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Suppose I have a rectangular brick of computer memory. My brick is very thin on one side and contains so much memory that it is just one bit away from being a black hole. According to the holographic limit, the amount of memory in the brick is given by the surface area.

Now lets say I have two of these memory bricks. I should now have twice as many bits. Suppose I stack one of these bricks on top of the other. When I stack my memory bricks along the fat side the surface area of the combined bricks is much less than twice the surface area of an individual brick. Therefore according to the holographic limit, the amount of memory in the stack must be less than twice the memory in an individual brick.

Why do half of my files disappear? What happened to the bits I thought I had?

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  • $\begingroup$ Hawking showed that the total area of the horizons of a collection of black holes (which is essentially what you have) always increases, even in processes like mergers (which is essentially what you're doing). So no, you don't have "much less than twice the surface area" than you did before, counterintuitive as that may seem. $\endgroup$ – probably_someone Jul 12 '18 at 0:22
  • $\begingroup$ I understand that black holes can merge in a way such that the event horizon is consistent with the holographic limit. But my question is about not-black-holes where there is no event horizon. How is the apparent paradox resolved in that case? Do certain shapes of matter (eg bricks) become impossible as the number of bits approach the holographic limit? $\endgroup$ – IIAOPSW Jul 12 '18 at 0:44
  • $\begingroup$ If you have two masses that are just short of becoming black holes, and you put them together, they will collapse into a black hole. I am sure that the answer has to do with this. $\endgroup$ – Nathaniel Jul 12 '18 at 3:55
  • $\begingroup$ (As a matter of fact, it's implausible that your memory bricks are rectangular, since they must be neutron stars already - there is no force strong enough to outweigh the gravitational attraction that's pulling them into spherical shapes. But this doesn't really affect the question much.) $\endgroup$ – Nathaniel Jul 12 '18 at 3:57
  • $\begingroup$ If you are 1 bit away from a black hole, you need a heavily dense object like a neutron star. You can't just ignore gravity. If you ignore gravity (QFT) the entropy scales with the volume of the system, but for gravitational theories, the entropy must be bounded by an area law similar to the Bekenstein-Hawking formula. $\endgroup$ – Panos C. Jul 12 '18 at 6:40

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