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My question relates to dark matter (DM). I understand that primordial black holes have been ruled out statistically as a candidate for DM. Could “dead” primordial black holes leave behind curved space-time (think in terms of an analogue to a “footprint” in space-time that is stable or is not stable but only dissipates over a long period of time) that accounts for gravitational lensing?

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    $\begingroup$ What is a dead black hole? $\endgroup$ – my2cts Apr 24 at 22:18
  • $\begingroup$ This is starting to get into speculative theory within a connection to a credible advancement of the theory, and it is also unclear what is meant by a "dead" black hole. $\endgroup$ – ohwilleke Apr 24 at 22:28
  • $\begingroup$ Since, as you know. Black holes have finite lives due to emission of Hawking radiation, when I say “dead” primordial black holes, I mean primordial black holes that have ceased to exist. $\endgroup$ – Andy Apr 24 at 23:01
  • $\begingroup$ @Andy Thanks, that's helpful in understanding what you mean $\endgroup$ – ohwilleke Apr 24 at 23:56
  • $\begingroup$ From your last comment where you clarify the "dead" , this is a scenario that cannot happen in General Relativity since the space time tensor depends on masses and energy . Your definition of "dead" cannot be accommodated within GR. The answer of Ben is the closest you can get to such a model. $\endgroup$ – anna v Apr 25 at 4:01
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I am presuming you are meaning a black hole remnant when you speak of a "dead black hole." A black hole remnant (BHR) reaches one or a few Planck masses and stops emitting Hawking radiation. I am not terribly motivated by the idea BHRs exist, but it has not been ruled out either.

The one problem I see with the idea is it would mean the universe started out with a very large entropy. The entropy of a black hole is given by the Bekenstein formula $S~=~k A/4\ell_p^2$, for $\ell_p~=~\sqrt{G\hbar/c^3}$ the Planck length. A black hole area is $A~=~4\pi r_s^2$ where the Schwarzschild radius is given by $r_s~=~2GM/c^2$. If you put all of this together you find black hole entropy is given by the number of Planck areas on its horizon $S~=~kN$, and a Planck unit of a black hole or a BHR has one unit. The universe has around $10^{55}kg$ of mass and a Planck unit of mass is around $10^{-8}kg$. If we assume most of this mass is in BHRs it then means $S~\simeq~k\times 10^{63}$. The inflationary universe expanded and O-region of around $10^{-26}$cm to around a meter. This means the early universe had entropy given by the horizon scale of this region and would be around $S~\simeq~k\times 10^{14}$. This is much lower.

The universe must have started out with a very low entropy. First off spacetime does not have a definition of thermal equilibrium. A hypothetical situation of a black hole sitting in a region with a black ground with the same temperature as the Hawking radiation $T~=~1/8\pi m$. There without units. The absorption or emission of a photon $m~\rightarrow~m~\pm~\delta m$ pushes the temperature away from that of the background. Stochastic quantum processes then will means the black hole either increases in mass or decreases in mass and evaporates away. Equilibrium, and frankly the quantum vacuum that has some analogues, are not well defined. Spacetime and the entire universe is then out of equilibrium.

Now one might argue these black holes came about after inflation and have since become remnant. However, there would have to be some mechanism that would have generated mini-black holes. We have no clear idea what that would be, and inflation is not conducive for BH formation. As a result I think the prospects for BHRs as dark matter is not good.

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    $\begingroup$ From the OP's last comment "I mean primordial black holes that have ceased to exist" I suspect that they mean primordial black holes that have totally evaporated, but which have left behind a "scar" of distorted spacetime. $\endgroup$ – PM 2Ring Apr 24 at 23:54

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