Collapsing Schwarzschild 's Cat

 kw: article reactions, black holes, primordial black holes, musings

A recent article at Space.com is titled, "Are planet-killing black holes hiding inside your cat?" I suppose the meme behind the title is Schrödinger's dead/not dead cat. The author, Robert Lea, quotes researcher Dejan Stojkovic as saying,

"But don't worry about a primordial black hole shooting through your cat, or you, for that matter. The team behind these findings says such an event would be non-lethal!"

The article includes an illustration of several sizes of black holes, from supermassive (a billion suns) to sub-proton size, which is in the range of theorized primordial black holes. In particular, a black hole with the "mass of an asteroid" is stated to be smaller than a proton. I suppose that depends on the asteroid; the term "asteroid" covers material ranging in size from a sand grain to a few hundred km.

It is stated that primordial black holes, if they exist, would be zipping about at near-light speed (nobody ever says why), so one would pass through you, or your cat, or Earth, very quickly; about a nanosecond, on your case. I thought of two ways a tiny black hole can cause harm. Firstly, the intense gravitational field "nearby" (we'll try to define that soon) could disrupt tissue; and the Hawking radiation that, we have learned, will eventually result in any black hole "evaporating" by emitting radiation and thus losing mass, could cook (or evaporate!) tissue it passes through.

I would expect these two phenomena to be significant in different regimes, viz:

1. A really small black hole, weighing, say less than a million metric tons (tonnes), will have a smaller reach, gravitationally, but its Hawking radiation will be stronger. If you are "near" such an object long enough, you may not suffer damage from the gravity, but you could get cooked.
2. A larger black hole will have much less Hawking radiation, but its gravitation reach will be greater. If you are "near" such an object long enough, its gravity can do great damage, but its radiation could be beneath notice.

Calculation time! I made much use of Victor T. Toth's Hawking Radiation Calculator. Here are relevant parameters for three possible black holes, one the size of a proton, one 100 times larger, and one 100 times smaller (in radius). The proton's radius is about 0.84 fm (femtometers), or 0.84x10^-15 m; we'll call this Rp.

• Radius in Rp:              100   1.0     0.01
• Mass, Million Tonnes:   56,600   566     5.66
• Temperature, Billion K:   2.17   217   21,700
• Heat, Billion Watts:  0.000111  1.11   11,100

Note that these all are really, really hot! To get a feel for their masses: Iron has a density of 7.9 Tonne/cubic m. A cube of iron weighing 5.66 million Tonnes would be 89.5 meters on a side; for 566 Tonnes, the size is 415 m, and for 56,600 million Tonnes, the size is 1,930 m, or more than a mile. That's getting to substantial asteroid size.

To illustrate how small a proton is in relation to a typical atom, atom radii are in the range of a tenth of a nanometer, or 100,000 fm. A black hole with a radius or 100,000 fm (a little smaller than an iron atom, for example) has these parameters:

• Mass, Million Tonnes:   67 million → 67 trillion Tonnes
• Temperature, K:        1.8 million
• Heat, Watts:               0.079

Note that this "bigger" black hole may be super-hot, but its radiation is negligible. Let's first focus on Gravity. For reference, "1 G" is 9.8 Nt/kg (Newtons per kilogram) at the surface of the Earth. The proton-sized black hole, weighing 566 million Tonnes, would exert a force of 378 Nt on a mass of one gram (such as a BB) at a distance of 1 cm. That's 38,600 G. Gravity scales as the square of 1/r, so within 1 mm of the black hole, anything there (cells in your body as it passes through?) would experience a force of 3.86 million G. Here, duration is everything. If the black hole's velocity is, say a third of the speed of light (or roughly 0.1 m per nanosecond), the time it takes to move one millimeter is about 10 picoseconds. That means that a random cell that is 1 mm off the center of the black hole's path will "see" a spike in force that rapidly changes direction through a 180° arc in the space of about 0.1 nanosecond, reaching nearly four billion G's.

I don't know how to describe the effect on the cell. It is unlikely to survive. It probably doesn't have time to be sucked into the black hole, but a cell that is "brushed by" (say, 1/100th mm) most certainly will be. The result will be a thin "soda straw" hole through the body, much less than 1 mm in diameter, but I don't know how much less.

How about the mass with a size of 100 proton radii? At 1 cm distance, the force would be 3.86 million G's. It is very likely that such a mass passing through you (or your cat) will leave a hole a substantial fraction of a cm across. It is similar to being hit by a 30 caliber rifle bullet, just much, much faster. If either of these masses were moving a lot more slowly, such as an orbital speed in the range of 30 km/sec, rather than 100,000 km/sec (1/3 of light speed), the breadth of destroyed tissue would be dozens to hundreds of times greater, and the diameter of the "soda straw" …? It's hard to comprehend. So let's not bother checking the atom-sized black hole, weighing in at 67 trillion Tonnes! If any exist, they could explain rare cases of disappearance, perhaps.

How about temperature? The hottest black hole is the smallest, and has an incredibly tiny surface area to radiate heat, but radiates 10,000 times as much heat as the proton-sized one. It can do so for a quarter of a million years. Its radiation, mostly X- and gamma rays, amounts to 11 trillion watts. If any of these were anywhere within a few light-years, we'd see them. Let's back off to the 100 Rp radius black hole, which radiates (still in X-rays and higher) 111,000 watts. If it is traveling at 1/3 c, it passes through you in 3-4 ns, leaving behind about 4 milliJoules, or some 4,000 ergs. Spread that out along the length of the path through your body, and it isn't much heat.

Now consider the middle mass, the proton-sized one. It radiates 1.11 billion watts. In the time given, it deposits around 4 Joules, or close to one calorie. Again, not much heating. So there is little "cooking" expected from really fast-moving primordial black holes. However, if their speed is closer to orbital speeds, the "dwell time" is several thousand times greater, and 4 joules becomes more than 10,000 joules, equal to a kilowatt for ten seconds. That'll burn a hole through you! It's not quite as powerful as a lightning strike, but it's getting in that range.

I started out thinking I could debunk the idea that primordial black holes aren't much danger. In certain circumstances they aren't, that's true, but what grounds to we have to assume they are going fast enough to pass through you, or me, or the nearest cat, without swallowing up a hurtful amount of stuff, and cooking much of what isn't sucked in?

Whatever speed they are moving, the smaller ones ought to be visible as sky-blue items that emit lots of X- and gamma radiation. With current instrumentation, we'd be hard pressed to determine their actual temperature. "Millions of degrees" just begins to describe it. So, I've actually presented a challenge to the idea that primordial black holes weighing less than about a half billion Tonnes exist at all.