How long could a planet or moon survive if it had an Earth mass black hole within it?

Question

How long could a planet or moon survive if it had an Earth mass black hole within it?

Hypothetical primordial black hole formed soon after the Big Bang, or low mass black holes formed later by some other hypothetical process, could have masses lower than stellar remnant black holes which should have at least three times the mass of the Sun.

The lower the mass of a sub stellar mass black hole, the faster it would evaporate via Hawking radiation, and the shorter the time it would last before exploding at the end.

It has been calculated that primordial black holes with a mass of about 1 times 10 to the 11th power kilograms would last for about the current lifetime of the universe before exploding.

The mass of Earth is about 5.9722 times 10 to the 24th power kilograms, which is about 5,972,200,000,000 times the minimum mass of about 1 times 10 to the 11th power necessary for a primordial low mass black hole to survive to the present.

Thus a world with an Earth mass black hole inside it would survive for much longer than the present age of the universe before the shrinking black hole evaported. And the black hole would probably not lose a signficant precentage of its mass within a few billion years.

But how long would a world with an Earth mass black hole inside it survive before it was swallowed up by the black hole?

Atoms and molecules and sub atomic particles of the world would be constantly falling inside the event horzon of the black hole, increasing its mass. As the mass of the black hole increased, its gravitational attraction on moleculesa and atoms outside its event horizon would increase, pulling them in faster. And As the mass of the black hole increased, the diameter of its event horizon would increase, and thus the surface of its event horizon would increase.

Those two processes would casue the rate of inflow into the black hole to increase exponentially, and the amount of matter left to the world outside the black hole to decrease exponentially.

So the Schwartzchild radius of the event horizon of a black hole with the mass of Earth would be about 8.87 millimeters, or about 0.3492 Inches. Thus the suface area of the event horizon of a black hole with the mass of Earth would be 8,769.6115 square millimeters.

That is a very small surface ara, but a very vast one compared to the sizes of molecules, atoms, and sub atomic particles.

So can anyone calculate how long an Earth mass black hole could remain inside a planet, moon, or other world before the rate of infall of the world's matter caused disasters on the surface of the world. Or can anyone point to a source of such calculations?

Answer 1

Billions of years

Assumptions:

I'm overlooking the "How did it get there" implied by the question. One moment it was suddenly there and the previous moment it wasn't.

I'm assuming your planet is also the size of Earth - so when the blackhole is "done" it's mass has doubled. The blackhole is in the exact centre so there's no orbital shenanigans - perfect equilibrium.

This is really a fluid dynamics question:

The blackhole is sitting in high pressure molten iron and consuming it, the upper bound of it's rate of consumption is going to be determined by high pressure liquid metal fluid dynamics - the blackhole swallows all the fluid it intersects, how quickly can the liquid attempt to replace the gap?

We don't know - these ultra high pressure ultra high temperature fluids don't get a lot of testing because we don't have lab equipment that can replicate these temperatures and pressures. There are papers and research on the topic (eg for metal casting) but they're multiple orders of magnitude below what we need. Liquids work towards filling the container, and they propagate the "information" they need to share to do that at the speed of sound in that liquid. To the best of our knowledge, the liquid iron will cross the event horizon at no faster than that rate.

The speed of sound in liquid iron is 3.8km/s.

Doesn't gravity somehow "outrank" fluid dynamics or something?

When looking at this problem - it's really easy to mistakenly think that gravity plays a big part here. Gravity is the weakest of the fundamental forces.

With the pressure at 3.6 million atmospheres accelerating the iron, we can essentially ignore gravity as a rounding error, but - If pressure, gravity or some other force tries to accelerate it faster then 3.8km/s - the fluid dynamics will fight back and the accelerating iron will collide with its neighbour atoms and and form turbulence patterns slowing it down, you'll also maybe get cavitation with pockets of gaseous or supercritical iron mucking up the flow. The spherical nature of the blackhole makes this effect much more pronounced - the flow chokes.

The Electroweak force causing the choking interaction is roughly $10^{25}$ times as strong as the Gravity accelerating the iron towards the hole. Gravity looses the battle against fluid dynamics by 25 orders of magnitude.

Put another way - the number of iron atoms at any given distance from the event horizon isn't constant - all atoms are accelerated to the blackhole by pressure / gravity but the increased density gradient approaching the event horizon as even more iron tries to accelerate into such a tight space deflects most of them away into eddies and turbulence.

3.8km/s is an upper bound from physics. Because of these fluid dynamic effects; I expect it to be lower instinctively due to inefficiencies in the motion but can't begin to prove it. I'd suggest reading the obligatory xkcd on the topic of forcing fluids through tiny holes at speeds faster than the speed of sound.

So how fast will the core go into the hole:

You've mentioned the surface area is 8,769 square mm. I calculate that's for a sphere of radius 26.42mm - the black hole will not get that big before destroying the planet. I've calculated it as 966 square mm from $4 \cdot \pi r^2$.

Your black hole has a surface area of 988 square mm, each square mm of surface area is consuming liquid iron at 3.8km/s, which is 3.8 litres per second per mm of surface area. 3.6kL of liquid Iron per second.

Wikipedia estimates that the core has $10^{23}$ kg of iron, and it's average density is about 12.9kg/L. So from this you're blackhole is consuming 46.4 tonnes per second of liquid iron, and it's got $10^{23}$ kg to get through before the core is gone. The core is 1/60th of the earths mass, so the blackhole will only grow by 1.6% from the consumption of the entire core - this is small enough that it can be effectively ignored for the purposes of this approximation and we can use the same consumption rate for the entire process and still be accurate to 2 significant figures.

This works out $2.1 \cdot 10^{18}$ seconds to consume the entire core, or $6.4 \cdot 10^{10}$ years - 64 trillion years to consume the entire core.

And when do we notice this on the surface?

How much of the core needs to disappear before we notice? That's hard to answer, as it'll be on a geological scale and difficult to differentiate from normal tectonic activity. 0.01% of the Earths core will be consumed in 6.4 billion years, shrinking the circumference by a few km over that period, but that's a rate slower than continental drift by 4 orders of magnitude. So as a rough approximation, for every 1 earthquake the blackhole causes, 1600 earthquakes are caused by tectonic plate activity.

Is there one in the Earth Now?

If the black hole had appeared 6.4 billion years ago in the middle of the Earth, I'm not sure we'd be able to detect it yet. We have some very precise measurements of things like Earth-Moon distance, but we don't have 500 years of these accurate records to compare and note a trend. All our precise time and distance measurements would include the existing blackhole effects in their calculations, and our estimate for the earths mass would be off by a factor of 2 - we'd just assume higher density in the core to explain the extra mass.

If one appeared today, would we notice it?

If it appeared today, we'd notice the time dilation pretty quickly (GPS would be off by 10s of meters almost instantly), and clocks in space would get out of sync and orbits would change - moon would change orbit noticeably - but we wouldn't be able to blame it for anything geological for at least 100 million years ("Hey that magnitude 3 earthquake last night that jiggled the glasses in the cupboard? 66% probability it was the blackhole!").