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I would know what would happen if we put a small part (let me assume 1 cm$^3$) of White Dwarf or Neutron Star on Earth. Will it explode or maybe it collapses to the Earth's core?

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A piece of a white dwarf would explode, because atoms would expand to their normal size. This would hold for outer layers of a neutron star, too. For inner parts of the neutron star, that's not quite clear, but that kind of matter would probably decay rapidly, also resulting in an explosion.

If the material of a neutron star could be kept stable, it would fall down to Earth's core, up to the other side of the Earth, and oscillate a while that way, before being slowed down by friction, and stopped close to Earth's center. (1 cm³ neutron star matter has a mass of about 400 million tons.)

One cm³ white dwarf matter would have a mass of about 1 ton. Assumed it could be kept stable, solid rock could withstand this load.

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Thank you for your clear and helpful explanation. I can accept this answer. –  Tom Feb 14 at 18:38

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