing of previous experiences. Therefore, they must learn Physics the old fashion way, by observations. Remember, they are at zero g outside the ring and as they enter the hole they experience a repulsive force pushing them back out of the hole. If they manage to get inside they feel a force of one g, perpendicular to the inside of the ring. If they picked up a rock and dropped it, they would measure an acceleration of 9.8m/sec^2, toward their feet and the inside of the ring.
Elementary Physics Disciples: Please note that your thinking is being baited to give you a broader application with mathematical proofs. First taking some elementary Trig and Algebra with a little Integral Calculus and the inverse square law and we easily prove that the gravitational field is zero inside the ring. Now use vector- tensor analysis to show a space compression and time dilation relationship. [This is where I can use your help] Please note with this example story an observer can view into another space-time dimension!
Anoman and Znoman are identical twins with identical synchronized Cesium Clocks living on opposite sides of a hollow planet along a diameter. Anoman sleeps with his clock on a table on the outside of the planet (1g), while Znoman sleeps with his clock just inside the hollow planet with (0g). Anoman's clock will run slower than Znoman's clock and Znoman will age faster. Now start the planet turning to the point where centripetal acceleration cancels gravitational acceleration on the surface (0g) and you now have (1g) in the interior. Znoman's Clock inside the planet now runs slower than Anoman's clock on the outside and Anoman will age faster.
Thanks, Are these observations valid? Your thoughts and feedback are greatly appreciated; for an old retired Physics/Mathematics college professor on a cold winter day: email@example.com