Followup question would be whether life could exist in the surface area of the progressively smaller circles some distance away from the core.
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1$\begingroup$ Welcome Abhay, please take our tour and refer to the help center as and when for guidance (also, don't forget the search facility at the top of the page to check for duplication - or just plane for interesting stuff). Enjoy the site. $\endgroup$– Escaped dental patient.Commented Aug 21, 2021 at 9:27
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$\begingroup$ The obvious answer is no, and this isn't the physics stack. But if you want to make this a worldbuilding question, ask how to get this shape. $\endgroup$– rekCommented Aug 21, 2021 at 13:03
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1$\begingroup$ Ever read Rochworld? A closely-orbiting double planet could somewhat approach the shape, although it would be more like a dumbbell of two egg-shapes rather than two opposing hemispheres. Not sure about the long-term stability either. $\endgroup$– BMFCommented Aug 21, 2021 at 14:56
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$\begingroup$ The mini version of Rocheworld: en.wikipedia.org/wiki/Contact_binary_(small_Solar_System_body) $\endgroup$– Anton SherwoodCommented Aug 22, 2021 at 2:40
2 Answers
A planet like this cannot form 'organically'; planetary formation as we know it always produces ellipsoids. That's just how gravity works; Nature tends to minimize potential energy and the sphere works best because everything is as close together as possible (with rotation causing them not to be perfectly spherical). Incidentally, it's also visible when you make snowballs; it's hard to produce anything which is relatively flat like your hourglass planet. (Or did you mean that the interior of the hourglass is solid too? It looks transparent in the picture.)
So it needs to be built with technology, i.e. it's more like a spaceship than a planet. If it's planet-sized, you need extremely strong material (much stronger than what we currently have) to withstand gravity and stress effects.
One of the main requirements for being a planet is that the gravity of the body must be strong enough to make the entire structure collapse into a sphere.
As the good old Wikipedia says:
A planet is an astronomical body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity.
While a planet can most definitely have very high mountains (possibly rising tens of dozens of kilometers above the surface of the planet for small planets), what you are suggesting are not mountains or anything remotely similar. You are suggesting that the entire crust be in semicircular shape and attached to the core through very short attachment points.
This sort of structure would be almost impossible to form naturally. And even if you did make it artificially, either the whole crust would later collapse on the core due to gravity later, or simply tear off the core if the gravity is too weak.